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Minimax Mean Squared Error for Low-order Network Interference under Bernoulli Assignment

Abstract

For fixed interaction order $β \geq 1$ and fixed assignment probability $p \in (0, 1)$ , this paper establishes a finite-population design-based minimax mean squared error frontier, up to constants depending on $(β, p)$ , for estimating the all-treated-versus-all-control effect under known bounded-degree low-order polynomial interference and common-probability Bernoulli assignment. The constants depend only on $(β, p)$ and are uniform in $n$ , $d$ , and $B$ ; they may deteriorate as $p$ approaches $0$ , $1$ , or a zero of the Bernoulli contrast. For radius $B$ , population size $n$ , and degree bound $d$ , the paper’s coefficient-mass and uniformly bounded-outcome minimax risks share the scale $B^{2} min {1, \frac{d A _{d}}{n}},$ where $A_{d}$ , the complete-block score energy, is the design second moment of the complete-block score used by the score-weighted neighborhood inverse-probability estimator (SNIPE, the shifted-neighbourhood inverse-probability estimator introduced by Cortez-Rodriguez et al. (2023)). Equivalently, $d A_{d} ≍_{β, p} d (k _{⋆} ( d , β , p ) d),$ where $k_{⋆} (d, β, p)$ , the largest interaction order with nonzero all-one-versus-all-zero Bernoulli contrast, is the exposed order in that contrast.

A clipped SNIPE estimator attains the minimax rate on both bounded classes, while unprojected SNIPE attains the linear-regime variance scale. Complete directed $d$ -blocks, with $⌊ n / d ⌋$ active blocks in the block construction of Definition 15, supply the matching lower-bound construction. When $d ∣ n$ , the same blocks give the exact worst-case risk $B^{2} A_{d} / m = B^{2} d A_{d} / n$ , where $m = n / d$ is the number of complete blocks, for unprojected SNIPE.

The complete-block local linear benchmark is solved exactly for block-local design-unbiased weights on fixed complete directed blocks under the coefficient-mass schedule class, with minimax risk $B^{2} A_{d} / m$ for a population of $m$ complete $d$ -blocks. For the fair-coin design, even-order Bernoulli contrasts cancel, $A_{d} = 4 \sum_{1 \leq r \leq m i n {β, d}, r odd} (r d)$ , and first-order interference yields the frontier $B^{2} min {1, d^{2} / n}$ .

Introduction

Randomized experiments on networks raise a basic design question: how much Bernoulli assignment variation is available for estimating a population-level contrast when each unit’s outcome may depend on its neighbors’ assignments? In the finite-population design-based tradition, potential outcomes are fixed and the assignment mechanism supplies the probability law for estimation (Horvitz et al., 1952; Rubin, 1978; Imbens et al., 2015). Interference changes the problem by making the all-treated-versus-all-control total treatment effect depend on exposure patterns induced by the graph, so the risk of an estimator reflects both local treatment-contrast geometry and overlap among neighborhoods.

This paper studies that question for known directed interference graphs with bounded in- and out-degree, common-probability Bernoulli assignment, and low-order polynomial potential outcomes. The estimand is the finite-population all-treated-versus-all-control effect $τ_{n}$ , the average difference between the all-one and all-zero assignments. The estimator observes the graph, the treatment assignment, and one realized outcome per unit. The two bounded classes considered here impose either a unitwise $ℓ_{1}$ coefficient-mass radius $B$ or a uniform potential-outcome radius $B$ ; Definitions 6 and 12 define these classes.

The main result is a matched minimax frontier. For fixed interaction order $β \geq 1$ and treatment probability $p \in (0, 1)$ , Theorems 1 and 2 establish $R_{n, ℓ_{1}}^{⋆} (d, β, p, B) ≍_{β, p} R_{n, \infty}^{⋆} (d, β, p, B) ≍_{β, p} B^{2} min {1, \frac{d A _{d}}{n}} .$ Here $A_{d}$ , the complete-block score energy in Definition 10, is the design second moment of the Bernoulli score representing the local all-treated-versus-all-control contrast on a $d$ -coordinate block. The same theorems compare this energy to the exposed binomial scale: $d A_{d} ≍_{β, p} d (k _{⋆} ( d , β , p ) d),$ where $k_{⋆} (d, β, p)$ is the largest exposed interaction order with nonzero Bernoulli contrast as defined in Definition 1.

The rate separates two sources of design difficulty. The first is local: $A_{d}$ is the squared norm of the Bernoulli representer for the block contrast. The second is graph-combinatorial: the extra factor $d$ is the out-degree overlap charge, arising because one assignment coordinate can enter the scores of as many as $d$ units under Assumption 2. The overlap count in Lemma 2 converts shared assignment monomials across neighborhoods into this single additional degree factor.

The estimator attaining the frontier is SNIPE with clipping. The unprojected statistic averages observed outcomes weighted by the unit score $g_{i, β, p}$ in Definitions 7 and 8. On the coefficient-mass class, projection to $[- B, B]$ supplies the saturated branch of the minimax bound; on the uniformly bounded-outcome class, projection to $[- 2 B, 2 B]$ gives the corresponding bounded-outcome estimator in Definition 14. The unprojected estimator is design-unbiased on both bounded classes and has worst-case mean squared error at most $B^{2} d A_{d} / n$ , giving the linear branch directly.

The lower bound comes from complete directed $d$ -blocks. The block family in Definition 15 perturbs compactly supported baseline outcomes in the direction of the normalized representer $h_{d} = g_{d} / A_{d}$ . The perturbation changes the total treatment effect by a controlled amount while the Hellinger distance between the two sign-indexed prior predictive laws remains bounded through the energy identity for $h_{d}$ . When $d ∣ n$ , Theorem 2 gives the exact complete-block worst-case risk of canonical unprojected SNIPE: $\frac{B ^{2} A _{d}}{m} = \frac{B ^{2} d A _{d}}{n},$ where $m = n / d$ is the number of complete blocks.

The complete-block local linear benchmark fixes complete directed block graphs, restricts attention to block-local linear weights satisfying the unbiasedness equations in Definition 17, and evaluates them over the coefficient-mass schedules in Definition 16. Theorem 3 shows that the minimax risk over this class is exactly $\frac{B ^{2} A _{d_{t}}}{m _{t}} .$ Its blockwise criterion $Ψ_{b, t}$ gives an exact worst-case risk decomposition, and the complete-block score $g_{d_{t}}$ attains the benchmark. The same result characterizes asymptotic optimality through blockwise excess risk and gives a sufficient representer-closeness condition in $L^{2} (P_{Z})$ .

The fair-coin specialization makes the role of the assignment probability explicit. At $p = 1/2$ , $Δ_{r} (1/2)$ vanishes for even $r$ and equals $2^{1 - r}$ for odd $r$ . Consequently, Theorem 4 gives $A_{d} = 4 1 \leq r \leq m i n {β, d} r odd \sum (r d) .$ For first-order interference, this reduces to $A_{d} = 4 d$ , so the minimax scale becomes $B^{2} min {1, d^{2} / n}$ , with exact complete-block unprojected SNIPE risk $4 B^{2} d^{2} / n$ when $d ∣ n$ .

In words, each neighborhood creates a local Bernoulli contrast problem, and $A_{d}$ measures its design cost. The graph then charges one additional factor $d$ because a treatment coordinate can enter many neighboring outcome equations. The frontier says that these two quantities, together with $n$ and the envelope $B$ , determine the finite-population design difficulty under the stated bounded low-order model.

The results connect to work on design-based causal inference under interference, including partial-interference, exposure-mapping, and network-experiment formulations (Sobel, 2006; Hudgens et al., 2008; Bowers et al., 2013; Ugander et al., 2013; Aronow et al., 2017; Eckles et al., 2017; Ogburn et al., 2017). They are closest to low-order SNIPE analysis under Bernoulli assignment (Cortez-Rodriguez et al., 2023) and to Riesz and pseudoinverse perspectives on linear estimation (Swaminathan et al., 2017; Harshaw et al., 2022). The frontier here gives the known-graph Bernoulli benchmark for bounded low-order polynomial interference, providing a calibration point for settings with graph uncertainty, clustered assignment, and covariate adjustment (Yu et al., 2022; Eichhorn et al., 2024; Wang et al., 2026).

The remainder of the paper is organized as follows. The related-work section positions the result within design-based causal inference under interference and the SNIPE literature. The setup section fixes the finite population, Bernoulli design, low-order interference model, bounded model classes, SNIPE scores, minimax risks, and block quantities. The main-results section states the two-class minimax frontier, the complete-block lower construction and clipped-SNIPE upper bounds, the bounded-outcome counterpart, the exact local linear benchmark, and the fair-coin specialization. The discussion interprets the block score energy and out-degree overlap charge and records limitations and future work. The appendices collect the mathematical proofs; the cited literature supplies econometric framing, terminology, and positioning.

This paper belongs to the design-based tradition in which potential outcomes are fixed features of a finite population and treatment assignment supplies the probability law for estimation. The Horvitz–Thompson construction (Horvitz et al., 1952) and the potential-outcome formulation of randomized causal effects (Rubin, 1978; Imbens et al., 2015) provide the baseline language for the finite-population Bernoulli randomization framework used here. Relative to that tradition, interference changes the estimand and the feasible information in the design problem: outcomes may depend on neighbors’ assignments, and the assignment mechanism must be read through the exposure structure induced by the graph.

A large literature studies causal inference when one unit’s treatment can affect another unit’s outcome. Early and foundational contributions include partial-interference and spillover formulations (Manski, 1993; Sobel, 2006; Hudgens et al., 2008), network and peer-effect models (Graham, 2008; Bramoullé et al., 2009; Tchetgen et al., 2010; Manski, 2013), and design-based estimands for network experiments (Bowers et al., 2013; Ugander et al., 2013; van der Laan, 2014; Liu et al., 2016; Aronow et al., 2017; Eckles et al., 2017; Ogburn et al., 2017). This work shares their finite-population orientation while specializing the outcome model to fixed low-order polynomial interference on a known bounded-degree graph. In that setting, the treatment assignments $Z_{j}$ are independent Bernoulli draws with common probability $p$ , the treatment probability, and the graph enters through the degree bound $d$ and neighborhoods $N_{i}$ .

The closest point of contact is low-order SNIPE estimation under Bernoulli assignment (Cortez-Rodriguez et al., 2023). In the present notation, their SNIPE analysis supplies design-unbiasedness and a stated worst-case worst-case variance upper bound whose degree dependence enters through $d^{β}$ , with constants depending on the interaction order and assignment probabilities, for bounded low-order polynomial interference. The present analysis characterizes the bounded-class design mean squared error in the low-order known-graph Bernoulli model through the complete-block score energy $A_{d}$ in Definition 10: for interaction order $β$ , the minimax scale is $B^{2} min {1, d A_{d} / n}$ , equivalently $B^{2} min {1, d (k _{⋆} d) / n}$ up to constants depending on $(β, p)$ , where $k_{⋆}$ is the exposed nonzero Bernoulli contrast order in Definition 1. Within this bounded-class setting, Theorems 1 and 2 provide the matched minimax lower construction, the attaining clipped SNIPE upper bound, the uniformly bounded-outcome extension, the exposed-binomial calibration, and exact complete-block constants. The comparison to the unrestricted class studied in Cortez-Rodriguez et al. (2023) is through that paper’s stated theorem, while the present frontier supplies the bounded-class calibration in the notation and model class used here. The attaining procedure is a clipped SNIPE estimator from Definitions 8 and 14, built from the unit score in Definition 7, under the finite-population Bernoulli model.

It is worth being precise about the scope of the comparison. Cortez-Rodriguez et al. (2023) study low-order neighborhood interference under Bernoulli assignment and establish design-unbiasedness of SNIPE together with a worst-case variance upper bound, in a setting that leaves the graph and the coefficient magnitudes free. The present paper fixes a smaller class — known graph, in- and out-degree bounded by $d$ , interaction order at most $β$ , and unitwise coefficient mass at most $B$ — and asks for the minimax risk over that class. Within it, the exposed-binomial scale $d (k _{⋆} d)$ takes the place of the $d^{β}$ factor carried by their stated bound, and the matching lower bound shows this rate is the correct one. The sharpening thus applies to the stated worst-case bound read on this restricted class; their own theorem, stated for their broader setting, stands as published, and the estimator compared against here is the same SNIPE family.

Sparse and local interference results provide a second comparison point. Work on exposure mappings, neighborhood interference, and asymptotic regimes with growing graphs develops conditions under which design-based estimators remain informative as the interference structure grows (Aronow et al., 2017; Athey et al., 2017; Baird et al., 2018; Basse et al., 2018; Basse et al., 2018; Jagadeesan et al., 2020; Sävje et al., 2021; Leung, 2022; Hu et al., 2022; Vazquez-Bare, 2023; Gao et al., 2025). The rate characterization here gives a finite-population minimax calibration for known bounded-degree low-order polynomial interference: the complexity contribution of the graph is summarized by the degree bound and the complete-block energy, while overlap across neighborhoods contributes one further degree factor. This connects local-interference intuition to a decision-theoretic benchmark for design mean squared error.

The complete-block calculations also relate to Riesz representer and pseudoinverse perspectives on linear estimation. The block-local problem can be viewed in the finite design space $P_{Z}$ , the Bernoulli assignment law, through a polynomial subspace and a contrast functional, with the normalized representer $h_{d}$ attaining the relevant block criterion. This geometry parallels the role of orthogonal scores and linear representers in semiparametric and design-based estimation, while remaining fully finite-dimensional under Bernoulli assignment. The exact complete-block local linear benchmark in the paper identifies the risk of block-local unbiased linear rules and supplies a sharp comparison class for SNIPE-type weights.

Recent work on unknown interference, covariate adjustment, and clustered or structured designs clarifies adjacent sources of statistical difficulty. Unknown or partially observed interference graphs introduce learning and robustness components (Yu et al., 2022; Eichhorn et al., 2024), covariate adjustment changes the information used by the estimator (Lin, 2013; Wang et al., 2026), and clustered designs modify the assignment law and exposure variation (Basse et al., 2018; Basse et al., 2018; Harshaw et al., 2022). The results here give the corresponding known-graph Bernoulli benchmark for bounded low-order polynomial interference, against which those additional design and information structures can be compared.

Setup and assumptions

All expectations in the main text are design expectations over the random assignment, with potential outcomes held fixed. Generic constants may depend on the fixed interaction order and assignment probability when explicitly subscripted by $(β, p)$ , and otherwise are numerical. Norms on finite coefficient arrays are the usual $ℓ_{1}$ or sup norms indicated by the displayed definitions. Asymptotic notation in the results keeps $β$ and $p$ fixed while allowing the population size and degree bound to vary; throughout this section, unit indices $i, j$ range over the finite population $V_{n}$ , and subset indices $S$ range over finite subsets of the relevant interference neighborhood.

The design is the finite-population Bernoulli randomization framework in Assumption 1: the graph and potential-outcome schedule are fixed, and the assignment supplies the probability law.

Assumption 1 [ass:bernoulli-design] (Bernoulli assignment).

Let $V_{n}$ be a finite population of $n := ∣ V_{n} ∣$ units, indexed by $i, j \in V_{n}$ , and let $p \in (0, 1)$ . For every unit $j \in V_{n}$ , $Pr (Z_{j} = 1) = p,$ and the variables ${Z_{j} : j \in V_{n}}$ are mutually independent. The assignment law $P_{Z}$ on ${0, 1}^{V_{n}}$ is $P_{Z} (z) = j \in V_{n} \prod p^{z_{j}} (1 - p)^{1 - z_{j}},$ and $E_{Z}$ denotes expectation under $P_{Z}$ .

⊢ Lean

Assumption 1 is the standard unit-level Bernoulli randomization condition used in low-order SNIPE analysis (Cortez-Rodriguez et al., 2023). It fixes the common treatment probability and the product assignment law, so all risk calculations below are design-based calculations under $P_{Z}$ .

Assumption 2 [ass:bounded-degree] (Bounded interference degree).

Let $G_{n} \subseteq V_{n} \times V_{n}$ be a directed interference graph, and write $N_{i} := {j \in V_{n} : (j, i) \in G_{n}}$ for the interference neighborhood of unit $i$ . Self-loops are allowed and count in both the in-degree and out-degree. The interference neighborhoods $N_{i}$ satisfy $i \in V_{n} max ∣ N_{i} ∣ \leq d and j \in V_{n} max {i \in V_{n} : j \in N_{i}} \leq d .$

⊢ Lean

Assumption 2 is the standard bounded in- and out-degree neighborhood interference condition, with self-loops counted, as in Cortez-Rodriguez et al. (2023). The in-degree bound limits how many assignment coordinates can enter a unit’s potential outcome, while the out-degree bound controls how many unit-level scores can share any one assignment coordinate.

Assumption 3 [ass:low-order] (Low-order polynomial outcomes).

For every unit $i \in V_{n}$ and every finite subset $S \subseteq V_{n}$ , the coefficient schedule satisfies $c_{i, S} = 0 whenever ∣ S ∣ > β .$

⊢ Lean

Assumption 3 is the standard bounded polynomial interaction order condition (Cortez-Rodriguez et al., 2023). It restricts the model class in Definition 6 to monomials of degree at most the interaction order $β$ , so the relevant Bernoulli contrasts are finite-order neighborhood contrasts.

The model is an exact finite-population polynomial schedule: once the assignment vector $z$ , graph, and coefficient schedule are fixed, $Y_{i} (z)$ is fixed. The reported risk is therefore the assignment-design mean squared error for that schedule. The procedure is permitted to use the interference graph $G_{n}$ , the interaction order $β$ , the assignment probability $p$ , and the radius $B$ ; the radius determines the projection interval in the clipped estimators.

Known directed graphs with self-loops are natural when interference neighborhoods are measured before assignment and a unit’s own treatment is allowed to affect its outcome. A common Bernoulli probability matches unit-randomized experiments run through a single assignment rule, while fixed $β$ describes regimes where the analyst treats low-order interactions as the structural complexity constraint as $n$ and $d$ vary.

Assumption 4 [ass:bounded-coefficient-mass] (Bounded coefficient mass).

For every unit $i \in V_{n}$ , the coefficient mass satisfies $S \subseteq N_{i} \sum ∣ c_{i, S} ∣ \leq B .$

⊢ Lean

Assumption 4 is the standard SNIPE $Y_{m a x}$ coefficient-mass normalization condition (Cortez-Rodriguez et al., 2023). The radius $B$ fixes the unitwise scale of the coefficient schedule and therefore the scale of the minimax mean squared error.

The next definitions collect the model class, the estimand, and the estimators. The exposed-order notation records which Bernoulli contrast orders contribute to all-treated-versus-all-control estimation under the interaction and degree restrictions.

Definition 1 [def:exposed-order] (Exposed order

\overset{ˉ}{β}_{d}

and

k_{⋆} (d, β, p)

For $r \geq 1$ , define the order- $r$ all-one-versus-all-zero Bernoulli contrast coefficient by $Δ_{r} (p) := (1 - p)^{r} - (- p)^{r} .$ For every $d \geq 0$ , define $\overset{ˉ}{β}_{d} := min {β, d} .$ In particular, $\overset{ˉ}{β}_{0} = 0.$ If $\overset{ˉ}{β}_{d} \geq 1$ , define $k_{⋆} (d, β, p) := max {r \in {1, \dots, \overset{ˉ}{β}_{d}} : Δ_{r} (p) \neq = 0} .$ Since $Δ_{1} (p) = 1$ , this maximum is well defined. If $\overset{ˉ}{β}_{d} = 0$ , equivalently $d = 0$ or $β = 0$ , define $k_{⋆} (d, β, p) := 0$ as a totality convention, rather than as an attained interaction order. Thus the maximum case applies exactly when $d \geq 1$ and $β \geq 1$ ; subsequent statements take $β \geq 1$ .

⊢ Lean

The order- $r$ Bernoulli contrast coefficient $Δ_{r} (p)$ is the all-one-versus-all-zero contrast induced by a centered $r$ -fold Bernoulli monomial. The exposed order $\overset{ˉ}{β}_{d}$ combines the polynomial order and the neighborhood size, while $k_{⋆} (d, β, p)$ identifies the largest exposed order with a nonzero Bernoulli contrast.

Definition 2 [def:zero-degree-conventions] (Zero-degree conventions

(0 0)

g_{0}

A_{0}

, and

h_{0}

For every $β \in N$ and $p \in R$ , the degree-zero conventions are $\overset{ˉ}{β}_{0} = 0, k_{⋆} (0, β, p) = 0, g_{0} = 0, A_{0} = 0, h_{0} = 0, h_{0, T} = 0 for every T \subseteq {1, \dots, 0} .$ Here $h_{0}$ is imposed directly as the zero representer, rather than formed as the quotient $g_{0} / A_{0}$ .

⊢ Lean

These conventions make the degree-zero boundary compatible with the score and energy notation used for positive degree. They keep later statements total in $d$ while the substantive interaction results take $β \geq 1$ .

Definition 3 [def:graph-class] (Graph class

G_{n, d}

$G_{n, d} := {G_{n} \in 2^{V_{n} \times V_{n}} : G_{n} satisfies A ss u m pt i o n 2} .$

⊢ Lean

The graph class $G_{n, d}$ collects the directed interference graphs satisfying the degree restriction in Assumption 2.

Definition 4 [def:coefficient-class] (Coefficient class

C_{n, d, β} (B)

$C_{n, d, β} (B) := {c = (c_{i, S})_{i, S} : c_{i, S} = 0 whenever S \neq \subseteq N_{i}, c satisfies A ss u m pt i o n s 3 an d 4} .$

⊢ Lean

The coefficient class $C_{n, d, β} (B)$ combines neighborhood support, low-order polynomial structure, and the unitwise $ℓ_{1}$ envelope. It is the coefficient-mass class used for the first minimax risk.

Definition 5 [def:potential-outcome] (Potential outcomes

Y_{i} (z)

and observed outcomes

Y_{i}^{obs}

For a coefficient schedule $c$ and an assignment $z \in {0, 1}^{V_{n}}$ , the potential outcome of unit $i \in V_{n}$ is $Y_{i} (z) := S \subseteq N_{i} \sum c_{i, S} j \in S \prod z_{j} .$ Evaluating the schedule at the realized assignment gives the observed outcome $Y_{i}^{obs} := Y_{i} (Z)$ .

⊢ Lean

The potential outcome $Y_{i} (z)$ is a raw-monomial polynomial in the assignments inside the unit’s neighborhood, and the observed outcome $Y_{i}^{obs}$ is its value at the realized assignment.

Definition 6 [def:model-class] (Model class

M_{n, d, β} (B)

Under Assumptions 2, 3, and 4, with coefficient schedules supported on the in-neighborhoods in the sense that $c_{i, S} = 0$ whenever $S \neq \subseteq N_{i}$ , define $M_{n, d, β} (B) := {(G_{n}, c) : G_{n} \in G_{n, d}, c \in C_{n, d, β} (B)} .$

⊢ Lean

The model class $M_{n, d, β} (B)$ pairs a known bounded-degree graph with an admissible low-order coefficient schedule. The estimator may use the graph, so the decision problem is graph-aware.

Definition 7 [def:snipe-score] (SNIPE score

g_{i, β, p}

$g_{i, β, p} (z) := r = 1 \sum \overset{ˉ}{β}_{d} \frac{Δ _{r} ( p )}{[ p ( 1 - p ) ] ^{r}} S \subseteq N_{i} ∣ S ∣ = r \sum j \in S \prod (z_{j} - p) .$

⊢ Lean

The unit-level score $g_{i, β, p}$ is built from centered Bernoulli monomials on the interference neighborhood. Its coefficients use the contrast factors in Definition 1, matching the all-treated-versus-all-control target for low-order monomials. When $∣ N_{i} ∣ < \overset{ˉ}{β}_{d}$ , the inner subset sum is empty for every order $r > ∣ N_{i} ∣$ . Equivalently, the same score can be read as summing over $1 \leq r \leq min {β, ∣ N_{i} ∣}$ for unit $i$ .

Definition 8 [def:snipe-estimator].

For each unit $i \in V_{n}$ , the observed outcome under the realized assignment is $Y_{i}^{obs} := Y_{i} (Z) .$ The unprojected SNIPE estimator is $τ_{n}^{SNIPE} := \frac{1}{n} i \in V_{n} \sum Y_{i}^{obs} g_{i, β, p} (Z) .$ The projected SNIPE estimator is $τ_{n}^{up} := Π_{[- B, B]} (τ_{n}^{SNIPE}),$ where $Π_{[- B, B]}$ is Euclidean projection onto $[- B, B]$ .

⊢ Lean

The estimator $τ_{n}^{SNIPE}$ averages observed outcomes weighted by their neighborhood scores. The projected version $τ_{n}^{up}$ keeps the estimate in the natural coefficient-mass range for the total effect.

Definition 9 [def:minimax-risk].

The finite-population all-treated-versus-all-control effect is $τ_{n} := \frac{1}{n} i \in V_{n} \sum (Y_{i} (1) - Y_{i} (0)),$ where $1$ and $0$ denote the all-one and all-zero assignments. The minimax risk is $R_{n}^{⋆} (d, β, p, B) := τ_{n} in f (G_{n}, c) \in M_{n, d, β} (B) sup E_{Z} [(τ_{n} - τ_{n})^{2}] .$ The infimum ranges over estimators $τ_{n}$ that are measurable functions of $G_{n}$ , $Z$ , and ${Y_{i}^{obs} : i \in V_{n}}$ .

⊢ Lean

The estimand $τ_{n}$ , the finite-population all-treated-versus-all-control total treatment effect, is evaluated on the fixed potential-outcome schedule. The risk $R_{n}^{⋆} (d, β, p, B)$ is the design-based minimax mean squared error over graph-aware estimators. Because the supremum ranges over the graph class $G_{n, d}$ , the frontier characterizes the hardest bounded-degree graph; for a given graph, achievable risk is governed by that graph’s own local energies and overlap counts, which the upper bound tracks unit by unit.

The block quantities used in the main theorems are the complete-block analogues of the unit score. They isolate the local Bernoulli energy before network overlap contributes additional degree dependence.

Definition 10 [def:block-score-energy] (Block score energy

g_{d}, A_{d}, h_{d}

For $d \geq 1$ and a $d$ -coordinate Bernoulli assignment vector $z$ , define $g_{d} (z) := r = 1 \sum \overset{ˉ}{β}_{d} \frac{Δ _{r} ( p )}{[ p ( 1 - p ) ] ^{r}} S \subseteq {1, \dots, d} ∣ S ∣ = r \sum j \in S \prod (z_{j} - p) .$ The complete-block score energy is $A_{d} := r = 1 \sum \overset{ˉ}{β}_{d} (r d) \frac{Δ _{r} ( p ) ^{2}}{[ p ( 1 - p ) ] ^{r}} .$ The normalized block representer is $h_{d} := \frac{g _{d}}{A _{d}},$ with raw-monomial expansion $h_{d} (z) = T \subseteq {1, \dots, d} \sum h_{d, T} j \in T \prod z_{j} .$

⊢ Lean

Here $g_{d}$ is the complete-block score on $d$ Bernoulli coordinates, $A_{d}$ is its design second moment, and $h_{d}$ is the normalized representer used in the block calculations. The raw-monomial coefficients $h_{d, T}$ also enter the least-favourable block construction below.

The paper also studies a uniformly bounded-outcome class. It uses the same graph and low-order structure, but the envelope is imposed directly on the potential-outcome functions.

Definition 11 [def:bounded-outcome-coefficient-class] (Bounded-outcome coefficient class

C_{n, d, β}^{\infty} (B)

$C_{n, d, β}^{\infty} (B) := {c : c_{i, S} = 0 whenever S \neq \subseteq N_{i}, c_{i, S} = 0 whenever ∣ S ∣ > β, z \in {0, 1}^{V_{n}} sup ∣ Y_{i} (z) ∣ \leq B for every i \in V_{n}} .$ The low-order restriction is the one in Assumption 3.

⊢ Lean

The class $C_{n, d, β}^{\infty} (B)$ places the radius $B$ on the realized range of each unit’s potential-outcome polynomial. This gives a parallel bounded class for comparing coefficient-mass and outcome-bounded formulations.

Definition 12 [def:bounded-outcome-model-class] (Bounded-outcome model class

M_{n, d, β}^{\infty} (B)

$M_{n, d, β}^{\infty} (B) := {(G_{n}, c) : G_{n} \in G_{n, d}, c \in C_{n, d, β}^{\infty} (B)} .$ The graph component satisfies Assumption 2, and the coefficient component satisfies the low-order condition in Assumption 3.

⊢ Lean

The model class $M_{n, d, β}^{\infty} (B)$ is the bounded-outcome counterpart of Definition 6. Both classes retain the same known graph and Bernoulli assignment structure.

Definition 13 [def:two-class-minimax-risks] (Two minimax risks

R_{n, ℓ_{1}}^{⋆}

and

R_{n, \infty}^{⋆}

$R_{n, ℓ_{1}}^{⋆} (d, β, p, B) := R_{n}^{⋆} (d, β, p, B),$ and $R_{n, \infty}^{⋆} (d, β, p, B) := τ_{n} in f (G_{n}, c) \in M_{n, d, β}^{\infty} (B) sup E_{Z} [(τ_{n} - τ_{n})^{2}],$ where $τ_{n}$ is measurable with respect to $(G_{n}, Z, {Y_{i}^{obs} : i \in V_{n}})$ .

⊢ Lean

The notation $R_{n, ℓ_{1}}^{⋆} (d, β, p, B)$ names the coefficient-mass minimax risk, while $R_{n, \infty}^{⋆} (d, β, p, B)$ names the bounded-outcome minimax risk. The estimator class is the same graph-aware design-based class in both risks.

Definition 14 [def:bounded-outcome-clipped-snipe] (Bounded-outcome clipped SNIPE

τ_{n, \infty}^{up}

$τ_{n, \infty}^{up} := Π_{[- 2 B, 2 B]} (τ_{n}^{SNIPE}),$ where $Π_{[- 2 B, 2 B]}$ is Euclidean projection onto $[- 2 B, 2 B]$ .

⊢ Lean

The bounded-outcome clipped estimator $τ_{n, \infty}^{up}$ uses the same unprojected SNIPE statistic as Definition 8 and projects it onto the total-effect range induced by uniformly bounded outcomes.

Finally, the lower-bound and block-calibration arguments use a complete-block least-favourable family. The construction fixes the block count, active population fraction, cosine-squared baseline density, and score-aligned perturbation used later in the proofs.

Definition 15 [def:block-family] (Block priors

Π_{σ}

δ (n, d, β, B, p)

, and

H^{2}

For $n, d, β \in N$ , $B > 0$ , $p \in R$ , $1 \leq d \leq n$ , and $σ \in {- 1, 1}$ , define $m := ⌊ \frac{n}{d} ⌋, q := m d, ρ := \frac{q}{n} .$ On the first $q$ units, form consecutive blocks $V_{b}$ , each of size $d$ . Each block contains every within-block directed arrow, including loops, and the remaining $n - q$ units are isolated.

Define $s := \frac{B}{2}$ and $f_{s} (u) := s^{- 1} cos^{2} (\frac{π u}{2 s}) 1 {∣ u ∣ \leq s} .$ Let $U_{1}, \dots, U_{m}$ be independent real draws with density $f_{s}$ . Define $H_{β, p} := r \geq 1 sup T \sum ∣ h_{r, T} ∣$ and $δ (n, d, β, B, p) := B min {(2 H_{β, p})^{- 1}, (4 π)^{- 1}} min {1, \frac{A _{d}}{m}} .$ Write $δ := δ (n, d, β, B, p)$ . The prior $Π_{σ}$ is the pushforward of the law of $(U_{1}, \dots, U_{m})$ under the following coefficient-schedule map. For each active unit $i \in V_{b}$ and within-block subset $T \subseteq V_{b}$ , $c_{i, T} := U_{b} 1 {T = \emptyset} + σ δ h_{d, T}, ∣ T ∣ \leq k_{⋆} .$ All coefficients for subsets not contained in the active block, and all coefficients of inactive units, are zero.

⊢ Lean

Definition 15 is the least-favourable block construction used to state the complete-block lower-bound construction. The quantities $m$ , $q$ , and $ρ$ describe how many units are active in complete directed $d$ -blocks; $f_{s}$ supplies a compactly supported baseline density; and the sign-indexed prior $Π_{σ}$ perturbs each active block in the direction of the normalized representer $h_{d}$ .

Discussion of the assumptions

Bounded degree with self-loops retained asks that each unit interfere with, and be interfered with by, at most $d$ others; it is satisfied by bounded-degree contact networks, by classroom, village, or market designs with capped group sizes, and by any graph obtained by truncating a sparse network at a known degree cap. It controls topology, while the coefficient-mass bound controls interference strength. A common assignment probability $p$ is normally a design choice: the experimenter selects it, and the frontier’s dependence on $p$ enters through the Bernoulli contrasts $Δ_{r} (p)$ , so the fair-coin case is a specialization of the common-probability design. The low-order condition fixes the functional form; the exposed order $k_{⋆}$ then records which interaction orders the all-treated-versus-all-control contrast can detect.

Main results

The main results characterize the design-based mean squared error in terms of the complete-block score energy $A_{d}$ and the neighborhood-overlap factor $d$ . Before the formal statements, the theorem hierarchy is:

Coefficient-mass minimax frontier. Theorem 1 gives matching lower and clipped-SNIPE upper bounds over $M_{n, d, β} (B)$ at the exposed-binomial scale.
Bounded-outcome extension. Theorem 2 compares the coefficient-mass and uniformly bounded-outcome risks and gives the attaining clipped estimator for the bounded-outcome envelope.
Exact complete-block SNIPE risk. Theorem 2 also evaluates canonical unprojected SNIPE exactly on complete directed blocks and globally for that estimator when $d ∣ n$ .
Local-linear benchmark. Theorem 3 solves the block-local unbiased linear problem on fixed complete directed blocks and characterizes its blockwise risk criterion.

The following map keeps those roles visible while also locating the fair-coin specialization.

Role	What is calibrated	Formal location
Minimax frontier	coefficient-mass lower bound and clipped-SNIPE upper bound	Theorem 1
Two-envelope comparison	coefficient-mass and uniformly bounded-outcome classes	Theorem 2
Estimator-risk equality	canonical unprojected SNIPE on complete directed blocks	Theorem 2
Linear benchmark	block-local design-unbiased linear weights	Definition 16, Definition 17, Definition 18, and Theorem 3
Fair-coin calibration	odd-order block energy and first-order frontier at $p = 1/2$	Theorem 4

Theorem 1 [thm:degree-frontier] (Degree frontier bounds).

Fix an interaction order $β \in N$ and an assignment probability $p \in R$ . Assume:

(Order.) $1 \leq β$ .
(Design probability.) $0 < p < 1$ .

Then there exist constants $c_{β, p}, C_{β, p} \in R$ such that $0 < c_{β, p} \leq C_{β, p} .$ For every $n, d \in N$ and $B \in R$ satisfying $1 \leq n$ , $d \leq n$ , and $0 \leq B$ , with $k_{⋆} (d, β, p)$ as in Definition 1 and $R_{n}^{⋆} (d, β, p, B)$ as in Definition 9, the following hold: $c_{β, p} B^{2} min {1, \frac{d ( k _{⋆} ( d , β , p ) d )}{n}} \leq R_{n}^{⋆} (d, β, p, B) .$ Moreover, $R_{n}^{⋆} (d, β, p, B) \leq (G, c) \in M_{n, d, β} (B) sup E_{Z} [(τ_{n}^{up} - τ_{n})^{2}] \leq C_{β, p} B^{2} min {1, \frac{d ( k _{⋆} ( d , β , p ) d )}{n}},$ where $M_{n, d, β} (B)$ is the model class in Definition 6 and $τ_{n}^{up}$ is SNIPE projected to $[- B, B]$ . The unprojected SNIPE estimator also satisfies $(G, c) \in M_{n, d, β} (B) sup E_{Z} [(τ_{n}^{SNIPE} - τ_{n})^{2}] \leq C_{β, p} B^{2} \frac{d ( k _{⋆} ( d , β , p ) d )}{n} .$

For every $(G, c) \in M_{n, d, β} (B)$ and unit $i$ , writing $N_{i} = {j : (j, i) \in G}$ , $r = 1 \sum m i n {β, ∣ N_{i} ∣} (r ∣ N _{i} ∣) \frac{Δ _{r} ( p ) ^{2}}{{ p ( 1 - p ) } ^{r}} \leq A_{d},$ where $A_{d} = r = 1 \sum m i n {β, d} (r d) \frac{Δ _{r} ( p ) ^{2}}{{ p ( 1 - p ) } ^{r}} .$ For every $(G, c) \in M_{n, d, β} (B)$ , every $r \in N$ , and every unit $i$ , if $1 \leq r$ , then $ℓ \in V_{n} \sum (r ∣ N _{i} \cap N _{ℓ} ∣) = S \subseteq N_{i} ∣ S ∣ = r \sum # {ℓ \in V_{n} : S \subseteq N_{ℓ}} \leq d (r ∣ N _{i} ∣) \leq d (r d) .$

Finally, if $0 < B$ and $1 \leq d$ , let $m = ⌊ \frac{n}{d} ⌋, ρ = \frac{m d}{n}, δ = δ (n, d, β, B, p) .$ For every $U : {1, \dots, m} \to R$ satisfying $∣ U_{b} ∣ \leq B /2$ for all $b$ , there are models $M_{+}$ and $M_{-}$ in $M_{n, d, β} (B)$ whose graph is the complete $d$ -block graph, whose coefficient schedules are the corresponding block schedules with signs $+ 1$ and $- 1$ , and whose total treatment effects obey $τ_{n} (M_{+}) = ρ δ, τ_{n} (M_{-}) = - ρ δ .$ The two block-prior densities with signs $+ 1$ and $- 1$ , relative to the block dominating measure, satisfy $H^{2} (Π_{+}, Π_{-}) \leq \frac{4 π ^{2} m δ ^{2}}{B ^{2} A _{d}} .$ The active fraction and block energy satisfy $\frac{1}{2} \leq ρ \leq 1, \frac{A _{d}}{m} = \frac{d A _{d} / n}{ρ} .$

⊢ Lean

The rate in Theorem 1 separates two forces. The local Bernoulli part is the complete-block energy $A_{d}$ , equivalently the exposed binomial scale in the theorem statement; the global network part is the single additional overlap charge $d$ , coming from the number of unit scores that can share a given $r$ -fold assignment set. The projection onto $[- B, B]$ supplies the saturated branch of the bound, while the unprojected estimator carries the linear-regime variance scale.

The next theorem states the simultaneous comparison between the coefficient-mass class and the uniformly bounded-outcome class. It uses the two minimax risks from Definition 13 and the bounded-outcome projection from Definition 14. The statement should be read in five parts: inclusion of the coefficient-mass class in the bounded-outcome class, the two-class minimax frontier, SNIPE unbiasedness and risk bounds, exact complete-block risks for unprojected SNIPE, and the block lower-bound construction with the exposed-binomial energy comparison.

Theorem 2 [thm:bounded-outcome-degree-frontier] (Bounded-outcome degree frontier).

Fix an interaction order $β \geq 1$ and a treatment probability $p \in (0, 1)$ . Then there are constants $c_{β, p}$ and $C_{β, p}$ such that $0 < c_{β, p} \leq C_{β, p}$ and the following statements hold for every $n, d \in N$ , every $B \in R$ , and every finite design $D$ on ${0, 1}^{V_{n}}$ satisfying the itemized conditions:

(Population and degree.) The population has size $n \geq 1$ , $V_{n} = {1, \dots, n}$ , and the degree index satisfies $d \leq n$ .
(Radius.) The radius satisfies $B \geq 0$ .
(Design.) The design $D$ is the product Bernoulli design with common treatment probability $p$ , as in Assumption 1.

Let $A_{d} = r = 1 \sum \overset{ˉ}{β}_{d} (r d) \frac{Δ _{r} ( p ) ^{2}}{{ p ( 1 - p ) } ^{r}},$ with $\overset{ˉ}{β}_{d}$ and $k_{⋆} (d, β, p)$ as in Definition 1. The coefficient-mass model $M_{n, d, β} (B)$ of Definition 6 is contained in the uniformly bounded-outcome model $M_{n, d, β}^{\infty} (B)$ of Definition 12. If $B > 0$ and $d \geq 1$ , this inclusion is strict: some graph and bounded-outcome coefficient schedule in $M_{n, d, β}^{\infty} (B)$ has no realization with the same graph and coefficient schedule in $M_{n, d, β} (B)$ .

The minimax risks of Definition 9 satisfy $c_{β, p} B^{2} min {1, \frac{d A _{d}}{n}} \leq R_{n, ℓ_{1}}^{⋆} (d, β, p, B) = R_{n}^{⋆} (d, β, p, B) \leq R_{n, \infty}^{⋆} (d, β, p, B) .$ For the bounded-outcome clipped SNIPE estimator $τ_{n, \infty}^{up}$ of Definition 14, $R_{n, \infty}^{⋆} (d, β, p, B) \leq (G, c) \in M_{n, d, β}^{\infty} (B) sup E_{D} [(τ_{n, \infty}^{up} - τ_{n} (G, c))^{2}] \leq C_{β, p} B^{2} min {1, \frac{d A _{d}}{n}} .$ The coefficient-class clipped SNIPE estimator $τ_{n}^{up}$ also satisfies $(G, c) \in M_{n, d, β} (B) sup E_{D} [(τ_{n}^{up} - τ_{n} (G, c))^{2}] \leq C_{β, p} B^{2} min {1, \frac{d A _{d}}{n}} .$

For every $(G, c) \in M_{n, d, β} (B)$ , the unprojected SNIPE estimator $τ_{n}^{SNIPE}$ is $D$ -unbiased for $τ_{n} (G, c)$ . The same $D$ -unbiasedness holds for every $(G, c) \in M_{n, d, β}^{\infty} (B)$ . Its worst-case mean squared error obeys $(G, c) \in M_{n, d, β} (B) sup E_{D} [(τ_{n}^{SNIPE} - τ_{n} (G, c))^{2}] \leq \frac{B ^{2} d A _{d}}{n},$ and $(G, c) \in M_{n, d, β}^{\infty} (B) sup E_{D} [(τ_{n}^{SNIPE} - τ_{n} (G, c))^{2}] \leq \frac{B ^{2} d A _{d}}{n} .$ For every unit $i$ , on both $M_{n, d, β} (B)$ and $M_{n, d, β}^{\infty} (B)$ , the local score energy is at most $A_{d}$ .

The complete-block energy is comparable to the exposed binomial term: $c_{β, p} d (k _{⋆} ( d , β , p ) d) \leq d A_{d} \leq C_{β, p} d (k _{⋆} ( d , β , p ) d) .$

If $d \geq 1$ and $d ∣ n$ , let $m = n / d$ and let the graph be the disjoint union of $m$ complete directed $d$ -blocks, including loops. Then canonical unprojected SNIPE has exact worst-case risk on the fixed block graph and globally on both classes: $M_{n, d, β} (B) on the fixed block graph sup E_{D} [(τ_{n}^{SNIPE} - τ_{n})^{2}] = \frac{B ^{2} A _{d}}{m},$ $M_{n, d, β}^{\infty} (B) on the fixed block graph sup E_{D} [(τ_{n}^{SNIPE} - τ_{n})^{2}] = \frac{B ^{2} A _{d}}{m},$ and $(G, c) \in M_{n, d, β} (B) sup E_{D} [(τ_{n}^{SNIPE} - τ_{n} (G, c))^{2}] = (G, c) \in M_{n, d, β}^{\infty} (B) sup E_{D} [(τ_{n}^{SNIPE} - τ_{n} (G, c))^{2}] = \frac{B ^{2} A _{d}}{m} = \frac{B ^{2} d A _{d}}{n} .$

If $B > 0$ and $d \geq 1$ , put $m = ⌊ n / d ⌋, ρ = \frac{m d}{n}, δ = δ (n, d, β, B, p) as in Definition 15 .$ For every vector $U = (U_{b})_{b = 1}^{m}$ with $∣ U_{b} ∣ \leq B /2$ for all $b$ , there are two coefficient-mass models on the block graph, with signs $+ 1$ and $- 1$ , whose coefficient schedules are the corresponding block schedules and whose total treatment effects are $τ_{n}^{+} = ρ δ, τ_{n}^{-} = - ρ δ .$ For the two associated block priors $Π_{+}$ and $Π_{-}$ , $H^{2} (Π_{+}, Π_{-}) \leq \frac{4 π ^{2} m δ ^{2}}{B ^{2} A _{d}},$ and the active share satisfies $\frac{1}{2} \leq ρ \leq 1, \frac{A _{d}}{m} = \frac{d A _{d} / n}{ρ} .$

⊢ Lean

Theorem 2 shows that the outcome-bounded envelope shares the same design difficulty as the coefficient-mass envelope, up to constants depending only on $(β, p)$ . The strict inclusion clause clarifies that this is a genuine comparison between different bounded classes. The complete-block equality identifies the variance contribution $B^{2} A_{d} / m = B^{2} d A_{d} / n$ exactly for canonical unprojected SNIPE, so the global rate can be read as complete-block energy multiplied by the block-to-population conversion.

The block-local benchmark restricts attention to fixed complete directed block graphs, coefficient-mass schedules on those graphs, and linear block-measurable weights satisfying exact design-unbiasedness equations. Fix a sequence indexed by $t$ with positive integers $n_{t}$ and $d_{t}$ satisfying $d_{t} ∣ n_{t}$ , put $m_{t} := n_{t} / d_{t}$ , and let $G_{t}$ be the complete directed $d_{t}$ -block graph on $V_{n_{t}}$ . The following definitions name the fixed-graph coefficient class, the admissible local linear estimators, and their minimax risk before the sharp benchmark theorem.

Definition 16 [def:local-linear-class] (Local class

M_{t} (G_{t})

Under Assumptions 3 and 4, on the fixed graph $G_{t}$ , define $M_{t} (G_{t}) := ⎩ ⎨ ⎧ c : c_{i, S} = 0 unless S \subseteq N_{i}^{G_{t}} and ∣ S ∣ \leq \overset{ˉ}{β}_{d_{t}}, S \subseteq N_{i}^{G_{t}} \sum ∣ c_{i, S} ∣ \leq B for every i \in V_{n_{t}} ⎭ ⎬ ⎫ .$

⊢ Lean

The class in Definition 16 fixes the complete-block graph and varies only the admissible coefficient schedule. This isolates the cost of block-local unbiased weighting from graph selection or graph heterogeneity.

Definition 17 [def:local-linear-estimator-class] (Block-local estimators

E_{t}^{loc, lin}

For each unit $i \in V_{n_{t}}$ , let $b (i)$ be the index of the complete block containing $i$ , and let $Z_{b (i)}$ be the assignment vector restricted to that block: $b (i) := the unique b such that i \in V_{b}, Z_{b (i)} := (Z_{j})_{j \in V_{b (i)}} .$ Let $L^{2} (P_{Z})$ denote the finite-design square-integrable functions under $P_{Z}$ . Define $E_{t}^{loc, lin} := ⎩ ⎨ ⎧ τ_{w, t} = \frac{1}{n _{t}} i \in V_{n_{t}} \sum w_{i, t} (Z_{b (i)}) Y_{i} (Z) : w_{i, t} \in L^{2} (P_{Z}) is measurable with respect to Z_{b (i)}, E_{Z} [w_{i, t}] = 0, E_{Z} [w_{i, t} \prod_{j \in S} Z_{j}] = 1 for every nonempty S \subseteq N_{i}^{G_{t}} with ∣ S ∣ \leq \overset{ˉ}{β}_{d_{t}} ⎭ ⎬ ⎫ .$

⊢ Lean

The class in Definition 17 encodes block-locality and unbiasedness through moment equations against every nonempty raw monomial up to $\overset{ˉ}{β}_{d_{t}}$ in the unit’s block. The finite-design space $L^{2} (P_{Z})$ is the natural Hilbert space for these block weights under the Bernoulli assignment law from Assumption 1.

Definition 18 [def:local-linear-risk] (Local linear risk

R_{t}^{loc, lin}

$R_{t}^{loc, lin} := τ_{w, t} \in E_{t}^{loc, lin} in f c \in M_{t} (G_{t}) sup E_{Z} [(τ_{w, t} - τ_{n_{t}})^{2}] .$ The supremum ranges over coefficient schedules in $M_{t} (G_{t})$ on the fixed graph $G_{t}$ .

⊢ Lean

The risk in Definition 18 is the complete-block linear benchmark against which SNIPE-type weights can be compared.

Theorem 3 [thm:sharp-local-linear-constant-and-representers] (Sharp local linear representers).

Fix an integer $β$ , a common Bernoulli treatment probability $p$ , and a radius $B$ . Suppose that:

(Smoothness order.) $β \geq 1$ .
(Assignment probability.) $p \in (0, 1)$ .
(Radius.) $B > 0$ .
(Block sequence.) For each index $t$ , $n_{t}$ and $d_{t}$ are positive integers with $d_{t} ∣ n_{t}$ , and $m_{t} := \frac{n _{t}}{d _{t}} .$
(Complete-block graph.) For each $t$ , $G_{t}$ is the directed graph on $V_{n_{t}}$ in which $j \to i$ exactly when $j$ and $i$ are active units in the same quotient class after division by $d_{t}$ ; equivalently, the active units form complete directed blocks with self-loops and inactive units are isolated.
(Assignment design.) The assignment design $D_{t}$ is the product Bernoulli design with common probability $p$ , as in Assumption 1.

Define the complete-block score energy by $A_{d} := r = 1 \sum \overset{ˉ}{β}_{d} (r d) \frac{Δ _{r} ( p ) ^{2}}{{ p ( 1 - p ) } ^{r}} .$ Then, for every $t$ , $R_{t}^{loc, lin} = \frac{B ^{2} A _{d_{t}}}{m _{t}} = \frac{B ^{2} d _{t} A _{d_{t}}}{n _{t}} .$

For local linear weights $w \in E_{t}^{loc, lin}$ , put $Z_{S} := \prod_{j \in S} Z_{j}$ . For each active block $V_{b}$ , define $Ψ_{b, t} (w) := (ε_{i}, S_{i})_{i \in V_{b}} ε_{i} \in {- 1, 1}, S_{i} \subseteq V_{b}, ∣ S_{i} ∣ \leq \overset{ˉ}{β}_{d_{t}} max E_{Z} (i \in V_{b} \sum ε_{i} {w_{i, t} (Z_{b (i)}) Z_{S_{i}} - 1 {S_{i} \neq = \emptyset}})^{2} .$ For every $t$ and every $w \in E_{t}^{loc, lin}$ , $c \in M_{t} (G_{t}) sup E_{Z} [(τ_{w, t} - τ_{n_{t}})^{2}] = \frac{B ^{2}}{n _{t}^{2}} b = 1 \sum m_{t} Ψ_{b, t} (w),$ and, for every active block $V_{b}$ , $d_{t}^{2} A_{d_{t}} \leq Ψ_{b, t} (w) .$

For any sequence $w_{t} \in E_{t}^{loc, lin}$ , $\frac{sup _{c \in M_{t} (G_{t})} E _{Z} [ ( τ _{w_{t}, t} - τ _{n_{t}} ) ^{2} ]}{B ^{2} A _{d_{t}} / m _{t}} ⟶ 1$ if and only if $\frac{1}{m _{t} d _{t}^{2} A _{d_{t}}} b = 1 \sum m_{t} {Ψ_{b, t} (w_{t}) - d_{t}^{2} A_{d_{t}}} ⟶ 0.$ Moreover, the representer condition $\frac{1}{n _{t} A _{d_{t}}} i \in V_{n_{t}} \sum E_{Z} [{w_{i, t} (Z_{b (i)}) - g_{d_{t}} (Z_{b (i)})}^{2}] ⟶ 0$ implies $\frac{sup _{c \in M_{t} (G_{t})} E _{Z} [ ( τ _{w_{t}, t} - τ _{n_{t}} ) ^{2} ]}{B ^{2} A _{d_{t}} / m _{t}} ⟶ 1.$

If $d_{t} \to \infty$ , then there exists a sequence $w_{t} \in E_{t}^{loc, lin}$ such that $\frac{sup _{c \in M_{t} (G_{t})} E _{Z} [ ( τ _{w_{t}, t} - τ _{n_{t}} ) ^{2} ]}{B ^{2} A _{d_{t}} / m _{t}} ⟶ 1,$ and, for all sufficiently large $t$ , $\frac{1}{n _{t} A _{d_{t}}} i \in V_{n_{t}} \sum E_{Z} [{w_{i, t} (Z_{b (i)}) - g_{d_{t}} (Z_{b (i)})}^{2}] = 2.$ For every permutation $π : V_{n_{t}} \to V_{n_{t}}$ that preserves block membership, meaning $b (π (i)) = b (i)$ for every $i$ , the corresponding relabeled normalized average squared distance from the complete-block SNIPE score is also equal to $2$ for all sufficiently large $t$ .

⊢ Lean

Theorem 3 gives an exact finite benchmark for local linear rules on complete blocks. $Ψ_{b, t}$ is the exact blockwise worst-case quadratic form: averaging it across blocks reproduces the worst-case risk, and its lower bound $d_{t}^{2} A_{d_{t}}$ yields the sharp risk $B^{2} A_{d_{t}} / m_{t}$ . The representer condition shows that average $L^{2} (P_{Z})$ closeness to the complete-block SNIPE score is sufficient for asymptotic optimality. The distance- $2$ construction matters because optimality can occur through the blockwise quadratic form even away from the representer in normalized $L^{2} (P_{Z})$ distance.

The fair-coin specialization makes the exposed-order calculation transparent. At $p = 1/2$ , even-order Bernoulli contrasts cancel and odd orders determine the energy; for first-order interference this gives the particularly simple $d^{2} / n$ rate.

Theorem 4 [thm:fair-coin-energy-frontier] (Fair-coin energy frontier).

For the fair-coin design $p = 1/2$ , the following two conclusions hold.

(Exact fair-coin contrasts.) For every $n, d, β \in N$ and $B \in R$ with $n \geq 1$ , $d \geq 1$ , $d \leq n$ , $β \geq 1$ , and $B \geq 0$ , the Bernoulli contrast coefficients satisfy, for every $r \geq 1$ , $Δ_{r} (1/2) = {2^{1 - r}, 0, r odd, r even .$ Consequently the complete-block score energy at interaction order $β$ is $A_{d} = 4 1 \leq r \leq \overset{ˉ}{β}_{d} r odd \sum (r d), \overset{ˉ}{β}_{d} = min {β, d},$ and the exposed order $k_{⋆} (d, β, 1/2)$ from Definition 1 is the largest odd integer at most $\overset{ˉ}{β}_{d}$ , with value $0$ when no such odd integer exists.
(First-order frontier.) There exist constants $c_{lower}, c_{upper} \in R$ with $0 < c_{lower} \leq c_{upper}$ such that, for every $n, d \in N$ and $B \in R$ with $n \geq 1$ , $d \geq 1$ , $d \leq n$ , and $B \geq 0$ , the order-one energy is $A_{d} = 4 d,$ and the minimax risks of Definition 9 satisfy $c_{lower} B^{2} min {1, \frac{d ^{2}}{n}} \leq R_{n}^{⋆} (d, 1, 1/2, B),$ $R_{n}^{⋆} (d, 1, 1/2, B) = R_{n, ℓ_{1}}^{⋆} (d, 1, 1/2, B) \leq R_{n, \infty}^{⋆} (d, 1, 1/2, B) \leq c_{upper} B^{2} min {1, \frac{d ^{2}}{n}} .$ If additionally $d ∣ n$ , then the unprojected SNIPE estimator $τ_{n}^{SNIPE}$ has exact worst-case mean squared error $M \in M_{n, d, 1} (B) sup E_{1/2, M} [(τ_{n}^{SNIPE} - τ_{n})^{2}] = \frac{4 B ^{2} d ^{2}}{n},$ and, over the uniformly bounded-outcome class, $M \in M_{n, d, 1}^{\infty} (B) sup E_{1/2, M} [(τ_{n}^{SNIPE} - τ_{n})^{2}] = \frac{4 B ^{2} d ^{2}}{n} .$

⊢ Lean

Theorem 4 gives the most legible additive-interference calibration of the general theory. For $p = 1/2$ and $β = 1$ , the energy identity $A_{d} = 4 d$ turns the general scale $B^{2} min {1, d A_{d} / n}$ into $B^{2} min {1, d^{2} / n}$ , with exact unprojected SNIPE risk $4 B^{2} d^{2} / n$ on complete blocks when $d ∣ n$ . This benchmark is the known-graph Bernoulli counterpart to problems where the interference structure, clustering, or adjustment information is part of the statistical design (Yu et al., 2022; Eichhorn et al., 2024; Wang et al., 2026).

Discussion and extensions

The rate characterization in Theorems 1 and 2 separates two sources of design difficulty. The first is the local Bernoulli score energy $A_{d}$ , which is determined by the assignment probability $p$ , the exposed interaction order, and the number of coordinates in one neighborhood. The second is the additional out-degree overlap charge $d$ , which enters because one assignment coordinate can appear in the scores of as many as $d$ units under Assumption 2. Together they yield the scale $B^{2} min {1, \frac{d A _{d}}{n}},$ with the equivalent exposed-order form $B^{2} min {1, \frac{d ( k _{⋆} ( d , β , p ) d )}{n}}$ up to constants depending on $(β, p)$ . The constants obtained from the block-energy comparison are interpreted for a fixed $(β, p)$ with $p \in (0, 1)$ . Under that fixed pair, the frontier is uniform in $n$ , $d$ , and $B$ .

For design planning, $A_{d}$ is the local price of extracting the all-treated-versus-all-control contrast from Bernoulli variation inside one neighborhood, while $d A_{d} / n$ is the population-level burden after neighborhood overlap. The linear branch $d A_{d} / n ≪ 1$ is the regime where unprojected SNIPE variance is informative. In the stated bounded classes, when $d A_{d} / n$ is bounded away from zero at the saturation scale, the minimax mean squared error remains of order $B^{2}$ up to constants depending on $(β, p)$ .

The energy $A_{d}$ gives the local price of recovering the all-treated-versus-all-control contrast from Bernoulli variation within a neighborhood. Its definition in Definition 10 sums the squared Bernoulli contrast coefficients across raw monomials up to the available order $\overset{ˉ}{β}_{d}$ . The comparison in Theorem 2 shows that, for fixed $(β, p)$ , this energy has the same order as $(k _{⋆} ( d , β , p ) d)$ . Thus the exposed nonzero contrast order determines how the local polynomial dimension enters the risk, while the assignment probability determines which orders are visible through the coefficients $Δ_{r} (p)$ .

The overlap factor has a different interpretation. It is a graph-combinatorial charge rather than a local representer charge. The covariance terms in SNIPE are organized by common assignment coordinates across neighborhoods, and Theorem 1 bounds the number of such overlaps by $d (r d)$ at each order $r$ . This is the point at which bounded out-degree matters for mean squared error: it controls how many unit-level score contributions can share a given monomial. Related local-interference analyses also emphasize neighborhood growth and overlap as central design features (Sävje et al., 2021; Leung, 2022; Hu et al., 2022; Gao et al., 2025); the characterization here converts that feature into the finite-population minimax factor $d A_{d} / n$ for known bounded-degree low-order polynomial interference.

Complete directed blocks provide the sharp calibration for the same expression. In the block construction of Definition 15, each active unit has the full $d$ -coordinate neighborhood, and the normalized representer direction produces least-favourable perturbations whose separation and affinity are governed by $A_{d} / m$ . When $d ∣ n$ , Theorem 2 gives the exact unprojected SNIPE worst-case risk $B^{2} A_{d} / m = B^{2} d A_{d} / n$ on complete blocks. The complete-block calculation matches the global upper bound’s degree dependence and gives the exact unprojected-SNIPE worst-case risk on the block graph.

The local linear benchmark in Theorem 3 gives a complementary finite-dimensional interpretation. On fixed complete directed block graphs under the coefficient-mass schedule class, the minimax risk over block-local unbiased linear estimators is exactly $B^{2} A_{d_{t}} / m_{t}$ . This exact constant characterizes the block-local unbiased linear class on fixed complete directed blocks under the coefficient-mass schedule class: it optimizes over weights satisfying the block-local moment equations against every nonempty raw monomial up to $\overset{ˉ}{β}_{d_{t}}$ . The clipped estimator, which uses projection, is the procedure that attains the saturated branch of the bounded-class frontier. The criterion $Ψ_{b, t} (w)$ identifies the blockwise extremal quantity that any proposed local linear weight must control, and the representer condition gives a sufficient route to the same benchmark through average $L^{2} (P_{Z})$ closeness to the complete-block SNIPE score. This places SNIPE within a broader local Riesz geometry while preserving the exact blockwise minimax constant.

The comparison between the coefficient-mass and uniformly bounded-outcome classes in Theorem 2 shows that the same degree dependence governs both boundedness formulations. The bounded-outcome class is larger, but the clipped SNIPE estimator projected to $[- 2 B, 2 B]$ attains the same $B^{2} min {1, d A_{d} / n}$ scale. This matters for applications in which the primitive restriction is naturally stated as a uniform potential-outcome envelope rather than an $ℓ_{1}$ coefficient-mass envelope: the design-based rate remains calibrated by the same local energy and overlap factor.

The fair-coin specialization in Theorem 4 makes the role of assignment probability especially transparent. At $p = 1/2$ , the even-order Bernoulli contrasts vanish and the energy is the sum over odd orders. For first-order interference, $A_{d} = 4 d$ , so the minimax scale becomes $B^{2} min {1, d^{2} / n}$ , with exact complete-block unprojected SNIPE risk $4 B^{2} d^{2} / n$ when $d ∣ n$ . This gives a simple known-graph Bernoulli benchmark for comparisons with clustered assignment, unknown interference, and covariate-adjusted designs (Eichhorn et al., 2024; Wang et al., 2026).

For fixed $(β, p)$ sequences with eventual positive degree and stable exposed order $k_{⋆}$ , the comparison $d A_{d} ≍_{β, p} d (k _{⋆} d)$ identifies a sufficient and rate-equivalent vanishing condition for the displayed minimax scale. The heuristic condition is $d (k _{⋆} d) = o (n)$ . Under that fixed-pair interpretation, this corresponds to degree growth on the order $d = o (n^{1/ (k_{⋆} + 1)})$ . For fair-coin assignment with first-order interactions $(β = 1)$ , the condition reads $d = o (n)$ . At the saturation scale, for fixed $(β, p)$ , the frontier identifies $B^{2}$ as the constant-order minimax scale.

Limitations and future work

The results establish design-based minimax mean squared error rates for known bounded-degree low-order polynomial interference under common-probability Bernoulli assignment, together with exact complete-block constants for the stated local linear class.

The framework treats potential outcomes as exact low-order polynomials of the assignment vector, so the reported risk is assignment-design mean squared error for the fixed schedule. Open directions include additive idiosyncratic outcome noise and the associated variance floor, exact leading constants over all measurable estimators beyond the complete-block and local linear benchmarks, heterogeneous treatment probabilities, variance estimation and studentization for the SNIPE family, limit laws under growing-degree regimes, unknown or partially observed graphs, clustered assignment, and covariate adjustment. These extensions would connect the benchmark developed here to settings studied in recent work on local interference, graph uncertainty, and adjusted experimental estimators (Sävje et al., 2021; Leung, 2022; Hu et al., 2022; Gao et al., 2025; Eichhorn et al., 2024; Wang et al., 2026).

Appendices

Proofs and auxiliary lemmas

This appendix collects the auxiliary objects used in the minimax and complete-block arguments. The first group of statements isolates the one-block Hilbert-space calculations: a polynomial perturbation program for least-favourable directions, a dual weight program for block-local unbiased scores, and the representer identity tying both programs to the complete-block energy.

The block-local Riesz formulation works in the finite design space induced by the product Bernoulli law in Assumption 1. For a complete $d$ -block, the low-order polynomial subspace $P_{d}$ contains raw monomials up to the exposed order, and the contrast $L_{d}$ evaluates the all-one-versus-all-zero difference on that block.

Definition 19 [def:perturbation-program] (Low-order block space

P_{d}

, contrast

L_{d}

, and program

PerturbProg_{d, β, p}

For $d \geq 1$ , let $P_{d}$ be the low-order polynomial subspace on $d$ Bernoulli coordinates, $P_{d} := ⎩ ⎨ ⎧ h (z) = S \subseteq {1, \dots, d} ∣ S ∣ \leq \overset{ˉ}{β}_{d} \sum a_{S} j \in S \prod z_{j} ⎭ ⎬ ⎫ .$ Define the all-one-versus-all-zero block contrast functional $L_{d} : P_{d} \to R$ by $L_{d} (h) := h (1, \dots, 1) - h (0, \dots, 0) .$ The perturbation program is $PerturbProg_{d, β, p} := h \in P_{d} min E_{Z} [h (Z)^{2}]$ subject to $L_{d} (h) = 1.$

⊢ Lean

Definition 19 identifies the minimum-energy direction that moves the block contrast by one unit. This is the direction used by the least-favourable complete-block construction in Definition 15: controlling its $L^{2} (P_{Z})$ energy controls the affinity between nearby prior-predictive laws while preserving a fixed separation in total treatment effects.

The dual calculation is a minimum-norm problem over square-integrable block weights. Its moment equations impose exact unbiasedness against every nonempty raw monomial that can enter a low-order block outcome.

Definition 20 [def:weight-program] (Weight program

WeightProg_{d, β, p}

For $d \geq 1$ , define $WeightProg_{d, β, p} := w \in L^{2} (P_{Z}) min E_{Z} [w (Z)^{2}]$ subject to $E_{Z} [w (Z)] = 0$ and, for every nonempty $S \subseteq {1, \dots, d}$ with $∣ S ∣ \leq \overset{ˉ}{β}_{d}$ , $E_{Z} w (Z) j \in S \prod Z_{j} = 1.$

⊢ Lean

Definition 20 is the block analogue of the SNIPE unbiasedness equations. The feasible weights reproduce the all-treated-versus-all-control contrast on every exposed raw monomial, so the value of the program is the smallest block variance compatible with those unbiasedness restrictions.

The next lemma supplies the exact representer calculation behind both programs. It also records the energy scale and a uniform raw-coefficient-mass bound for the normalized representer coefficient $h_{d, T}$ , which is used in the block perturbations.

Lemma 1 [lem:block-energy-representer] (Block representer energy).

Fix an integer $β \geq 1$ and a Bernoulli treatment probability $p \in (0, 1)$ . Let $k_{⋆} (d, β, p)$ be the exposed order in Definition 1. For each $d \geq 1$ , write $A_{d} = r = 1 \sum \overset{ˉ}{β}_{d} (r d) \frac{Δ _{r} ( p ) ^{2}}{{ p ( 1 - p ) } ^{r}} .$ Let $g_{d}$ denote the complete-block centered contrast score, $h_{d} := g_{d} / A_{d}$ , and let $h_{d, T}$ be the coefficient of the raw monomial indexed by $T \subseteq {1, \dots, d}$ in $h_{d}$ .

Then there exist constants $c_{1}, c_{2}, H \in R$ , depending only on $β$ and $p$ , such that $0 < c_{1} \leq c_{2}$ , $0 < H$ , and for every $d \geq 1$ :

(Energy scale.) $c_{1} (k _{⋆} ( d , β , p ) d) \leq A_{d} \leq c_{2} (k _{⋆} ( d , β , p ) d) .$
(Product Bernoulli identities.) For every finite design $D$ on ${0, 1}^{d}$ that is product Bernoulli with common probability $p$ as in Assumption 1, and every $f \in P_{d}$ , $E_{D} [g_{d} (Z) f (Z)] = L_{d} (f) .$ Moreover, $L_{d} (h_{d}) = 1, E_{D} [h_{d} (Z)^{2}] = A_{d}^{- 1} .$
(Perturbation program.) The representer $h_{d}$ is feasible for the perturbation program in Definition 19. For every feasible perturbation $h$ , $A_{d}^{- 1} \leq E_{D} [h (Z)^{2}],$ with equality if and only if $h = h_{d}$ . Consequently, for any proof witnesses of $0 \leq p$ and $p \leq 1$ , $PerturbProg_{d, β, p} = A_{d}^{- 1} .$
(Weight program.) For every weight $w$ feasible for the weight program in Definition 20 under $D$ , $A_{d} \leq E_{D} [w (Z)^{2}],$ with equality if and only if $w = g_{d}$ . Consequently, for any proof witnesses of $0 \leq p$ and $p \leq 1$ , $WeightProg_{d, β, p} = A_{d} .$
(Coefficient mass.) $T \subseteq {1, \dots, d} \sum ∣ h_{d, T} ∣ \leq H .$

⊢ Lean

Proof of Lemma 1.

Fix $β \geq 1$ and $p \in (0, 1)$ . For every integer $r \geq 1$ write $e_{r} (p) := \frac{Δ _{r} ( p ) ^{2}}{[ p ( 1 - p ) ] ^{r}}, m_{r} (p) := \frac{∣ Δ _{r} ( p ) ∣}{[ p ( 1 - p ) ] ^{r}} (1 + p)^{r},$ and define the three constants of the statement by $c_{1} := min {e_{r} (p) : 1 \leq r \leq β, Δ_{r} (p) \neq = 0}, c_{2} := 2^{β} r = 1 \sum β e_{r} (p), H := c_{1} + \frac{2 ^{β} \sum _{r = 1}^{β} m _{r} ( p )}{c _{1}} .$ Since $Δ_{1} (p) = (1 - p) - (- p) = 1 \neq = 0$ , the index set defining $c_{1}$ contains $r = 1$ and is therefore a nonempty finite set; on it $Δ_{r} (p)^{2} > 0$ and $[p (1 - p)]^{r} > 0$ , so $c_{1}$ is a minimum of finitely many strictly positive numbers and $c_{1} > 0$ . Each $m_{r} (p) \geq 0$ for the same reason, so $H \geq c_{1} > 0$ .

Step 1 (exposed orders). Fix $d \geq 1$ and write $k_{⋆} := k_{⋆} (d, β, p)$ . Since $β \geq 1$ and $d \geq 1$ we have $\overset{ˉ}{β}_{d} = min {β, d} \geq 1$ , and $Δ_{1} (p) = 1 \neq = 0$ shows that the set ${r \in {1, \dots, \overset{ˉ}{β}_{d}} : Δ_{r} (p) \neq = 0}$ of Definition 1 is nonempty. Hence $k_{⋆}$ is the maximum of that set, so $1 \leq k_{⋆} \leq \overset{ˉ}{β}_{d}, Δ_{k_{⋆}} (p) \neq = 0, and r \leq k_{⋆} for every r \in {1, \dots, \overset{ˉ}{β}_{d}} with Δ_{r} (p) \neq = 0.$ In particular $k_{⋆} \leq β$ and $k_{⋆} \leq d$ .

Step 2 (binomial domination at fixed order). For all integers $r \leq k \leq d$ with $k \leq β$ , $(r d) \leq 2^{β} (k d) .$ Indeed, counting the pairs $R \subseteq K \subseteq {1, \dots, d}$ with $∣ R ∣ = r$ and $∣ K ∣ = k$ in two ways gives $(r d) (k - r d - r) = (k d) (r k)$ ; the left factor $(k - r d - r) \geq 1$ and $(r k) \leq 2^{k} \leq 2^{β}$ .

Step 3 (energy scale). By Definition 10, $A_{d} = r = 1 \sum \overset{ˉ}{β}_{d} (r d) e_{r} (p),$ a sum of nonnegative terms. Discarding all summands except $r = k_{⋆}$ , which is legitimate by Step 1, $A_{d} \geq (k _{⋆} d) e_{k_{⋆}} (p) \geq c_{1} (k _{⋆} d),$ the second inequality because $1 \leq k_{⋆} \leq β$ and $Δ_{k_{⋆}} (p) \neq = 0$ , so $e_{k_{⋆}} (p)$ is one of the numbers over which $c_{1}$ is the minimum. For the upper bound fix $r \in {1, \dots, \overset{ˉ}{β}_{d}}$ . If $Δ_{r} (p) = 0$ then $e_{r} (p) = 0$ and both sides of the next display vanish; if $Δ_{r} (p) \neq = 0$ then $r \leq k_{⋆} \leq min {β, d}$ by Step 1 and Step 2 applies. Either way $(r d) e_{r} (p) \leq 2^{β} (k _{⋆} d) e_{r} (p) .$ Summing over $r \in {1, \dots, \overset{ˉ}{β}_{d}}$ and then enlarging the index range from ${1, \dots, \overset{ˉ}{β}_{d}}$ to ${1, \dots, β}$ , which only adds nonnegative terms since $\overset{ˉ}{β}_{d} \leq β$ , $A_{d} \leq 2^{β} (k _{⋆} d) r = 1 \sum β e_{r} (p) = c_{2} (k _{⋆} d) .$ This is the energy-scale claim.

Step 4 ( $c_{1} \leq c_{2}$ ). Apply Step 3 at $d = 1$ and set $K := (k _{⋆} ( 1 , β , p ) 1)$ . Step 1 gives $k_{⋆} (1, β, p) \leq \overset{ˉ}{β}_{1} \leq 1$ , so $K \geq 1 > 0$ , and $c_{1} K \leq A_{1} \leq c_{2} K .$ Dividing by $K > 0$ gives $c_{1} \leq c_{2}$ .

Step 5 (design moments). Fix $d \geq 1$ and let $D$ be a finite design on ${0, 1}^{d}$ satisfying the common- $p$ product Bernoulli condition of Assumption 1. Then $D$ assigns to each $z \in {0, 1}^{d}$ the mass $\prod_{j = 1}^{d} p^{z_{j}} (1 - p)^{1 - z_{j}} > 0$ , so for every $f : {0, 1}^{d} \to R$ , $E_{D} [f (Z)^{2}] = 0 if and only if f (z) = 0 for every z \in {0, 1}^{d} .$

Moreover, factorizing the expectation over the independent coordinates, for every nonempty $S \subseteq {1, \dots, d}$ and every $T \subseteq {1, \dots, d}$ , $E_{D} j \in S \prod (Z_{j} - p) j \in T \prod Z_{j} = {[p (1 - p)]^{∣ S ∣} p^{∣ T ∣ - ∣ S ∣}, 0, S \subseteq T, otherwise,$ $E_{D} j \in S \prod (Z_{j} - p) j \in T \prod (Z_{j} - p) = {[p (1 - p)]^{∣ S ∣}, 0, S = T, otherwise .$ In the first identity a coordinate in $S \cap T$ contributes $E [(Z_{j} - p) Z_{j}] = p (1 - p)$ , a coordinate of $T$ outside $S$ contributes $E [Z_{j}] = p$ , a coordinate of $S$ outside $T$ contributes $E [Z_{j} - p] = 0$ , and all remaining coordinates contribute $1$ ; in the second identity a coordinate in $S \cap T$ contributes $E [(Z_{j} - p)^{2}] = p (1 - p)$ and a coordinate of the symmetric difference contributes $0$ .

Step 6 (the score represents $L_{d}$ ). For every integer $t \geq 1$ , $r = 1 \sum t (r t) Δ_{r} (p) p^{t - r} = 1,$ because $\sum_{r = 0}^{t} (r t) (1 - p)^{r} p^{t - r} = [(1 - p) + p]^{t} = 1$ and $\sum_{r = 0}^{t} (r t) (- p)^{r} p^{t - r} = (- p + p)^{t} = 0$ , while the two $r = 0$ terms cancel since $Δ_{0} (p) = 0$ . Let $T \subseteq {1, \dots, d}$ be nonempty with $∣ T ∣ \leq \overset{ˉ}{β}_{d}$ . Inserting the definition of $g_{d}$ from Definition 10 into the first moment identity of Step 5, and using that exactly $(r ∣ T ∣)$ of the sets $S$ with $∣ S ∣ = r$ satisfy $S \subseteq T$ , $E_{D} g_{d} (Z) j \in T \prod Z_{j} = r = 1 \sum \overset{ˉ}{β}_{d} \frac{Δ _{r} ( p )}{[ p ( 1 - p ) ] ^{r}} (r ∣ T ∣) [p (1 - p)]^{r} p^{∣ T ∣ - r} = r = 1 \sum \overset{ˉ}{β}_{d} (r ∣ T ∣) Δ_{r} (p) p^{∣ T ∣ - r} = 1,$ where the last equality is the previous display with $t = ∣ T ∣ \geq 1$ : the summands with $∣ T ∣ < r \leq \overset{ˉ}{β}_{d}$ vanish because $(r ∣ T ∣) = 0$ , so the sum runs effectively over $1 \leq r \leq ∣ T ∣$ . Taking $T = \emptyset$ in the same moment identity, where no nonempty $S$ satisfies $S \subseteq T$ , $E_{D} [g_{d} (Z)] = 0.$

Now $L_{d} (z \mapsto \prod_{j \in T} z_{j}) = 1$ for nonempty $T$ and $L_{d} (1) = 0$ , so the two displays state precisely that $E_{D} [g_{d} (Z) f (Z)] = L_{d} (f)$ whenever $f$ is a raw monomial of degree at most $\overset{ˉ}{β}_{d}$ . Both sides are linear in $f$ , and Definition 19 defines $P_{d}$ as the span of these raw monomials; hence for every $f \in P_{d}$ , $E_{D} [g_{d} (Z) f (Z)] = L_{d} (f) .$

Step 7 (energy identities and positivity). Evaluating $g_{d}$ at the all-treated and at the all-control assignment, each of the $(r d)$ sets $S$ with $∣ S ∣ = r$ contributes $(1 - p)^{r}$ and $(- p)^{r}$ respectively, so $L_{d} (g_{d}) = r = 1 \sum \overset{ˉ}{β}_{d} \frac{Δ _{r} ( p )}{[ p ( 1 - p ) ] ^{r}} (r d) [(1 - p)^{r} - (- p)^{r}] = r = 1 \sum \overset{ˉ}{β}_{d} (r d) e_{r} (p) = A_{d} .$

By the second moment identity of Step 5 the centered monomials appearing in $g_{d}$ are pairwise orthogonal and the one indexed by $S$ has second moment $[p (1 - p)]^{∣ S ∣}$ , so $E_{D} [g_{d} (Z)^{2}] = r = 1 \sum \overset{ˉ}{β}_{d} (r d) \frac{Δ _{r} ( p ) ^{2}}{[ p ( 1 - p ) ] ^{2 r}} [p (1 - p)]^{r} = A_{d} .$

Keeping only the $r = 1$ summand of $A_{d}$ and using $Δ_{1} (p) = 1$ , $A_{d} \geq (1 d) \frac{Δ _{1} ( p ) ^{2}}{p ( 1 - p )} = \frac{d}{p ( 1 - p )} > 0.$

Since $h_{d} = g_{d} / A_{d}$ with $A_{d} > 0$ , dividing the two preceding identities by $A_{d}$ and by $A_{d}^{2}$ gives $L_{d} (h_{d}) = 1, E_{D} [h_{d} (Z)^{2}] = A_{d}^{- 1} .$

Step 8 (the score lies in $P_{d}$ ). Expanding the product coordinate by coordinate, for every $S \subseteq {1, \dots, d}$ and every $z$ , $j \in S \prod (z_{j} - p) = T \subseteq S \sum (- p)^{∣ S ∣ - ∣ T ∣} j \in T \prod z_{j} .$

Every raw monomial occurring on the right has degree $∣ T ∣ \leq ∣ S ∣ \leq \overset{ˉ}{β}_{d}$ when $∣ S ∣ \leq \overset{ˉ}{β}_{d}$ , hence lies in $P_{d}$ . As $g_{d}$ is a finite linear combination of centered monomials with $∣ S ∣ \leq \overset{ˉ}{β}_{d}$ , it follows that $g_{d} \in P_{d}$ and therefore $h_{d} = g_{d} / A_{d} \in P_{d}$ . Together with $L_{d} (h_{d}) = 1$ from Step 7, $h_{d}$ is feasible for the program of Definition 19.

Step 9 (perturbation optimality). Let $h \in P_{d}$ satisfy $L_{d} (h) = 1$ . Step 6 applied to $f = h$ gives $E_{D} [g_{d} (Z) h (Z)] = 1$ , and Step 7 gives $E_{D} [g_{d} (Z)^{2}] = A_{d}$ . Expanding the square, $E_{D} [(A_{d} h (Z) - g_{d} (Z))^{2}] = A_{d}^{2} E_{D} [h (Z)^{2}] - 2 A_{d} E_{D} [g_{d} (Z) h (Z)] + E_{D} [g_{d} (Z)^{2}] = A_{d}^{2} E_{D} [h (Z)^{2}] - A_{d} .$ The left-hand side is nonnegative and $A_{d} > 0$ , so $A_{d}^{2} E_{D} [h (Z)^{2}] \geq A_{d}$ , that is, $A_{d}^{- 1} \leq E_{D} [h (Z)^{2}] .$ Equality holds if and only if the displayed square has expectation zero, which by the faithfulness statement of Step 5 happens if and only if $A_{d} h (z) = g_{d} (z)$ for every $z$ , that is, if and only if $h = h_{d}$ . Conversely $h = h_{d}$ attains the value $A_{d}^{- 1}$ by Step 7.

Step 10 (weight optimality). Let $w$ be feasible for Definition 20 under $D$ , that is, $E_{D} [w (Z)] = 0$ and $E_{D} [w (Z) \prod_{j \in S} Z_{j}] = 1$ for every nonempty $S \subseteq {1, \dots, d}$ with $∣ S ∣ \leq \overset{ˉ}{β}_{d}$ . Since $L_{d} (1) = 0$ and $L_{d} (z \mapsto \prod_{j \in S} z_{j}) = 1$ for nonempty $S$ , these constraints say that $E_{D} [w (Z) f (Z)] = L_{d} (f)$ on every raw monomial $f$ of degree at most $\overset{ˉ}{β}_{d}$ ; both sides are linear in $f$ , and Definition 19 defines $P_{d}$ as the span of those raw monomials, so for every $f \in P_{d}$ , $E_{D} [w (Z) f (Z)] = L_{d} (f) .$

Taking $f = g_{d}$ , which lies in $P_{d}$ by Step 8, and using $L_{d} (g_{d}) = A_{d}$ and $E_{D} [g_{d} (Z)^{2}] = A_{d}$ from Step 7, $E_{D} [(w (Z) - g_{d} (Z))^{2}] = E_{D} [w (Z)^{2}] - 2 E_{D} [w (Z) g_{d} (Z)] + E_{D} [g_{d} (Z)^{2}] = E_{D} [w (Z)^{2}] - A_{d} .$ Nonnegativity of the left-hand side gives $A_{d} \leq E_{D} [w (Z)^{2}],$ and equality forces the displayed square to have expectation zero, hence $w = g_{d}$ by the faithfulness statement of Step 5. Conversely, the two identities of Step 6 are exactly the feasibility constraints for $w = g_{d}$ , and $E_{D} [g_{d} (Z)^{2}] = A_{d}$ , so $g_{d}$ is feasible and attains the value $A_{d}$ .

Step 11 (program values). Both programs are infima of sets of expectations of squares, hence of sets of nonnegative reals, and both sets are nonempty: $h_{d}$ is feasible for the perturbation program with value $A_{d}^{- 1}$ by Steps 7 and 8, and $g_{d}$ is feasible for the weight program with value $A_{d}$ by Step 10. Therefore $PerturbProg_{d, β, p} \leq A_{d}^{- 1}, WeightProg_{d, β, p} \leq A_{d},$ while Steps 9 and 10 bound every value in the two feasible sets below by $A_{d}^{- 1}$ and by $A_{d}$ respectively, giving the reverse inequalities. Hence $PerturbProg_{d, β, p} = A_{d}^{- 1}, WeightProg_{d, β, p} = A_{d} .$

Step 12 (coefficient envelope). Expanding each centered monomial of $g_{d}$ by the identity of Step 8, collecting the coefficient of $\prod_{j \in T} z_{j}$ , and dividing by $A_{d}$ , the raw coefficients of $h_{d}$ in Definition 10 are $h_{d, T} = A_{d}^{- 1} r = 1 \sum \overset{ˉ}{β}_{d} S \subseteq {1, \dots, d}, ∣ S ∣ = r T \subseteq S \sum \frac{Δ _{r} ( p )}{[ p ( 1 - p ) ] ^{r}} (- p)^{r - ∣ T ∣} .$

Applying the triangle inequality inside each $h_{d, T}$ and then exchanging the order of summation, so that each $S$ with $∣ S ∣ = r$ is paired with the subsets $T \subseteq S$ , $T \subseteq {1, \dots, d} \sum ∣ h_{d, T} ∣ \leq A_{d}^{- 1} r = 1 \sum \overset{ˉ}{β}_{d} S \subseteq {1, \dots, d} ∣ S ∣ = r \sum T \subseteq S \sum \frac{∣ Δ _{r} ( p ) ∣}{[ p ( 1 - p ) ] ^{r}} p^{r - ∣ T ∣} = A_{d}^{- 1} r = 1 \sum \overset{ˉ}{β}_{d} (r d) m_{r} (p),$ using $\sum_{T \subseteq S} p^{∣ S ∣ - ∣ T ∣} = (1 + p)^{∣ S ∣} = (1 + p)^{r}$ and the fact that there are $(r d)$ sets $S$ with $∣ S ∣ = r$ . Exactly as in Step 3 — orders with $Δ_{r} (p) = 0$ contribute zero, orders with $Δ_{r} (p) \neq = 0$ satisfy $r \leq k_{⋆}$ and hence $(r d) \leq 2^{β} (k _{⋆} d)$ by Step 2, and the index range may then be enlarged to ${1, \dots, β}$ — $r = 1 \sum \overset{ˉ}{β}_{d} (r d) m_{r} (p) \leq 2^{β} (k _{⋆} d) r = 1 \sum β m_{r} (p) .$ Combining this with the lower energy bound $c_{1} (k _{⋆} d) \leq A_{d}$ of Step 3, with $A_{d} > 0$ and $(k _{⋆} d) > 0$ , $T \subseteq {1, \dots, d} \sum ∣ h_{d, T} ∣ \leq \frac{2 ^{β} ( k _{⋆} d ) \sum _{r = 1}^{β} m _{r} ( p )}{A _{d}} \leq \frac{2 ^{β} \sum _{r = 1}^{β} m _{r} ( p )}{c _{1}} \leq H,$ the last step because $H$ exceeds that quotient by $c_{1} > 0$ .

∎

Lemma 1 is the algebraic core of the appendix. It establishes that the normalized complete-block score is the unique minimum-energy perturbation with unit block contrast, while the unnormalized score is the unique minimum-energy unbiased weight. The same calculation gives $A_{d} ≍_{β, p} (k _{⋆} ( d , β , p ) d)$ , which is the local energy comparison used in Theorems 1 and 2.

The global SNIPE variance calculation needs one graph-combinatorial ingredient. The covariance terms are organized by shared assignment subsets across neighborhoods, and the next lemma converts those shared subsets into a single out-degree charge.

Lemma 2 [lem:overlap-count] (Overlap count bound).

Let $V_{n}$ be the finite population and unit index set, let $G$ be a directed interference graph on $V_{n}$ , and write $N_{i} = {j \in V_{n} : G (j, i)} .$ Suppose:

(Bounded degree.) The graph $G$ satisfies the bounded-degree condition with bound $d$ in Assumption 2.
(Order.) The integer $r$ satisfies $r \geq 1$ .
(Unit.) The unit $i$ belongs to $V_{n}$ .

Then $l \in V_{n} \sum (r ∣ N _{i} \cap N _{l} ∣) = S \subseteq N_{i} ∣ S ∣ = r \sum # {l \in V_{n} : S \subseteq N_{l}}$ and $S \subseteq N_{i} ∣ S ∣ = r \sum # {l \in V_{n} : S \subseteq N_{l}} \leq d (r ∣ N _{i} ∣) \leq d (r d) .$

⊢ Lean

Proof of Lemma 2.

Let $F := {S \subseteq N_{i} : ∣ S ∣ = r}$ be the family of $r$ -element subsets of the neighborhood $N_{i}$ , so that the middle expression in the statement is $\sum_{S \in F} # {l \in V_{n} : S \subseteq N_{l}}$ .

Step 1: a pointwise counting identity. Fix $l \in V_{n}$ . A finite set $S$ satisfies $S \subseteq N_{i} \cap N_{l}$ if and only if $S \subseteq N_{i}$ and $S \subseteq N_{l}$ , so the $r$ -element subsets of $N_{i} \cap N_{l}$ are exactly the members of $F$ that are contained in $N_{l}$ : ${S \subseteq N_{i} \cap N_{l} : ∣ S ∣ = r} = {S \in F : S \subseteq N_{l}} .$ The number of $r$ -element subsets of a finite set $A$ is $(r ∣ A ∣)$ . Taking cardinalities of the two sides and writing the cardinality of the right-hand side as a sum of indicators gives $(r ∣ N _{i} \cap N _{l} ∣) = # {S \in F : S \subseteq N_{l}} = S \in F \sum 1 {S \subseteq N_{l}} .$

Step 2: the double-count identity. Summing the display of Step 1 over $l \in V_{n}$ and exchanging the two finite sums, $l \in V_{n} \sum (r ∣ N _{i} \cap N _{l} ∣) = l \in V_{n} \sum S \in F \sum 1 {S \subseteq N_{l}} = S \in F \sum l \in V_{n} \sum 1 {S \subseteq N_{l}} = S \in F \sum # {l \in V_{n} : S \subseteq N_{l}},$ the last equality because an indicator sum over $l \in V_{n}$ counts the $l$ satisfying the condition. This is the equality asserted in the statement.

Step 3: each subset is reused at most $d$ times. Fix $S \in F$ . Since $∣ S ∣ = r$ and $1 \leq r$ , the set $S$ is nonempty; fix any $j \in S$ . If $S \subseteq N_{l}$ then in particular $j \in N_{l}$ , so ${l \in V_{n} : S \subseteq N_{l}} \subseteq {l \in V_{n} : j \in N_{l}} .$ The out-degree half of Assumption 2, applied at the coordinate $j$ , states that $# {l \in V_{n} : j \in N_{l}} \leq d$ . Combining this with monotonicity of cardinality under inclusion, $# {l \in V_{n} : S \subseteq N_{l}} \leq # {l \in V_{n} : j \in N_{l}} \leq d .$

Step 4: the first inequality. The family $F$ is precisely the family of $r$ -element subsets of $N_{i}$ , hence $∣ F ∣ = (r ∣ N _{i} ∣) .$ Summing the bound of Step 3 over $S \in F$ therefore gives $S \in F \sum # {l \in V_{n} : S \subseteq N_{l}} \leq S \in F \sum d = d ∣ F ∣ = d (r ∣ N _{i} ∣) .$

Step 5: the second inequality. The in-degree half of Assumption 2, applied at the unit $i$ , gives $∣ N_{i} ∣ \leq d$ . Since $m \mapsto (r m)$ is nondecreasing in $m$ , we get $(r ∣ N _{i} ∣) \leq (r d)$ , and multiplying this inequality by the nonnegative integer $d$ yields $d (r ∣ N _{i} ∣) \leq d (r d) .$

Chaining the identity of Step 2 with the inequalities of Steps 4 and 5 proves the claim.

∎

Lemma 2 explains where the extra factor $d$ enters the upper bounds. The binomial term $(r ∣ N _{i} ∣)$ is already represented in the local block energy, while the count of units whose neighborhoods contain the same $r$ -set is controlled by the out-degree part of Assumption 2. This is the covariance-counting step used to pass from block energy to the global $d A_{d} / n$ scale.

The remaining auxiliary statements support the lower bound. They place the two sign-indexed complete-block priors from Definition 15 on a common dominating measure and bound the unhalved squared Hellinger distance between their densities.

Lemma 3 [lem:block-prior-dominating-measure] (Block prior integrates to one).

Use the notation for the Bernoulli assignment law and finite population from Assumption 1, the block size from Assumption 2, the interaction order from Assumption 3, and the radius from Assumption 4. Let $n, d, β \in N$ and $B, p, σ \in R$ , and set $m = ⌊ n / d ⌋$ . Suppose that:

(Population and blocks.) $1 \leq n$ and $1 \leq d \leq n$ .
(Radius.) $B > 0$ , and let $s = B /2$ .
(Design probability.) $p \in (0, 1)$ .
(Prior sign.) $σ \in {- 1, + 1}$ .

Let $f_{s}$ be the cosine-squared baseline density from Definition 15, let $δ$ be the perturbation amplitude from Definition 15, and let $h_{d}$ be the normalized complete-block representer from Definition 10. Define the block dominating measure by $μ_{n, d} = count_{{0, 1}^{V_{n}}} \otimes λ^{m} .$ For $(z, u) \in {0, 1}^{V_{n}} \times R^{m}$ , define the sign- $σ$ block prior density by $π_{σ} (z, u) = (j \in V_{n} \prod p^{z_{j}} (1 - p)^{1 - z_{j}}) b = 1 \prod m f_{s} (u_{b} - σ δ h_{d} (z_{V_{b}})),$ where $V_{1}, \dots, V_{m}$ are the complete $d$ -blocks of Definition 15. Then $\int_{{0, 1}^{V_{n}} \times R^{m}} π_{σ} (z, u) d μ_{n, d} (z, u) = 1.$ Thus $π_{σ}$ is a probability density with respect to the common block dominating measure $μ_{n, d}$ .

⊢ Lean

Proof of Lemma 3.

Put $a (z) := j \in V_{n} \prod p^{z_{j}} (1 - p)^{1 - z_{j}}, c_{b} (z) := σ δ h_{d} (z_{V_{b}}), s := B /2.$ Then $s > 0$ . The cosine-squared baseline satisfies $\int_{R} f_{s} (u) d u = s^{- 1} \int_{- s}^{s} cos^{2} (\frac{π u}{2 s}) d u = 1,$ by the change of variables $x = π u / (2 s)$ and $\int_{- π /2}^{π /2} cos^{2} x d x = π /2$ . Translation invariance of Lebesgue measure gives, for every real $c$ , $\int_{R} f_{s} (u - c) d u = 1.$

For a fixed assignment $z$ , the conditional density in the block coordinates is $u ⟼ b = 1 \prod m f_{s} (u_{b} - c_{b} (z)) .$ Each coordinate factor is integrable, and finite-product Fubini gives $\int_{R^{m}} b = 1 \prod m f_{s} (u_{b} - c_{b} (z)) d u = b = 1 \prod m \int_{R} f_{s} (u - c_{b} (z)) d u = 1.$ Consequently, $\int_{R^{m}} π_{σ} (z, u) d u = a (z) .$

The measurability and integrability needed to apply product Fubini follow from measurability of $f_{s}$ , finiteness of the assignment space, and integrability of the translated coordinate densities. Hence $\int_{{0, 1}^{V_{n}} \times R^{m}} π_{σ} (z, u) d μ_{n, d} (z, u) = z \in {0, 1}^{V_{n}} \sum \int_{R^{m}} π_{σ} (z, u) d u = z \in {0, 1}^{V_{n}} \sum a (z) .$

Finally, the Bernoulli product weights sum to one: $z \in {0, 1}^{V_{n}} \sum j \in V_{n} \prod p^{z_{j}} (1 - p)^{1 - z_{j}} = j \in V_{n} \prod {p + (1 - p)} = 1.$ This proves that the displayed integral is equal to $1$ .

∎

Lemma 3 makes the prior comparison a calculation between two ordinary densities. The product Bernoulli assignment component is the same under both signs, while the observed block coefficients are shifted by the score-aligned perturbation from Definition 15.

Lemma 4 [lem:block-prior-hellinger-bound] (Block prior Hellinger bound).

Let $n, d, β$ be integers and let $B, p \in R$ . Suppose that:

(Design probability.) The Bernoulli design notation and treatment probability $p$ are as in Assumption 1, with $0 < p < 1$ .
(Block size.) The degree parameter $d$ is as in Assumption 2, with $1 \leq d \leq n$ and $1 \leq n$ .
(Interaction order.) The interaction-order parameter $β$ is as in Assumption 3, with $β \geq 1$ .
(Radius.) The radius $B$ is as in Assumption 4, with $B > 0$ .
(Block priors.) Let $μ_{n, d}$ , $π_{+ 1}$ , and $π_{- 1}$ be the block dominating measure and the two signed block prior densities from Lemma 3.
(Amplitude and energy.) Let $m := ⌊ \frac{n}{d} ⌋, δ := B min {(2 H_{β, p})^{- 1}, (4 π)^{- 1}} min {1, \frac{A _{d}}{m}},$ where $m$ and $δ$ are the complete-block count and perturbation amplitude from Definition 15, and $A_{d}$ is the complete-block score energy from Definition 10.

Then the unhalved squared Hellinger distance $H^{2} (π_{+ 1}, π_{- 1}) := \int (π_{+ 1} - π_{- 1})^{2} d μ_{n, d}$ satisfies $H^{2} (π_{+ 1}, π_{- 1}) \leq \frac{4 π ^{2} m δ ^{2}}{B ^{2} A _{d}} .$

⊢ Lean

Proof of Lemma 4.

Let $s := B /2, δ := B min {(2 H_{β, p})^{- 1}, (4 π)^{- 1}} min {1, \frac{A _{d}}{m}},$ and, for an assignment $z$ , write $r_{b} (z) := h_{d} (z_{V_{b}}) .$ For the two conditional block-coordinate densities define $C_{z}^{+} (u) := b = 1 \prod m f_{s} (u_{b} - δ r_{b} (z)), C_{z}^{-} (u) := b = 1 \prod m f_{s} (u_{b} + δ r_{b} (z)),$ so that $π_{+ 1} (z, u) = P_{Z} (z) C_{z}^{+} (u), π_{- 1} (z, u) = P_{Z} (z) C_{z}^{-} (u) .$

Since $P_{Z} (z) \geq 0$ , $(π_{+ 1} (z, u) - π_{- 1} (z, u))^{2} = P_{Z} (z) (C_{z}^{+} (u) - C_{z}^{-} (u))^{2} .$ Applying product Fubini over the counting measure in $z$ and Lebesgue measure in $u$ gives $H^{2} (π_{+ 1}, π_{- 1}) = z \in {0, 1}^{V_{n}} \sum P_{Z} (z) H^{2} (C_{z}^{+}, C_{z}^{-}) .$

We next bound the conditional Hellinger distance. For fixed $z$ , set $f_{b} (u) := f_{s} (u - δ r_{b} (z)), g_{b} (u) := f_{s} (u + δ r_{b} (z)) .$ The coordinate densities are nonnegative, integrable, and normalized: $f_{b} (u) \geq 0, g_{b} (u) \geq 0, \int f_{b} (u) d u = \int g_{b} (u) d u = 1.$

Let $α_{b} := \int_{R} f_{b} (u) g_{b} (u) d u .$ The identity $H^{2} (C_{z}^{+}, C_{z}^{-}) = 2 (1 - b = 1 \prod m α_{b})$ follows by expanding $(\prod_{b} f_{b} - \prod_{b} g_{b})^{2}$ , using $\prod_{b} f_{b} g_{b} = \prod_{b} f_{b} g_{b}$ , and applying finite-product Fubini. Moreover $0 \leq α_{b} \leq 1$ , where the upper bound is Cauchy–Schwarz together with the two normalization identities. Therefore $H^{2} (C_{z}^{+}, C_{z}^{-}) \leq 2 b = 1 \sum m (1 - α_{b}) .$

For any $a, b \in R$ , the cosine translate affinity obeys $1 - \int_{R} f_{s} (u - a) f_{s} (u - b) d u \leq \frac{π ^{2} ( a - b ) ^{2}}{8 s ^{2}} .$ Indeed, if $s \leq ∣ a - b ∣$ , the left side is at most $1$ , while $π^{2} (a - b)^{2} / (8 s^{2}) \geq 1$ . If $∣ a - b ∣ < s$ , order the shifts as $x \leq y$ , put $t := y - x, q := \frac{π t}{2 s},$ and compute the overlap integral as $\int f_{s} (u - x) f_{s} (u - y) d u = (1 - \frac{t}{2 s}) cos q + \frac{sin q}{π} .$ Since $0 \leq q < π /2$ and $q \leq tan q$ , the right-hand side is at least $cos q$ . Thus $1 - \int f_{s} (u - x) f_{s} (u - y) d u \leq 1 - cos q \leq \frac{q ^{2}}{2} = \frac{π ^{2} ( x - y ) ^{2}}{8 s ^{2}},$ and symmetry gives the displayed bound.

Applying this with $a = δ r_{b} (z)$ and $b = - δ r_{b} (z)$ yields $1 - α_{b} \leq \frac{π ^{2} ( 2 δ r _{b} ( z ) ) ^{2}}{8 s ^{2}} .$ Therefore, since $s = B /2$ , $H^{2} (C_{z}^{+}, C_{z}^{-}) \leq 2 b = 1 \sum m \frac{π ^{2} ( 2 δ r _{b} ( z ) ) ^{2}}{8 s ^{2}} = \frac{4 π ^{2} δ ^{2}}{B ^{2}} b = 1 \sum m r_{b} (z)^{2} .$

Substituting the conditional bound into the outer sum gives $H^{2} (π_{+ 1}, π_{- 1}) \leq \frac{4 π ^{2} δ ^{2}}{B ^{2}} z \in {0, 1}^{V_{n}} \sum P_{Z} (z) b = 1 \sum m h_{d} (z_{V_{b}})^{2} .$

It remains to evaluate the Bernoulli average. The restriction $Z_{V_{b}}$ has the $d$ -coordinate product Bernoulli law with probability $p$ , because $P_{Z}$ is the product law from Assumption 1. By the energy identity in Lemma 1, $E [h_{d} (Z_{V_{b}})^{2}] = A_{d}^{- 1} .$ Summing over the $m$ blocks gives $z \in {0, 1}^{V_{n}} \sum P_{Z} (z) b = 1 \sum m h_{d} (z_{V_{b}})^{2} = \frac{m}{A _{d}} .$

Combining the last two displays, $H^{2} (π_{+ 1}, π_{- 1}) \leq \frac{4 π ^{2} δ ^{2}}{B ^{2}} \frac{m}{A _{d}} = \frac{4 π ^{2} m δ ^{2}}{B ^{2} A _{d}} .$ This is the claimed bound.

∎

Lemma 4 supplies the affinity control for the two-point prior comparison. Combined with the total-effect separation stated in Theorems 1 and 2, it yields the saturated lower-bound branch through the complete-block construction and the linear branch through the choice of perturbation amplitude.

Together, these auxiliary statements organize the theorem proofs. The upper bounds in Theorems 1 and 2 use the product-Bernoulli orthogonality in Lemma 1 to identify SNIPE unbiasedness and then apply Lemma 2 to bound the covariance sum by $d A_{d} / n$ . The lower bounds use the complete-block family in Definition 15, the coefficient-mass control in Lemma 1, and the density and Hellinger calculations in Lemmas 3 and 4. The complete-block local linear theorem is the finite-dimensional dual counterpart: Definition 20 and Lemma 1 identify the least possible block energy for unbiased weights, and Theorem 3 aggregates that block criterion across complete blocks.

Verification note

This appendix records the Lean verification scope for the mathematical results stated in the paper.

The Lean 4 development verifies the finite-design algebra, displayed formal statements, and displayed proofs used in the paper. The checked portion includes the Bernoulli design algebra in Assumption 1, the bounded-degree and low-order polynomial constructions in Assumption 2, Assumption 3, Definition 5, and Definition 6, the SNIPE score and estimator identities in Definitions 7 and 8, the minimax-risk definitions in Definitions 9 and 13, the complete-block score energy in Definition 10, and the least-favourable block family in Definition 15. It also verifies the displayed theorem and auxiliary statements in Lemma 1, Lemma 2, Lemma 3, Lemma 4, Theorem 1, Theorem 3, Theorem 2, and Theorem 4.

The checked statements include the SNIPE unbiasedness claims, the displayed worst-case mean squared error and risk bounds, the exact complete-block risk identities, the block-prior density and Hellinger calculations in Lemmas 3 and 4, and the perturbation and weight programs in Definitions 19 and 20 through the representer result in Lemma 1. The verification contract records no external formal dependencies for the displayed objects. The cited literature supplies the econometric framing, terminology, and scholarly positioning around these formal statements, including the randomized-experiment, potential-outcomes, design-based, SNIPE, and low-order interference contexts (Horvitz et al., 1952; Rubin, 1978; Imbens et al., 2015; Aronow et al., 2017; Cortez-Rodriguez et al., 2023; Harshaw et al., 2022).

The Lean development accompanying this paper is the module tree CausalSmith/Experimentation/EXP_SnipeDegreeFrontier_Research of the CausalSmith repository, at commit 3e2bcb6. It builds with lake build against the pinned toolchain recorded there, and every displayed statement carries the declaration name it was checked against, so each can be located and rechecked individually.

Proofs of the main results

Proof of Theorem 1.

Apply Theorem 2 with the fixed $β$ and $p$ . It gives constants $α_{β, p}, Γ_{β, p} \in R$ , depending only on $(β, p)$ , such that $0 < α_{β, p} \leq Γ_{β, p} .$ Set $c_{β, p} := min {1, α_{β, p}^{2}}, C_{β, p} := max {1, Γ_{β, p}^{2}} .$ Then $0 < c_{β, p}$ and $c_{β, p} \leq C_{β, p}$ . Now fix $n, d \in N$ and $B \in R$ with $1 \leq n$ , $d \leq n$ , and $0 \leq B$ . Let $D_{n, p}$ be the product Bernoulli law on ${0, 1}^{V_{n}}$ , $D_{n, p} (z) = j \in V_{n} \prod p^{z_{j}} (1 - p)^{1 - z_{j}} .$ This is the product Bernoulli design with common probability $p$ , so Theorem 2 applies to $D_{n, p}$ . Write $x_{n, d} := \frac{d ( k _{⋆} ( d , β , p ) d )}{n}, y_{n, d} := \frac{d A _{d}}{n} .$ The same conclusion supplies $α_{β, p} d (k _{⋆} ( d , β , p ) d) \leq d A_{d} \leq Γ_{β, p} d (k _{⋆} ( d , β , p ) d) .$ Since $n \geq 1$ , division by $n$ gives $α_{β, p} x_{n, d} \leq y_{n, d} \leq Γ_{β, p} x_{n, d} .$ The case $d = 0$ is included in these displayed inequalities; then $x_{n, d} = y_{n, d} = 0$ .
The elementary rescalings used to pass from $y_{n, d}$ to $x_{n, d}$ are $c_{β, p} min {1, x_{n, d}} \leq α_{β, p} min {1, α_{β, p} x_{n, d}},$ and $Γ_{β, p} min {1, Γ_{β, p} x_{n, d}} \leq C_{β, p} min {1, x_{n, d}} .$ For the first inequality, split into $x_{n, d} \leq 1$ and $x_{n, d} > 1$ , and inside each case split again according as $α_{β, p} x_{n, d} \leq 1$ or $α_{β, p} x_{n, d} > 1$ ; use $c_{β, p} \leq α_{β, p}^{2}$ , $c_{β, p} \leq α_{β, p}$ , and $x_{n, d} \geq 0$ . The second inequality follows from the same two splits, using $Γ_{β, p} \leq C_{β, p}, Γ_{β, p}^{2} \leq C_{β, p}, x_{n, d} \geq 0.$ Together with $α_{β, p} x_{n, d} \leq y_{n, d} \leq Γ_{β, p} x_{n, d}$ , these imply $c_{β, p} min {1, x_{n, d}} \leq α_{β, p} min {1, y_{n, d}},$ and $Γ_{β, p} min {1, y_{n, d}} \leq C_{β, p} min {1, x_{n, d}} .$
Let $R_{n, ℓ_{1}}^{⋆} (d, β, p, B)$ denote the coefficient-mass minimax risk in Definition 13. Theorem 2 gives $α_{β, p} B^{2} min {1, y_{n, d}} \leq R_{n, ℓ_{1}}^{⋆} (d, β, p, B), R_{n, ℓ_{1}}^{⋆} (d, β, p, B) = R_{n}^{⋆} (d, β, p, B) .$ Combining this with the first rescaling yields $c_{β, p} B^{2} min {1, x_{n, d}} \leq R_{n}^{⋆} (d, β, p, B),$ which is the lower bound.
The same application of Theorem 2 gives the projected-SNIPE upper bound at the energy scale, $(G, c) \in M_{n, d, β} (B) sup E_{Z} [(τ_{n}^{up} - τ_{n})^{2}] \leq Γ_{β, p} B^{2} min {1, y_{n, d}} .$ Using the second rescaling gives $(G, c) \in M_{n, d, β} (B) sup E_{Z} [(τ_{n}^{up} - τ_{n})^{2}] \leq C_{β, p} B^{2} min {1, x_{n, d}} .$
It remains to place the minimax risk below the projected-SNIPE risk. The model class is nonempty: the graph with no edges and the identically zero coefficient schedule belongs to $M_{n, d, β} (B)$ , since its degree and coefficient mass are both zero and $0 \leq B$ . Also, for every $(G, c) \in M_{n, d, β} (B)$ , every unit $i$ , and every assignment $z$ , $∣ Y_{i} (z) ∣ = S \subseteq N_{i} \sum c_{i, S} j \in S \prod z_{j} \leq S \subseteq N_{i} \sum ∣ c_{i, S} ∣ j \in S \prod z_{j} \leq S \subseteq N_{i} \sum ∣ c_{i, S} ∣ \leq B,$ because each $z_{j} \in {0, 1}$ . Hence, since $n \geq 1$ , $∣ τ_{n} ∣ = \frac{1}{n} i \in V_{n} \sum {Y_{i} (1) - Y_{i} (0)} \leq 2 B .$ The projection defining $τ_{n}^{up}$ gives $∣ τ_{n}^{up} ∣ \leq B,$ and therefore $(τ_{n}^{up} - τ_{n})^{2} \leq 9 B^{2}$ pointwise. Finite-space measurability is immediate, so $τ_{n}^{up}$ is an admissible estimator for the coefficient-mass minimax problem.
Since $R_{n}^{⋆} (d, β, p, B)$ is the infimum of worst risks over admissible estimators and the preceding step shows that $τ_{n}^{up}$ is admissible, $R_{n}^{⋆} (d, β, p, B) \leq (G, c) \in M_{n, d, β} (B) sup E_{Z} [(τ_{n}^{up} - τ_{n})^{2}] .$
For the unprojected SNIPE estimator, Theorem 2 gives $(G, c) \in M_{n, d, β} (B) sup E_{Z} [(τ_{n}^{SNIPE} - τ_{n})^{2}] \leq B^{2} y_{n, d} .$ Using $y_{n, d} \leq Γ_{β, p} x_{n, d}$ and $Γ_{β, p} \leq C_{β, p}$ , $(G, c) \in M_{n, d, β} (B) sup E_{Z} [(τ_{n}^{SNIPE} - τ_{n})^{2}] \leq C_{β, p} B^{2} x_{n, d},$ which is the stated unprojected bound.
For the unitwise energy statement, define, for $(G, c) \in M_{n, d, β} (B)$ and $i \in V_{n}$ , $A_{i} := r = 1 \sum m i n {β, ∣ N_{i} ∣} (r ∣ N _{i} ∣) \frac{Δ _{r} ( p ) ^{2}}{{ p ( 1 - p ) } ^{r}} .$ The local-energy conclusion of Theorem 2 gives $A_{i} \leq A_{d},$ which is exactly $r = 1 \sum m i n {β, ∣ N_{i} ∣} (r ∣ N _{i} ∣) \frac{Δ _{r} ( p ) ^{2}}{{ p ( 1 - p ) } ^{r}} \leq A_{d} .$
Fix $(G, c) \in M_{n, d, β} (B)$ , $i \in V_{n}$ , and $r \in N$ with $1 \leq r$ . Put $S_{i, r} := {S \subseteq N_{i} : ∣ S ∣ = r} .$ Since membership in $M_{n, d, β} (B)$ includes the bounded-degree condition with bound $d$ , Lemma 2 gives $ℓ \in V_{n} \sum (r ∣ N _{i} \cap N _{ℓ} ∣) = S \in S_{i, r} \sum # {ℓ \in V_{n} : S \subseteq N_{ℓ}},$ and $S \in S_{i, r} \sum # {ℓ \in V_{n} : S \subseteq N_{ℓ}} \leq d (r ∣ N _{i} ∣) \leq d (r d) .$
Finally assume $0 < B$ and $1 \leq d$ , and set $m = ⌊ \frac{n}{d} ⌋, ρ = \frac{m d}{n}, δ = δ (n, d, β, B, p) .$ For every $U : {1, \dots, m} \to R$ satisfying $∣ U_{b} ∣ \leq B /2$ , the block-mixture conclusion of Theorem 2 supplies models $M_{+}$ and $M_{-}$ in $M_{n, d, β} (B)$ with complete $d$ -block graph and the corresponding sign- $+ 1$ and sign- $- 1$ block schedules. Their total treatment effects satisfy $τ_{n} (M_{+}) = ρ δ, τ_{n} (M_{-}) = - ρ δ .$ The same conclusion gives the Hellinger bound $H^{2} (Π_{+}, Π_{-}) \leq \frac{4 π ^{2} m δ ^{2}}{B ^{2} A _{d}},$ and the active-fraction identities $\frac{1}{2} \leq ρ \leq 1, \frac{A _{d}}{m} = \frac{d A _{d} / n}{ρ} .$ These are precisely the remaining assertions.

∎

Proof of Theorem 3.

Throughout, when $t$ is fixed we write $n = n_{t}$ , $d = d_{t}$ , $m = m_{t}$ , $A = A_{d_{t}}$ and $\overset{ˉ}{β} = \overset{ˉ}{β}_{d_{t}} = min {β, d}$ . The population $V_{n}$ is partitioned into the blocks $V_{1}, \dots, V_{m}$ , each of size $d$ ; $b (i)$ is the block containing $i$ , $Z_{b} = (Z_{j})_{j \in V_{b}}$ , and $Z_{S} = \prod_{j \in S} Z_{j}$ . Since $G_{t}$ is the directed complete-block graph, $N_{i}^{G_{t}} = V_{b (i)} for every i \in V_{n}, n = m d .$ The clauses are proved in the order in which their ingredients are established: the exact worst-case formula, then the per-block lower bound, then the minimax constant, then the two asymptotic criteria, then the witness sequence.

We first record the three block-score identities that every step uses. The block score $g_{d}$ of Definition 10 lies in $P_{d}$ , and Lemma 1 gives, under Assumption 1, $E_{Z} [g_{d} (Z_{b})] = 0, E_{Z} [g_{d} (Z_{b}) Z_{S}] = 1 for every \emptyset \neq = S \subseteq V_{b} with ∣ S ∣ \leq \overset{ˉ}{β},$ $E_{Z} [g_{d} (Z_{b})^{2}] = L_{d} (g_{d}) = A > 0.$ The first two displays are the representer identity $E_{Z} [g_{d} (Z) f (Z)] = L_{d} (f)$ at $f \equiv 1$ and at $f = Z_{S}$ ; the third combines that identity at $f = g_{d}$ with $E_{Z} [h_{d} (Z)^{2}] = A^{- 1}$ and $g_{d} = A h_{d}$ , and the strict positivity is the energy-scale bound $A \geq c_{1} (k _{⋆} ( d , β , p ) d) > 0$ .

The exact worst-case risk formula. Fix $t$ and $w \in E_{t}^{loc, lin}$ . For $c \in M_{t} (G_{t})$ the potential outcomes are $Y_{i} (Z) = \sum_{S \subseteq V_{b (i)}, ∣ S ∣ \leq \overset{ˉ}{β}} c_{i, S} Z_{S}$ , and the target is the all-treated versus all-control contrast $τ_{n} = \frac{1}{n} i \in V_{n} \sum {Y_{i} (1_{n}) - Y_{i} (0_{n})} = \frac{1}{n} i \in V_{n} \sum S \subseteq V_{b (i)} ∣ S ∣ \leq \overset{ˉ}{β} \sum c_{i, S} 1 (S \neq = \emptyset) .$ Subtracting this from $τ_{w, t} = n^{- 1} \sum_{i} w_{i, t} (Z_{b (i)}) Y_{i} (Z)$ and grouping units by block, $τ_{w, t} - τ_{n} = \frac{1}{n} b = 1 \sum m e_{b} (c; Z), e_{b} (c; Z) := i \in V_{b} \sum S \subseteq V_{b} ∣ S ∣ \leq \overset{ˉ}{β} \sum c_{i, S} {w_{i, t} (Z_{b}) Z_{S} - 1 (S \neq = \emptyset)} .$

Each $e_{b} (c; Z)$ is a function of $Z_{b}$ alone, and the moment restrictions defining $E_{t}^{loc, lin}$ give $E_{Z} [e_{b} (c; Z)] = 0$ termwise. Distinct blocks use disjoint assignment coordinates, so under Assumption 1 the variables $e_{1} (c; Z), \dots, e_{m} (c; Z)$ are independent and the cross terms vanish: $E_{Z} [(τ_{w, t} - τ_{n})^{2}] = \frac{1}{n ^{2}} b = 1 \sum m E_{Z} [e_{b} (c; Z)^{2}] .$

For a fixed block, $c \mapsto e_{b} (c; Z)$ is linear in the rows $(c_{i, S})_{S}$ and the only constraint $M_{t} (G_{t})$ places on a row is $\sum_{S} ∣ c_{i, S} ∣ \leq B$ with $S \subseteq V_{b}$ and $∣ S ∣ \leq \overset{ˉ}{β}$ . The criterion $c \mapsto E_{Z} [e_{b} (c; Z)^{2}]$ is convex in each row, so its maximum over the row simplex of radius $B$ is attained at a vertex $c_{i, S} = ε_{i} B 1 (S = S_{i})$ with $ε_{i} \in {- 1, 1}$ , for which $e_{b} (c; Z) = B \sum_{i \in V_{b}} ε_{i} {w_{i, t} (Z_{b}) Z_{S_{i}} - 1 (S_{i} \neq = \emptyset)}$ . Hence $c \in M_{t} (G_{t}) sup E_{Z} [e_{b} (c; Z)^{2}] = B^{2} Ψ_{b, t} (w) .$

The row constraint is unitwise, so the maximizing vertices may be chosen block by block and assembled into one schedule in $M_{t} (G_{t})$ that attains every block maximum simultaneously. Therefore $c \in M_{t} (G_{t}) sup E_{Z} [(τ_{w, t} - τ_{n})^{2}] = \frac{B ^{2}}{n ^{2}} b = 1 \sum m Ψ_{b, t} (w),$ which is the asserted worst-case identity.
The per-block lower bound. Fix $t$ , $w \in E_{t}^{loc, lin}$ and a block $V_{b}$ . Every $i \in V_{b}$ has $N_{i}^{G_{t}} = V_{b}$ , so the single function $g_{d} (Z_{b})$ is the common block score for all of them. Expand it in the raw monomials it is spanned by, $g_{d} (Z_{b}) = T \subseteq V_{b} ∣ T ∣ \leq \overset{ˉ}{β} \sum a_{T} Z_{T}, T \subseteq V_{b} ∣ T ∣ \leq \overset{ˉ}{β} \sum a_{T} = g_{d} (1_{d}), a_{\emptyset} = g_{d} (0_{d}) .$ Applying $E_{Z} [w_{i, t}] = 0$ to the term $T = \emptyset$ and $E_{Z} [w_{i, t} Z_{T}] = 1$ to every other term gives, for every $i \in V_{b}$ , $E_{Z} [w_{i, t} (Z_{b}) g_{d} (Z_{b})] = \emptyset \neq = T \subseteq V_{b} ∣ T ∣ \leq \overset{ˉ}{β} \sum a_{T} = g_{d} (1_{d}) - g_{d} (0_{d}) = L_{d} (g_{d}) = A .$

Summing over the $d$ units of $V_{b}$ and using $E_{Z} [g_{d} (Z_{b})^{2}] = A$ , the expansion of a nonnegative square gives $0 \leq E_{Z} {i \in V_{b} \sum w_{i, t} (Z_{b}) - d g_{d} (Z_{b})}^{2} = E_{Z} {i \in V_{b} \sum w_{i, t} (Z_{b})}^{2} - 2 d (d A) + d^{2} A,$ that is, $d^{2} A \leq E_{Z} {i \in V_{b} \sum w_{i, t} (Z_{b})}^{2} .$ The choice $ε_{i} = 1$ and $S_{i} = \emptyset$ for every $i \in V_{b}$ is admissible in $Ψ_{b, t}$ , since $\emptyset \subseteq V_{b}$ and $∣ \emptyset ∣ = 0 \leq \overset{ˉ}{β}$ , and it produces exactly that expectation; as $Ψ_{b, t}$ is a maximum over admissible choices, $E_{Z} {i \in V_{b} \sum w_{i, t} (Z_{b})}^{2} \leq Ψ_{b, t} (w), hence d_{t}^{2} A_{d_{t}} \leq Ψ_{b, t} (w) .$
The exact minimax constant. Define the canonical block-local weights $w_{i, t}^{0} (Z) := g_{d} (Z_{b (i)}),$ which are the SNIPE scores of Definition 7 on $G_{t}$ because $N_{i}^{G_{t}} = V_{b (i)}$ has exactly $d$ elements. They depend on $Z$ only through $Z_{b (i)}$ , and the three block-score identities recorded above are precisely the centering and unit raw-moment restrictions, so $w^{0} \in E_{t}^{loc, lin}$ .

Fix a block $V_{b}$ and an admissible choice $(ε_{i}, S_{i})_{i \in V_{b}}$ in $Ψ_{b, t}$ . All units of $V_{b}$ carry the same weight $g_{d} (Z_{b})$ , so $i \in V_{b} \sum ε_{i} {w_{i, t}^{0} (Z_{b}) Z_{S_{i}} - 1 (S_{i} \neq = \emptyset)} = g_{d} (Z_{b}) F (Z_{b}) - Λ,$ $F (Z_{b}) := i \in V_{b} \sum ε_{i} Z_{S_{i}}, Λ := i \in V_{b} \sum ε_{i} 1 (S_{i} \neq = \emptyset) .$ The block-score identities give $E_{Z} [g_{d} (Z_{b}) F (Z_{b})] = Λ$ , whence $E_{Z} [(g_{d} (Z_{b}) F (Z_{b}) - Λ)^{2}] = E_{Z} [(g_{d} (Z_{b}) F (Z_{b}))^{2}] - Λ^{2} \leq E_{Z} [(g_{d} (Z_{b}) F (Z_{b}))^{2}] .$ Since $0 \leq Z_{S_{i}} \leq 1$ , $∣ ε_{i} ∣ = 1$ and $∣ V_{b} ∣ = d$ , we have $∣ F (Z_{b}) ∣ \leq d$ pointwise, so $E_{Z} [(g_{d} (Z_{b}) F (Z_{b}))^{2}] \leq d^{2} E_{Z} [g_{d} (Z_{b})^{2}] = d^{2} A .$ Maximizing over admissible choices gives the upper bound, and the empty-monomial construction gives the reverse inequality for the same canonical weights, so $Ψ_{b, t} (w^{0}) \leq d^{2} A, and Ψ_{b, t} (w^{0}) = d^{2} A .$

By the worst-case identity and the per-block lower bound, every $w \in E_{t}^{loc, lin}$ satisfies $c \in M_{t} (G_{t}) sup E_{Z} [(τ_{w, t} - τ_{n})^{2}] = \frac{B ^{2}}{n ^{2}} b = 1 \sum m Ψ_{b, t} (w) \geq \frac{B ^{2}}{n ^{2}} m d^{2} A = \frac{B ^{2} A}{m},$ the last equality because $n = m d$ ; and the same computation with $Ψ_{b, t} (w^{0}) = d^{2} A$ turns the inequality into an equality for $w^{0}$ . Taking the infimum over $E_{t}^{loc, lin}$ , $R_{t}^{loc, lin} = \frac{B ^{2} A _{d_{t}}}{m _{t}} = \frac{B ^{2} d _{t} A _{d_{t}}}{n _{t}},$ where the second equality is again $n_{t} = m_{t} d_{t}$ .
The risk ratio equals one plus the normalized block excess. For a sequence $w_{t} \in E_{t}^{loc, lin}$ set $Y_{t} := \frac{1}{m _{t} d _{t}^{2} A _{d_{t}}} b = 1 \sum m_{t} {Ψ_{b, t} (w_{t}) - d_{t}^{2} A_{d_{t}}} .$ Dividing the worst-case identity by $B^{2} A_{d_{t}} / m_{t}$ and using $n_{t}^{2} = m_{t}^{2} d_{t}^{2}$ , $\frac{sup _{c \in M_{t} (G_{t})} E _{Z} [( τ _{w_{t}, t} - τ _{n_{t}} ) ^{2} ]}{B ^{2} A _{d_{t}} / m _{t}} = \frac{m _{t}}{n _{t}^{2} A _{d_{t}}} b = 1 \sum m_{t} Ψ_{b, t} (w_{t}) = \frac{1}{m _{t} d _{t}^{2} A _{d_{t}}} b = 1 \sum m_{t} Ψ_{b, t} (w_{t}) = 1 + Y_{t} .$

Adding, respectively subtracting, the constant sequence $1$ therefore shows that the left-hand ratio converges to $1$ if and only if $Y_{t} \to 0$ , which is the asserted equivalence.
Score closeness implies attainment. Put $δ_{i, t} := w_{i, t} - g_{d_{t}}$ on the block of $i$ , and set $E_{b, t} := i \in V_{b} \sum E_{Z} [δ_{i, t} (Z_{b})^{2}], E_{t} := b = 1 \sum m_{t} E_{b, t} = i \in V_{n_{t}} \sum E_{Z} [{w_{i, t} (Z_{b (i)}) - g_{d_{t}} (Z_{b (i)})}^{2}],$ $X_{t} := \frac{E _{t}}{n _{t} A _{d_{t}}} = \frac{1}{n _{t} A _{d_{t}}} i \in V_{n_{t}} \sum E_{Z} [{w_{i, t} (Z_{b (i)}) - g_{d_{t}} (Z_{b (i)})}^{2}],$ so that the hypothesis of this clause is $X_{t} \to 0$ . The per-block lower bound gives $Y_{t} \geq 0$ for every $t$ .

Fix $t$ , a block $V_{b}$ and an admissible choice $(ε_{i}, S_{i})_{i \in V_{b}}$ . Splitting each weight into its canonical part and its deviation, $i \in V_{b} \sum ε_{i} {w_{i, t} (Z_{b}) Z_{S_{i}} - 1 (S_{i} \neq = \emptyset)} = f (Z_{b}) + v (Z_{b}),$ $f (Z_{b}) := i \in V_{b} \sum ε_{i} {g_{d_{t}} (Z_{b}) Z_{S_{i}} - 1 (S_{i} \neq = \emptyset)}, v (Z_{b}) := i \in V_{b} \sum ε_{i} δ_{i, t} (Z_{b}) Z_{S_{i}} .$ For the canonical part, the calculation with the weights $w^{0}$ gives $E_{Z} [f (Z_{b})^{2}] \leq d_{t}^{2} A_{d_{t}}$ . For $v$ , the Cauchy–Schwarz inequality over the $d_{t}$ summands together with $0 \leq Z_{S_{i}} \leq 1$ and $∣ ε_{i} ∣ = 1$ gives $v (Z_{b})^{2} \leq d_{t} \sum_{i \in V_{b}} δ_{i, t} (Z_{b})^{2}$ pointwise. Thus $E_{Z} [f (Z_{b})^{2}] \leq d_{t}^{2} A_{d_{t}}, E_{Z} [v (Z_{b})^{2}] \leq d_{t} E_{b, t} .$

Bounding the cross term by Cauchy–Schwarz, $E_{Z} [(f + v)^{2}] \leq E_{Z} [f^{2}] + 2 E_{Z} [f^{2}] E_{Z} [v^{2}] + E_{Z} [v^{2}]$ , and maximizing over admissible choices, $Ψ_{b, t} (w_{t}) - d_{t}^{2} A_{d_{t}} \leq 2 d_{t}^{2} A_{d_{t}} d_{t} E_{b, t} + d_{t} E_{b, t} .$

Summing over $b$ and applying Cauchy–Schwarz once more, $\sum_{b = 1}^{m_{t}} d_{t} E_{b, t} \leq m_{t} d_{t} E_{t}$ , so $b = 1 \sum m_{t} {Ψ_{b, t} (w_{t}) - d_{t}^{2} A_{d_{t}}} \leq 2 d_{t}^{2} A_{d_{t}} m_{t} d_{t} E_{t} + d_{t} E_{t} .$

Dividing by $m_{t} d_{t}^{2} A_{d_{t}}$ and using $n_{t} = m_{t} d_{t}$ , $\frac{2 d _{t}^{2} A _{d_{t}} m _{t} d _{t} E _{t} + d _{t} E _{t}}{m _{t} d _{t}^{2} A _{d_{t}}} = 2 \frac{E _{t}}{n _{t} A _{d_{t}}} + \frac{E _{t}}{n _{t} A _{d_{t}}} = 2 X_{t} + X_{t},$

so that $0 \leq Y_{t} \leq 2 X_{t} + X_{t} .$ If $X_{t} \to 0$ then the right-hand bound tends to $0$ by continuity of the square root, so $Y_{t} \to 0$ by squeezing. Since the risk ratio is $1 + Y_{t}$ , this is the asserted convergence of the risk ratio to $1$ .
An optimal sequence at normalized distance two. Assume $d_{t} \to \infty$ . For every $t$ and each block $V_{b}$ , fix a distinguished unit $i_{b}^{⋆} \in V_{b}$ , and define the perturbation by $ψ_{b} (Z_{b}) := ⎩ ⎨ ⎧ {\frac{2 d _{t} A _{d_{t}}}{[ p ( 1 - p ) ] ^{d_{t}}}}^{1/2} \prod_{j \in V_{b}} (Z_{j} - p), 0, if β < d_{t}, if β \geq d_{t} .$ Set, for every $t$ , $w_{i, t} (Z) := g_{d_{t}} (Z_{b (i)}) + 1 (i = i_{b (i)}^{⋆}) ψ_{b (i)} (Z_{b (i)}) .$ Thus the sequence is defined on the whole index set; at indices with $β \geq d_{t}$ it coincides with the canonical block-score weights.

Admissibility. Each $ψ_{b}$ is a function of $Z_{b}$ alone. If $β \geq d_{t}$ , then $ψ_{b} = 0$ , and the block-score identities verify the centering and unit raw-moment conditions. If $β < d_{t}$ , then $\overset{ˉ}{β}_{d_{t}} = β$ , and for every $S \subseteq V_{b}$ with $∣ S ∣ \leq β$ , including $S = \emptyset$ , at least one index $j \in V_{b}$ outside $S$ occurs in the product. Under Assumption 1, that coordinate is independent of $Z_{S}$ and of the remaining factors and satisfies $E_{Z} [Z_{j} - p] = 0$ ; hence $E_{Z} (j \in V_{b} \prod (Z_{j} - p)) Z_{S} = 0 for every S \subseteq V_{b} with ∣ S ∣ < d_{t} .$

So the perturbation changes neither the centering nor the unit raw moments inherited from $g_{d_{t}}$ , and $w_{t} \in E_{t}^{loc, lin}$ for every $t$ .

Since $d_{t} \to \infty$ , the inequality $β < d_{t}$ holds for all sufficiently large $t$ . All remaining computations are on this tail, where the first branch in the definition of $ψ_{b}$ applies.

Perturbation energy. By independence and $E_{Z} [(Z_{j} - p)^{2}] = p (1 - p)$ , $E_{Z} (j \in V_{b} \prod (Z_{j} - p))^{2} = [p (1 - p)]^{d_{t}}, hence E_{Z} [ψ_{b} (Z_{b})^{2}] = 2 d_{t} A_{d_{t}} .$

Block bound and the risk ratio. Fix a block $V_{b}$ and an admissible choice, and split the inner sum into canonical and perturbation parts. Because the perturbation is carried by the single unit $i_{b}^{⋆}$ , the deviation part reduces to one summand, $v (Z_{b}) = ε_{i_{b}^{⋆}} ψ_{b} (Z_{b}) Z_{S_{i_{b}^{⋆}}}$ , and $0 \leq Z_{S_{i_{b}^{⋆}}} \leq 1$ gives $E_{Z} [v (Z_{b})^{2}] \leq E_{Z} [ψ_{b} (Z_{b})^{2}] = 2 d_{t} A_{d_{t}}$ ; the canonical part obeys $E_{Z} [f (Z_{b})^{2}] \leq d_{t}^{2} A_{d_{t}}$ . The same Cauchy–Schwarz bound on the cross term therefore yields, for all sufficiently large $t$ and every $b$ , $Ψ_{b, t} (w_{t}) \leq d_{t}^{2} A_{d_{t}} + 2 d_{t}^{2} A_{d_{t}} 2 d_{t} A_{d_{t}} + 2 d_{t} A_{d_{t}} .$

Dividing the excess by $d_{t}^{2} A_{d_{t}}$ gives $2 2/ d_{t} + 2/ d_{t}$ for each block, hence the same bound for $Y_{t}$ , while $Y_{t} \geq 0$ by the per-block lower bound. Since the risk ratio is $1 + Y_{t}$ , $1 \leq \frac{sup _{c \in M_{t} (G_{t})} E _{Z} [( τ _{w_{t}, t} - τ _{n_{t}} ) ^{2} ]}{B ^{2} A _{d_{t}} / m _{t}} \leq 1 + 2 \frac{2}{d _{t}} + \frac{2}{d _{t}}$ for all sufficiently large $t$ . Since $d_{t} \to \infty$ , squeezing proves the asserted convergence of the risk ratio to $1$ .

Normalized distance. For the same sufficiently large $t$ , the deviation $w_{i, t} - g_{d_{t}}$ equals $ψ_{b}$ at $i = i_{b}^{⋆}$ and vanishes at every other unit, so $i \in V_{n_{t}} \sum E_{Z} [{w_{i, t} (Z_{b (i)}) - g_{d_{t}} (Z_{b (i)})}^{2}] = b = 1 \sum m_{t} E_{Z} [ψ_{b} (Z_{b})^{2}] = 2 m_{t} d_{t} A_{d_{t}},$ and dividing by $n_{t} A_{d_{t}} = m_{t} d_{t} A_{d_{t}}$ gives the value $2$ .

Invariance under block-preserving relabeling. Let $π$ be a bijection of $V_{n_{t}}$ with $b (π (i)) = b (i)$ for every $i$ , and take $t$ in the same sufficiently large tail. Then $π$ maps each block onto itself and therefore permutes the coordinates of $Z_{b}$ among themselves. The relabeled distance moves the score term only: at the unit $i$ it compares the weight $w_{i, t}$ , read at the unit $i$ and at the original assignment, with the canonical score of the relabeled unit $π (i)$ , whose neighborhood is again $V_{b (i)}$ , read at the relabeled assignment whose $j$ th coordinate is $Z_{π^{- 1} (j)}$ . Since $g_{d_{t}}$ is a symmetric function of the coordinates of its block, that relabeled score is the original one, $g_{d_{t}} ((Z_{π^{- 1} (j)})_{j \in V_{b (i)}}) = g_{d_{t}} (Z_{b (i)}) for every i \in V_{n_{t}} .$

Each summand of the normalized distance is therefore unchanged, and substituting this equality into the normalized-distance expression gives $\frac{1}{n _{t} A _{d_{t}}} i \in V_{n_{t}} \sum E_{Z} [{w_{i, t} (Z_{b (i)}) - g_{d_{t}} ((Z_{π^{- 1} (j)})_{j \in V_{b (i)}})}^{2}] = 2,$ which is the asserted invariance.

∎

Proof of Theorem 2.

The two constants. By Lemma 1 there are constants $c_{1}, c_{2}, K$ , depending only on $(β, p)$ , with $0 < c_{1} \leq c_{2}$ and $0 < K$ , such that for every $d \geq 1$ $c_{1} (k _{⋆} ( d , β , p ) d) \leq A_{d} \leq c_{2} (k _{⋆} ( d , β , p ) d), T \subseteq {1, \dots, d} \sum ∣ h_{d, T} ∣ \leq K .$ In particular the quantity $H_{β, p} = sup_{d \geq 1} \sum_{T} ∣ h_{d, T} ∣$ of Definition 15 is finite, with $H_{β, p} \leq K$ . It is also at least one: evaluating the raw expansion of Definition 10 at the all-treated and the all-control assignment of a single-coordinate block and subtracting gives $1 = L_{1} (h_{1}) = \emptyset \neq = T \subseteq {1} \sum h_{1, T} \leq T \subseteq {1} \sum ∣ h_{1, T} ∣ \leq H_{β, p},$ where $L_{1} (h_{1}) = 1$ is the representer identity of Lemma 1. Now set $κ := min {(2 H_{β, p})^{- 1}, (4 π)^{- 1}}, c_{β, p} := min {c_{1}, \frac{κ ^{2}}{16}}, C_{β, p} := max {16, c_{2}} .$ Since $H_{β, p} \geq 1 > 0$ and $π > 0$ , both entries in the definition of $κ$ are positive, so $κ > 0$ and therefore $c_{β, p} > 0$ . Also $c_{β, p} \leq c_{1} \leq c_{2} \leq C_{β, p} .$ With this $κ$ , the perturbation amplitude of Definition 15 reads $δ = B κ η, η := min {1, A_{d} / m} .$
Standing data. Fix $n \geq 1$ , a design $D$ satisfying Assumption 1 with probability $p$ , an integer $d$ with $d \leq n$ , and $B \geq 0$ . Since $p (1 - p) > 0$ , every summand of $A_{d}$ in Definition 10 is nonnegative, so $A_{d} \geq 0$ ; and by Definition 2 $A_{0} = 0$ . For $d \geq 1$ the order $r = 1$ contributes $(1 d) Δ_{1} (p)^{2} / [p (1 - p)] = d / [p (1 - p)] > 0$ to $A_{d}$ , so $0 < A_{d} (d \geq 1) .$
Model comparison. Let $(G_{n}, c) \in M_{n, d, β} (B)$ . Every raw monomial takes values in $[0, 1]$ , so for every unit $i$ and every assignment $z$ $∣ Y_{i} (z) ∣ = S \subseteq N_{i} \sum c_{i, S} j \in S \prod z_{j} \leq S \subseteq N_{i} \sum ∣ c_{i, S} ∣ \leq B,$ the last step being Assumption 4. The remaining requirements of Definition 11 — neighborhood support and Assumption 3 — are already part of Definition 4, and the graph requirement is unchanged. Hence the same pair $(G_{n}, c)$ lies in $M_{n, d, β}^{\infty} (B)$ : the containment holds carrier by carrier.

Now let $B > 0$ and $d \geq 1$ . Pick a unit $i_{0} \in V_{n}$ (possible since $n \geq 1$ ), let $G^{0}$ consist of the single loop $i_{0} \to i_{0}$ , and set $c_{i_{0}, \emptyset}^{0} := B, c_{i_{0}, {i_{0}}}^{0} := - 2 B,$ with all other coefficients zero. Then $N_{i_{0}} = {i_{0}}$ and $N_{i} = \emptyset$ for $i \neq = i_{0}$ , so both degrees in Assumption 2 are at most $1 \leq d$ , and the only nonzero coefficients are indexed by sets of cardinality at most $1 \leq β$ . The induced outcomes are $Y_{i_{0}} (z) = B - 2 B z_{i_{0}} \in {B, - B}, Y_{i} (z) = 0 (i \neq = i_{0}),$ so $(G^{0}, c^{0}) \in M_{n, d, β}^{\infty} (B)$ . Its coefficient mass at $i_{0}$ is $S \subseteq N_{i_{0}} \sum ∣ c_{i_{0}, S}^{0} ∣ = B + 2 B = 3 B > B,$ so no member of $M_{n, d, β} (B)$ has graph $G^{0}$ and schedule $c^{0}$ , and the containment is strict.
Local energy. Let $(G_{n}, c)$ belong to either model class and let $i \in V_{n}$ . By Assumption 2, $∣ N_{i} ∣ \leq d$ , hence $min {β, ∣ N_{i} ∣} \leq \overset{ˉ}{β}_{d}$ and $(r ∣ N _{i} ∣) \leq (r d)$ for every $r$ . Since each term $Δ_{r} (p)^{2} / [p (1 - p)]^{r}$ is nonnegative, $A_{i} = r = 1 \sum m i n {β, ∣ N_{i} ∣} (r ∣ N _{i} ∣) \frac{Δ _{r} ( p ) ^{2}}{[ p ( 1 - p ) ] ^{r}} \leq r = 1 \sum \overset{ˉ}{β}_{d} (r d) \frac{Δ _{r} ( p ) ^{2}}{[ p ( 1 - p ) ] ^{r}} = A_{d} .$ The estimate uses only the degree bound of the carrier graph, so it holds verbatim on $M_{n, d, β} (B)$ and on $M_{n, d, β}^{\infty} (B)$ .
Unbiasedness of SNIPE. Under Assumption 1 the design $D$ is the common- $p$ product Bernoulli law, so the centered monomials in Definition 7 obey, for every unit $i$ and every nonempty $S \subseteq N_{i}$ with $∣ S ∣ \leq min {β, ∣ N_{i} ∣}$ , $E_{Z} g_{i, β, p} (Z) j \in S \prod Z_{j} = 1, E_{Z} [g_{i, β, p} (Z)] = 0.$ The first identity collapses the double sum in $g_{i, β, p}$ to the binomial contrast $\sum_{r = 1}^{∣ S ∣} (r ∣ S ∣) Δ_{r} (p) p^{∣ S ∣ - r} = 1$ ; the second is the vanishing design mean of every nonconstant centered monomial. Expanding $Y_{i}$ over its raw monomials, using Assumption 3 to discard the terms with $∣ S ∣ > β$ , and applying the two identities term by term gives $E_{Z} [g_{i, β, p} (Z) Y_{i} (Z)] = \emptyset \neq = S \subseteq N_{i} \sum c_{i, S} = Y_{i} (1) - Y_{i} (0) .$ Averaging over $i \in V_{n}$ yields $E_{Z} [τ_{n}^{SNIPE}] = τ_{n}$ at every model of $M_{n, d, β} (B)$ and of $M_{n, d, β}^{\infty} (B)$ , since only the low-order and support properties of the schedule were used.
Unclipped SNIPE risk. Fix a model in either class and write $W_{i} (z) := Y_{i} (z) g_{i, β, p} (z), F_{i} (z) := W_{i} (z) - E_{Z} [W_{i} (Z)] .$ Each $F_{i}$ has design mean zero and depends on the assignment only through ${Z_{j} : j \in N_{i}}$ , because both $Y_{i}$ and $g_{i, β, p}$ do. Expanding each $F_{i}$ in the orthogonal basis of centered monomials $\prod_{j \in S} (Z_{j} - p)$ , only subsets $S \subseteq N_{i}$ receive a nonzero coefficient, and the empty subset receives none. For a fixed nonempty $S$ , the units that can contribute are those with $S \subseteq N_{i}$ , and there are at most $d$ of them: choosing $j \in S$ places every such unit in the out-neighborhood of $j$ , which is the out-degree count in Lemma 2. Applying Cauchy–Schwarz to each such group of at most $d$ coefficients and summing over $S$ gives $E_{Z} [(i \in V_{n} \sum F_{i} (Z))^{2}] \leq d i \in V_{n} \sum E_{Z} [F_{i} (Z)^{2}] .$

For each unit, $∣ Y_{i} (z) ∣ \leq B$ — by the coefficient-mass envelope derived from Definitions 6 and 5 on $M_{n, d, β} (B)$ , and by Definition 12 on $M_{n, d, β}^{\infty} (B)$ — while the exact score energy is $E_{Z} [g_{i, β, p} (Z)^{2}] = A_{i}$ . Hence, using the local energy comparison above, $E_{Z} [F_{i} (Z)^{2}] \leq E_{Z} [W_{i} (Z)^{2}] \leq B^{2} E_{Z} [g_{i, β, p} (Z)^{2}] = B^{2} A_{i} \leq B^{2} A_{d} .$

The unbiasedness identity above makes the mean squared error equal to the variance, and $E_{Z} [(τ_{n}^{SNIPE} - τ_{n})^{2}] = \frac{1}{n ^{2}} E_{Z} [(i \in V_{n} \sum F_{i} (Z))^{2}] \leq \frac{d}{n ^{2}} \cdot n \cdot B^{2} A_{d} = B^{2} \frac{d A _{d}}{n} .$ Taking the supremum over each class, $M_{n, d, β} (B) sup E_{Z} [(τ_{n}^{SNIPE} - τ_{n})^{2}] \leq B^{2} \frac{d A _{d}}{n}, M_{n, d, β}^{\infty} (B) sup E_{Z} [(τ_{n}^{SNIPE} - τ_{n})^{2}] \leq B^{2} \frac{d A _{d}}{n} .$
Target envelopes and clipping. On $M_{n, d, β} (B)$ the all-treated versus all-control contrast of a single unit is $Y_{i} (1) - Y_{i} (0) = \sum_{\emptyset \neq = S \subseteq N_{i}} c_{i, S}$ , so by Assumption 4 $∣ τ_{n} ∣ \leq \frac{1}{n} i \in V_{n} \sum \emptyset \neq = S \subseteq N_{i} \sum ∣ c_{i, S} ∣ \leq B,$ whereas on $M_{n, d, β}^{\infty} (B)$ the uniform outcome envelope gives $∣ τ_{n} ∣ \leq \frac{1}{n} i \in V_{n} \sum (∣ Y_{i} (1) ∣ + ∣ Y_{i} (0) ∣) \leq 2 B .$

For $R \geq 0$ and any target $t$ with $∣ t ∣ \leq R$ , Euclidean projection onto $[- R, R]$ is a contraction towards $t$ : $(Π_{[- R, R]} (x) - t)^{2} \leq (x - t)^{2} for every x \in R .$ Applying this with $R = B$ , $t = τ_{n}$ on $M_{n, d, β} (B)$ , and with $R = 2 B$ , $t = τ_{n}$ on $M_{n, d, β}^{\infty} (B)$ , gives modelwise $E_{Z} [(τ_{n}^{up} - τ_{n})^{2}] \leq E_{Z} [(τ_{n}^{SNIPE} - τ_{n})^{2}], E_{Z} [(τ_{n, \infty}^{up} - τ_{n})^{2}] \leq E_{Z} [(τ_{n}^{SNIPE} - τ_{n})^{2}] .$

The projections also give saturated bounds that do not involve the design at all: $∣ τ_{n}^{up} ∣ \leq B$ together with $∣ τ_{n} ∣ \leq B$ yields $(τ_{n}^{up} - τ_{n})^{2} \leq 4 B^{2}$ pointwise, and $∣ τ_{n, \infty}^{up} ∣ \leq 2 B$ together with $∣ τ_{n} ∣ \leq 2 B$ yields $(τ_{n, \infty}^{up} - τ_{n})^{2} \leq 16 B^{2}$ pointwise. Using the unprojected SNIPE risk bound above when $d A_{d} / n \leq 1$ and the saturated bounds when $d A_{d} / n > 1$ , $M_{n, d, β} (B) sup E_{Z} [(τ_{n}^{up} - τ_{n})^{2}] \leq 4 B^{2} min {1, \frac{d A _{d}}{n}},$ $M_{n, d, β}^{\infty} (B) sup E_{Z} [(τ_{n, \infty}^{up} - τ_{n})^{2}] \leq 16 B^{2} min {1, \frac{d A _{d}}{n}} .$ Since $4 \leq 16 \leq C_{β, p}$ and $B^{2} min {1, d A_{d} / n} \geq 0$ , both right-hand sides are at most $C_{β, p} B^{2} min {1, d A_{d} / n}$ , which proves the coefficient-class clipped bound of the theorem.
Ordering of the two minimax risks. The identity $R_{n, ℓ_{1}}^{⋆} (d, β, p, B) = R_{n}^{⋆} (d, β, p, B)$ is the definition of the coefficient-mass risk in Definition 13.

The carrier-preserving inclusion established from Definitions 6 and 12 sends each $(G_{n}, c) \in M_{n, d, β} (B)$ to a member of $M_{n, d, β}^{\infty} (B)$ with the same graph and the same schedule, hence with the same observed outcomes and the same target; the modelwise risk of any estimator is therefore unchanged by the embedding. Consequently, for every graph-aware estimator $τ_{n}$ , $M_{n, d, β} (B) sup E_{Z} [(τ_{n} - τ_{n})^{2}] \leq M_{n, d, β}^{\infty} (B) sup E_{Z} [(τ_{n} - τ_{n})^{2}] .$ The infima in Definitions 9 and 13 range over the same graph-, assignment-, and observed-outcome-measurable estimators. Taking the infimum over that common estimator class gives $R_{n, ℓ_{1}}^{⋆} (d, β, p, B) \leq R_{n, \infty}^{⋆} (d, β, p, B) .$

The estimator $τ_{n, \infty}^{up}$ of Definition 14 is measurable in the graph, assignment, and observed outcomes. Therefore it is one of the estimators over which the bounded-outcome infimum in Definition 13 is taken. The infimum is at most the worst-case risk of this estimator, and the clipped-risk bound above gives $R_{n, \infty}^{⋆} (d, β, p, B) \leq M_{n, d, β}^{\infty} (B) sup E_{Z} [(τ_{n, \infty}^{up} - τ_{n})^{2}] \leq C_{β, p} B^{2} min {1, \frac{d A _{d}}{n}} .$
The least-favourable block family. Assume $B > 0$ and $d \geq 1$ , and use the notation of Definition 15: $m = ⌊ n / d ⌋$ , $q = m d$ , $ρ = q / n$ , $s = B /2$ , and $δ = B κ η$ , as specified in Definition 15 with the constant $κ$ fixed above. Since $1 \leq d \leq n$ we have $m \geq 1$ . Fix $U = (U_{1}, \dots, U_{m})$ with $∣ U_{b} ∣ \leq B /2$ for all $b$ , and let $σ \in {- 1, + 1}$ .

Membership. The signed schedule of Definition 15 assigns to each active unit $i \in V_{b}$ the coefficients $c_{i, T} = U_{b} 1 {T = \emptyset} + σ δ h_{d, T}$ for $T \subseteq V_{b}$ , and zero to every other coefficient. It is supported in $N_{i} = V_{b}$ on the complete-block graph, and it vanishes for $∣ T ∣ > β$ because $h_{d, T} = 0$ there, so Assumption 3 holds; the complete-block graph has both degrees equal to $d$ , so Assumption 2 holds. For the mass, $η \leq 1$ and $κ \leq (2 H_{β, p})^{- 1}$ give $T \subseteq N_{i} \sum ∣ c_{i, T} ∣ \leq ∣ U_{b} ∣ + ∣ δ ∣ T \sum ∣ h_{d, T} ∣ \leq \frac{B}{2} + B κ H_{β, p} \leq \frac{B}{2} + \frac{B}{2} = B .$ Hence both signed schedules define models $M_{+}$ and $M_{-}$ in $M_{n, d, β} (B)$ carried by the complete-block graph.

Targets. On an active unit the baseline $U_{b}$ cancels in the all-treated versus all-control contrast, and $L_{d} (h_{d}) = h_{d} (1_{d}) - h_{d} (0_{d}) = 1$ by Lemma 1, so $Y_{i} (1) - Y_{i} (0) = σ δ (i active), Y_{i} (1) - Y_{i} (0) = 0 (i inactive) .$ There are $q = m d$ active units, so $τ_{n} (M_{σ}) = \frac{q}{n} σ δ = σ ρ δ, i.e. τ_{n} (M_{+}) = ρ δ, τ_{n} (M_{-}) = - ρ δ .$

Hellinger distance. Every unit of an active block shares the block’s outcome value, so the observation reduces to the pair $(Z, (Y_{b})_{b \leq m})$ , whose law under $Π_{σ}$ has density $Pr (Z = z) b = 1 \prod m f_{s} (y_{b} - σ δ h_{d} (z_{V_{b}}))$ against counting measure on assignments times $m$ -dimensional Lebesgue measure. That reference measure is the block dominating measure $μ_{n, d}$ and the displayed function is the sign- $σ$ block prior density $π_{σ}$ of Lemma 3, which integrates to one against $μ_{n, d}$ ; hence $Π_{+}$ and $Π_{-}$ are probability laws with the common dominating measure $μ_{n, d}$ , and their squared Hellinger distance may be computed from the two densities. Conditionally on $Z = z$ , the two laws $Π_{+}$ and $Π_{-}$ are products of $m$ translates of $f_{s}$ with separations $2 δ h_{d} (z_{V_{b}})$ , and the cosine-squared density has quadratic affinity defect $1 - \int f_{s} (u - a) f_{s} (u - a^{'}) d u \leq \frac{π ^{2} ( a - a ^{'} ) ^{2}}{8 s ^{2}} .$ Summing the defect over the $m$ coordinates and using $H^{2} = 2 (1 - affinity)$ with $s = B /2$ gives the conditional bound $H^{2} (Π_{+} (\cdot ∣ z), Π_{-} (\cdot ∣ z)) \leq \frac{4 π ^{2} δ ^{2}}{B ^{2}} b = 1 \sum m h_{d} (z_{V_{b}})^{2} .$ Because the assignment marginal is common to the two laws, the unconditional squared Hellinger distance is the design average of the conditional ones. Each block sees an independent copy of the Bernoulli law, so $E_{Z} [h_{d} (Z_{V_{b}})^{2}] = A_{d}^{- 1}$ by Lemma 1, whence $E_{Z} [\sum_{b} h_{d} (Z_{V_{b}})^{2}] = m / A_{d}$ and $H^{2} (Π_{+}, Π_{-}) \leq \frac{4 π ^{2} m δ ^{2}}{B ^{2} A _{d}} .$ This is exactly the bound of Lemma 4, whose amplitude $δ$ and complete-block count $m$ are the ones fixed above.

Active share. From $q = m d \leq n$ we get $ρ \leq 1$ . Writing $n = q + (n - q)$ with $n - q < d \leq q$ (the second inequality because $m \geq 1$ ) gives $n < 2 q$ , hence $\frac{1}{2} \leq ρ \leq 1.$ Finally, $q = m d$ and $ρ = q / n$ give $ρ A_{d} / m = d A_{d} / n$ , that is, $\frac{A _{d}}{m} = \frac{d A _{d} / n}{ρ} .$
Lower frontier. If $d = 0$ then $A_{0} = 0$ by Definition 2 and the left side of the claimed bound is $0$ ; if $B = 0$ it is $0$ as well. In both cases the bound holds because the minimax risk is nonnegative: every modelwise risk is a mean squared error, the zero estimator is admissible so the infimum is over a nonempty set, and the empty graph with the zero schedule is a member of $M_{n, d, β} (B)$ .

Assume now $B > 0$ and $d \geq 1$ , and keep the notation of Definition 15. Since $η \leq A_{d} / m$ we have $η^{2} \leq A_{d} / m$ , i.e. $m η^{2} \leq A_{d},$ and since $κ \leq (4 π)^{- 1}$ we have $16 π^{2} κ^{2} \leq 1$ , i.e. $4 π^{2} κ^{2} \leq \frac{1}{4}$ . Substituting $δ = B κ η$ into the preceding Hellinger bound, $H^{2} (Π_{+}, Π_{-}) \leq \frac{4 π ^{2} m δ ^{2}}{B ^{2} A _{d}} = 4 π^{2} κ^{2} \frac{m η ^{2}}{A _{d}} \leq \frac{1}{4} \cdot \frac{m η ^{2}}{A _{d}} \leq \frac{1}{4} .$ In the unhalved convention the total variation distance is bounded by the Hellinger distance, so $A sup Π_{+} (A) - Π_{-} (A) \leq H^{2} (Π_{+}, Π_{-}) \leq \frac{1}{2} .$ The two displayed block targets $\pm ρ δ$ are $2 ρ δ$ apart. Let $τ_{n}$ be any admissible estimator and let $T$ be its value read on the block construction, a measurable function of $(Z, (Y_{b})_{b \leq m})$ . Le Cam’s two-point bound gives $max {Π_{+} (∣ T - ρ δ ∣ \geq ρ δ), Π_{-} (∣ T + ρ δ ∣ \geq ρ δ)} \geq \frac{1 - \frac{1}{2}}{2} = \frac{1}{4} .$

Each of the two probabilities is dominated by the worst-case risk. Indeed, for each sign $σ$ and each baseline vector $U$ in the support of the prior, the block schedule described in Definition 15 is a member of $M_{n, d, β} (B)$ whose target is $σ ρ δ$ , and pointwise in the assignment $(ρ δ)^{2} 1 {∣ τ_{n} - σ ρ δ ∣ \geq ρ δ} \leq (τ_{n} - σ ρ δ)^{2},$ so taking design expectations and then averaging over $U$ — a probability law — gives $(ρ δ)^{2} Π_{σ} (∣ T - σ ρ δ ∣ \geq ρ δ) \leq M_{n, d, β} (B) sup E_{Z} [(τ_{n} - τ_{n})^{2}] .$ Combining the last two displays, every admissible estimator has worst-case risk at least $(ρ δ)^{2} /4$ , and taking the infimum over admissible estimators, $\frac{( ρ δ ) ^{2}}{4} \leq R_{n, ℓ_{1}}^{⋆} (d, β, p, B) .$

It remains to convert the scale. By the displayed identity $A_{d} / m = (d A_{d} / n) / ρ$ and $ρ \leq 1$ , $\frac{d A _{d}}{n} = ρ \frac{A _{d}}{m} \leq \frac{A _{d}}{m}, hence min {1, \frac{d A _{d}}{n}} \leq min {1, \frac{A _{d}}{m}} = η^{2},$ the last equality because $η = min {1, A_{d} / m}$ . Also $ρ \geq \frac{1}{2}$ gives $ρ^{2} /4 \geq 1/16$ . Therefore, using $c_{β, p} \leq κ^{2} /16$ , $c_{β, p} B^{2} min {1, \frac{d A _{d}}{n}} \leq \frac{κ ^{2}}{16} B^{2} η^{2} \leq \frac{ρ ^{2}}{4} B^{2} κ^{2} η^{2} = \frac{( ρB κ η ) ^{2}}{4} = \frac{( ρ δ ) ^{2}}{4} \leq R_{n, ℓ_{1}}^{⋆} (d, β, p, B) .$
Binomial scale. Let $d \geq 1$ and write $k_{⋆} = k_{⋆} (d, β, p)$ . Multiplying the energy scale of Lemma 1 by $d \geq 0$ and using the inequalities $c_{β, p} \leq c_{1}$ and $c_{2} \leq C_{β, p}$ fixed with the constants above, $c_{β, p} d (k _{⋆} d) \leq c_{1} d (k _{⋆} d) \leq d A_{d} \leq c_{2} d (k _{⋆} d) \leq C_{β, p} d (k _{⋆} d) .$ For $d = 0$ all three quantities vanish, since the factor $d$ is zero.
Exact complete-block risk. Assume $d \geq 1$ and $d ∣ n$ , put $m = n / d$ , and let $G$ be the disjoint union of $m$ complete directed $d$ -blocks with loops. Since $n = m d$ , $\frac{B ^{2} A _{d}}{m} = B^{2} \frac{d A _{d}}{n} .$

Suppose first $B > 0$ . On $G$ every unit has $N_{i} = V_{b (i)}$ with $∣ N_{i} ∣ = d$ , so the SNIPE score of Definition 7 is exactly the complete-block score of Definition 10 read on that unit’s block, $g_{i, β, p} (z) = g_{d} (z_{V_{b (i)}}),$ and consequently $τ_{n}^{SNIPE} = τ_{w, t}$ for the block-local weights $w_{i, t} := g_{d}$ , which satisfy the mean-zero and raw-moment restrictions of Definition 17 by Lemma 1. Apply Theorem 3 at the index with $n_{t} = n$ , $d_{t} = d$ , $m_{t} = m$ and $G_{t} = G$ . It gives the exact blockwise representation of the fixed-graph worst-case risk and the lower bound $d^{2} A_{d} \leq Ψ_{b, t} (w)$ for every admissible $w$ . For the canonical choice $w_{i, t} = g_{d}$ the matching upper bound $Ψ_{b, t} (w) \leq d^{2} A_{d}$ holds on every block $V_{b}$ . Indeed, fix a block $V_{b}$ and a signed choice $(ε_{i}, S_{i})_{i \in V_{b}}$ admissible in the maximum defining $Ψ_{b, t}$ . The $d$ summands are not copies of one another — they carry different subsets $S_{i}$ and different signs $ε_{i}$ — but every unit of the block carries the same weight $g_{d} (Z_{b})$ , which therefore factors out: $i \in V_{b} \sum ε_{i} {g_{d} (Z_{b}) Z_{S_{i}} - 1 (S_{i} \neq = \emptyset)} = g_{d} (Z_{b}) Φ (Z_{b}) - Λ,$ $Φ (Z_{b}) := i \in V_{b} \sum ε_{i} Z_{S_{i}}, Λ := i \in V_{b} \sum ε_{i} 1 (S_{i} \neq = \emptyset) .$ The representer identities of Lemma 1 give $E_{Z} [g_{d} (Z_{b}) Z_{S}] = 1$ for every nonempty $S \subseteq V_{b}$ with $∣ S ∣ \leq \overset{ˉ}{β}_{d}$ and $E_{Z} [g_{d} (Z_{b})] = 0$ , so $E_{Z} [g_{d} (Z_{b}) Φ (Z_{b})] = Λ$ and expanding the square removes the centering term, $E_{Z} [(g_{d} (Z_{b}) Φ (Z_{b}) - Λ)^{2}] = E_{Z} [(g_{d} (Z_{b}) Φ (Z_{b}))^{2}] - Λ^{2} \leq E_{Z} [(g_{d} (Z_{b}) Φ (Z_{b}))^{2}] .$ Each of the $d$ terms of $Φ$ has modulus at most one, since $0 \leq Z_{S_{i}} \leq 1$ and $∣ ε_{i} ∣ = 1$ , so $∣Φ (Z_{b}) ∣ \leq d$ pointwise and therefore $E_{Z} [(g_{d} (Z_{b}) Φ (Z_{b}))^{2}] \leq d^{2} E_{Z} [g_{d} (Z_{b})^{2}] = d^{2} A_{d},$ the last equality being the exact block-score energy $E_{Z} [g_{d} (Z)^{2}] = A_{d}$ of Lemma 1. Maximizing over admissible choices gives the displayed upper bound.

Hence $Ψ_{b, t} (w) = d^{2} A_{d}$ on every block, and $c \in M_{t} (G) sup E_{Z} [(τ_{n}^{SNIPE} - τ_{n})^{2}] = \frac{B ^{2}}{n ^{2}} b = 1 \sum m Ψ_{b, t} (w) = \frac{B ^{2} m d ^{2} A _{d}}{n ^{2}} = \frac{B ^{2} A _{d}}{m} .$

If instead $B = 0$ , the only admissible schedule on any graph has zero coefficient mass, so $τ_{n} = 0$ , while the unprojected SNIPE risk bound above bounds every modelwise SNIPE risk by $B^{2} d A_{d} / n = 0$ ; as design mean squared errors are nonnegative, the fixed-graph coefficient-class risk again equals $0 = B^{2} A_{d} / m$ .

For all $B \geq 0$ the four risks now coincide. Restricting the supremum to a single graph can only decrease it, so the fixed-graph coefficient-class risk is at most the global coefficient-class risk, and the fixed-graph bounded-outcome risk is at most the global bounded-outcome risk. The carrier-preserving inclusion above embeds each coefficient-mass model on $G$ in the bounded-outcome class without changing the graph, the schedule, or the risk, so the fixed-graph coefficient-class risk is at most the fixed-graph bounded-outcome risk. The unprojected SNIPE risk bound above bounds both global risks by $B^{2} d A_{d} / n = B^{2} A_{d} / m$ , which is the value just computed for the fixed-graph coefficient-class risk. The chain of inequalities therefore closes, and $c \in M_{t} (G) sup E_{Z} [(τ_{n}^{SNIPE} - τ_{n})^{2}] = (G_{n}, c) \in M_{n, d, β}^{\infty} (B) G_{n} = G sup E_{Z} [(τ_{n}^{SNIPE} - τ_{n})^{2}] = M_{n, d, β} (B) sup E_{Z} [(τ_{n}^{SNIPE} - τ_{n})^{2}]$ $= M_{n, d, β}^{\infty} (B) sup E_{Z} [(τ_{n}^{SNIPE} - τ_{n})^{2}] = \frac{B ^{2} A _{d}}{m} = B^{2} \frac{d A _{d}}{n} .$

The model comparison, the unbiasedness identity, the worst-case bound $B^{2} d A_{d} / n$ for unclipped SNIPE, the clipped and minimax frontier bounds, the local energy comparison, the binomial scale, the exact complete-block risk, and the least-favourable block family together give the six assertions of the theorem. The constants $c_{β, p}$ and $C_{β, p}$ were fixed above independently of $n$ , $D$ , $d$ and $B$ .

∎

Proof of Theorem 4.

Fair-coin contrasts. Fix $n, d, β \in N$ and $B \in R$ with $n \geq 1$ , $1 \leq d \leq n$ , $β \geq 1$ , and $B \geq 0$ . The contrast, energy, and exposed-order identities involve only $d$ and $β$ . The contrast coefficient is $Δ_{r} (p) = (1 - p)^{r} - (- p)^{r}$ , so at $p = 1/2$ every $r \geq 1$ satisfies $Δ_{r} (1/2) = (\frac{1}{2})^{r} - (- \frac{1}{2})^{r} = 2^{- r} - (- 1)^{r} 2^{- r} = 2^{- r} {1 - (- 1)^{r}} .$ Since $1 - (- 1)^{r}$ equals $2$ for odd $r$ and $0$ for even $r$ , $Δ_{r} (1/2) = {2^{1 - r}, 0, r odd, r even .$
Fair-coin energy. By Definition 10 at $p = 1/2$ , the design variance factor is $[p (1 - p)]^{r} = (1/4)^{r}$ , so $A_{d} = r = 1 \sum \overset{ˉ}{β}_{d} (r d) \frac{Δ _{r} ( 1/2 ) ^{2}}{[ \frac{1}{2} ( 1 - \frac{1}{2} ) ] ^{r}} .$ For every $r$ with $1 \leq r \leq \overset{ˉ}{β}_{d}$ , the fair-coin contrast formula gives $(r d) \frac{Δ _{r} ( 1/2 ) ^{2}}{[ \frac{1}{2} ( 1 - \frac{1}{2} ) ] ^{r}} = {4 (r d), 0, r odd, r even,$ because for odd $r$ the numerator is $(r d) (2^{1 - r})^{2} = 4 (r d) 4^{- r}$ while the denominator is $(1/4)^{r} = 4^{- r}$ , and for even $r$ the numerator vanishes. Summing the surviving terms, $A_{d} = 4 1 \leq r \leq \overset{ˉ}{β}_{d} r odd \sum (r d) .$
Fair-coin exposed order. The index set in Definition 1 satisfies ${r \in {1, \dots, \overset{ˉ}{β}_{d}} : Δ_{r} (1/2) \neq = 0} = {r \in {1, \dots, \overset{ˉ}{β}_{d}} : r odd},$ since $Δ_{r} (1/2) = 2^{1 - r} \neq = 0$ for odd $r$ and $Δ_{r} (1/2) = 0$ for even $r$ . Hence $k_{⋆} (d, β, 1/2)$ is the maximum of the odd integers in ${1, \dots, \overset{ˉ}{β}_{d}}$ when that set is nonempty, and equals $0$ otherwise, which is the value assigned by the zero-degree convention in Definition 1.
Choice of constants. Apply Theorem 2 with interaction order $β = 1$ , which satisfies $β \geq 1$ , and treatment probability $p = 1/2$ , which satisfies $0 < p < 1$ . It supplies constants $c_{1, 1/2}$ and $C_{1, 1/2}$ with $0 < c_{1, 1/2} \leq C_{1, 1/2} .$ Set $c_{lower} := c_{1, 1/2}, c_{upper} := 4 C_{1, 1/2} .$ Then $c_{lower} > 0$ , and since $0 < C_{1, 1/2}$ gives $C_{1, 1/2} \leq 4 C_{1, 1/2}$ , $0 < c_{lower} \leq C_{1, 1/2} \leq 4 C_{1, 1/2} = c_{upper} .$
Design. Fix $n, d \in N$ and $B \in R$ with $n \geq 1$ , $1 \leq d \leq n$ , and $B \geq 0$ , and take for $P_{Z}$ the product Bernoulli law on $V_{n}$ with common treatment probability $1/2$ , that is, $Pr (Z_{j} = 1) = \frac{1}{2} for every j \in V_{n},$ with ${Z_{j} : j \in V_{n}}$ mutually independent. Since $0 < 1/2 < 1$ , this design satisfies Assumption 1 at $p = 1/2$ , so the conclusions of Theorem 2 are available at $(n, d, B)$ with $β = 1$ and $p = 1/2$ .
Degree-one energy. With $β = 1$ and $d \geq 1$ the exposed order of Definition 1 is $\overset{ˉ}{β}_{d} = min {1, d} = 1,$ so the odd integers in ${1, \dots, \overset{ˉ}{β}_{d}}$ are the single index $r = 1$ . The fair-coin energy formula gives $A_{d} = 4 (1 d) = 4 d .$
Reduction of the rate argument. Put $x := \frac{d ^{2}}{n} .$ Since $n \geq 1$ , we have $x \geq 0$ , and the identity $A_{d} = 4 d$ gives $\frac{d A _{d}}{n} = \frac{d ( 4 d )}{n} = 4 x .$

Two elementary comparisons of truncated arguments are used below. First, $x \leq 4 x$ because $x \geq 0$ , and truncation at $1$ is monotone, so $min {1, x} \leq min {1, 4 x} .$

Second, $min {1, 4 x} \leq 4 min {1, x} .$ Indeed, if $x \leq 1$ then $min {1, x} = x$ , and either $4 x \leq 1$ , in which case $min {1, 4 x} = 4 x = 4 min {1, x}$ , or $4 x > 1$ , in which case $min {1, 4 x} = 1 \leq 4 x = 4 min {1, x}$ ; while if $x > 1$ then $min {1, x} = 1$ and $min {1, 4 x} \leq 1 \leq 4 = 4 min {1, x}$ .
Lower bound. Because $c_{lower} > 0$ and $B \geq 0$ , $0 \leq c_{lower} B^{2},$ so multiplying $min {1, x} \leq min {1, 4 x}$ by this factor preserves the inequality. The lower frontier bound of Theorem 2 at $β = 1$ , $p = 1/2$ reads $c_{lower} B^{2} min {1, \frac{d A _{d}}{n}} \leq R_{n, ℓ_{1}}^{⋆} (d, 1, 1/2, B),$ and the same result identifies $R_{n, ℓ_{1}}^{⋆} (d, 1, 1/2, B) = R_{n}^{⋆} (d, 1, 1/2, B) .$ Using $x = d^{2} / n$ and $d A_{d} / n = 4 x$ , $c_{lower} B^{2} min {1, \frac{d ^{2}}{n}} = c_{lower} B^{2} min {1, x} \leq c_{lower} B^{2} min {1, 4 x} = c_{lower} B^{2} min {1, \frac{d A _{d}}{n}} \leq R_{n}^{⋆} (d, 1, 1/2, B) .$
Class comparison. The carrier inclusion recorded in Theorem 2 supplies $R_{n, ℓ_{1}}^{⋆} (d, 1, 1/2, B) \leq R_{n, \infty}^{⋆} (d, 1, 1/2, B) .$
Upper bound. The clipped bounded-outcome estimator $τ_{n, \infty}^{up}$ of Definition 14 is one admissible competitor in the bounded-outcome minimax problem, and Theorem 2 bounds its worst-case risk, giving $R_{n, \infty}^{⋆} (d, 1, 1/2, B) \leq (G, c) \in M_{n, d, 1}^{\infty} (B) sup E_{Z} [(τ_{n, \infty}^{up} - τ_{n} (G, c))^{2}] \leq C_{1, 1/2} B^{2} min {1, \frac{d A _{d}}{n}} .$

Since $C_{1, 1/2} > 0$ and $B \geq 0$ , the factor $C_{1, 1/2} B^{2}$ is nonnegative, so multiplying $min {1, 4 x} \leq 4 min {1, x}$ by it preserves the inequality. With $d A_{d} / n = 4 x$ , this yields $C_{1, 1/2} B^{2} min {1, \frac{d A _{d}}{n}} = C_{1, 1/2} B^{2} min {1, 4 x} \leq 4 C_{1, 1/2} B^{2} min {1, x} = c_{upper} B^{2} min {1, \frac{d ^{2}}{n}},$ and therefore $R_{n, \infty}^{⋆} (d, 1, 1/2, B) \leq c_{upper} B^{2} min {1, \frac{d ^{2}}{n}} .$ Combining the lower bound, the equality of the two coefficient-class risk notations, the class comparison, and the upper bound gives the asserted chain $c_{lower} B^{2} min {1, \frac{d ^{2}}{n}} \leq R_{n}^{⋆} (d, 1, 1/2, B) = R_{n, ℓ_{1}}^{⋆} (d, 1, 1/2, B) \leq R_{n, \infty}^{⋆} (d, 1, 1/2, B) \leq c_{upper} B^{2} min {1, \frac{d ^{2}}{n}} .$
Exact complete-block identities. Assume in addition that $d ∣ n$ and set $m := n / d$ , the number of complete $d$ -blocks. The exact-block-risk clause of Theorem 2 gives, for the unprojected SNIPE estimator of Definition 8, $(G, c) \in M_{n, d, 1} (B) sup E_{Z} [(τ_{n}^{SNIPE} - τ_{n} (G, c))^{2}] = \frac{B ^{2} A _{d}}{m} = (G, c) \in M_{n, d, 1}^{\infty} (B) sup E_{Z} [(τ_{n}^{SNIPE} - τ_{n} (G, c))^{2}],$ together with the identity $\frac{B ^{2} A _{d}}{m} = \frac{B ^{2} d A _{d}}{n} .$ Substituting $A_{d} = 4 d$ , $\frac{B ^{2} d A _{d}}{n} = \frac{B ^{2} d ( 4 d )}{n} = \frac{4 B ^{2} d ^{2}}{n},$ which is the asserted common value of both worst-case risks.

∎

References

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Formal layer — Lean correspondence

Every object below is formalized in Lean 4. Click a verified row to view its statement; the rest link to the Lean source.

Setups (5)

Setup S-1 — finite-population design-based randomization world: a finite unit type V with a known directed interference graph (self-loops allowed) and the unit-level product Bernoulli assignment law on Ω = V → Bool, carried by FiniteDesign with the finite-sum E/Var/Cov algebra. — This is the single highest-leverage reuse decision: the paper's only randomness is the unit-level product Bernoulli assignment, which is exactly bernoulliDesign p hp0 hp1 : FiniteDesign (V → Bool) (constant p i = p). Cross-coordinate independence — the engine of every orthogonality, block-independence and variance computation in the paper — is then STRUCTURAL (prodDesign), not an assumption: FiniteDesign.E_prod_apply, E_prod_apply₂, Cov_prod_apply_of_ne, Var_prod_apply, Var_prod_linear_comb (ProductVariance) and the block generalizations E_prod_block_mul, Cov_prod_disjoint_zero (ProductBlock) give the disjoint-block factorization the complete-directed-block arguments need. \mathbb E_Z is FiniteDesign.E of this design; P_Z is the design itself. The graph is a plain V → V → Prop with DecidablePred, neighborhoods N i : Finset V; Causalean's DAG is NOT usable (self-loops are required and counted in both degrees), recorded as bypass-justified.
Setup S-2 — fixed-potential-outcome (noiseless) low-order polynomial interference world: a coefficient schedule c : (i : V) → Finset V → ℝ supported on subsets of N i of size ≤ β, the induced potential outcome Y i z = ∑_{S ⊆ N i} c i S ∏_{j ∈ S} z j, the observed outcome Y i Z, the finite-population TTE contrast, and the two model classes (ℓ¹ coefficient-mass and uniformly bounded-outcome). — Search trace: read ../doc/API.md §18 Experimentation/ (grepped '^## ' index, then the DesignBased/ExposureMappingInterference/TwoStageInterference/UnknownInterference tables in full); scanned Causalean/Experimentation/DesignBased/{PotentialOutcome,Exposure,Estimators,HT/*}.lean for 'potential outcome', 'exposure mapping', 'polynomial interference', 'low order', 'coefficient schedule'; scanned Causalean/Experimentation/UnknownInterference/Basic.lean for the whole-assignment PO type. Result: not found — Causalean's design-based PO layers are exposure-mapping-valued (ProperlySpecified : ∀ i z, yr i z = y i (expo i z), Yobs y f θ i z = y i (expo i z), an ABSTRACT finite exposure θ) or fully unstructured (UnknownInterference.y i z); neither carries the RAW-MONOMIAL POLYNOMIAL structure Y i z = ∑_{S ⊆ N i} c i S ∏_{j∈S} z j that every result here is about (the coefficient schedule c IS the model parameter, the interaction order β is the model complexity, and the ℓ¹ coefficient-mass envelope is a class restriction on c). Bypass-justified: the exposure layer is at the wrong abstraction (it hides the coefficients we must vary). The bridge fact that this PO factors through z restricted to N i is recorded and proved locally (one lemma), so DesignBased.PotentialOutcome.ProperlySpecified remains available as an instance if a later artifact wants it.
Setup S-3 — design-based decision-theoretic world: GRAPH-AWARE estimators measurable in the full observed data (G, Z, (Y_i^obs)), their design mean squared error at a fixed model, the worst case over a model class, and the minimax (inf over estimators) risk. — REVISED (round 2, FLAG-world-arity): an estimator is GRAPH-AWARE — Estimator V := (V → V → Bool) → (V → Bool) → (V → ℝ) → ℝ (known graph, assignment, observed outcome vector), matching the core's \widehat\tau_n symbol (refs: [G_n, Z, Y_i^{obs}]) and def:minimax-risk's 'measurable in (G_n, Z, {Y_i^obs})'. The graph is NOT fixed inside the risk functional: riskAt COMPOSES the single global estimator with the model's own graph, riskAt p M est = (bernoulliDesign …).mse (fun z => est (edgeFn M) z (obsOutcome M z)) (tte M), where edgeFn M := fun j i => decide (M.edge j i) uses the class structure's existing decEdge field. The inf–sup is therefore the core's: ONE estimator faces every (G_n, c) in the class, and snipeEstimator β p : Estimator V (which reads its neighborhoods off its graph argument) is one admissible global competitor, so the middle term of thm:bounded-outcome-degree-frontier is literally worstRisk … (snipeClipped B β p). The per-model risk is exactly D.mse est μ = D.E (fun z => (est z - μ)^2) with D = bernoulliDesign, μ = τ_n; mse_eq_var_add_bias_sq, mse_eq_var_of_unbiased, var_le_mse, Unbiased are all reused verbatim (the unclipped-SNIPE branch is mse = Var after unbiasedness; the clipping branch uses mse_nonneg + a pointwise projection bound). Measurability is vacuous on the finite assignment space and is imposed only on the outcome argument, and only in the converse. worstRisk/minimaxRisk are sSup/sInf over the model class / estimator class (build-inline, see def:minimax-risk). Surveyed and bypass-justified: Causalean.Experimentation.DesignBased.worstRisk / IsMinimaxOn (Optimality/Minimax.lean) minimize over DESIGNS at a fixed estimator, not over ESTIMATORS at a fixed design — the wrong quantifier order for this paper; ProbabilityTheory.minimaxRisk (Mathlib Decision/Risk) is kernel/loss-shaped over a general Kernel-valued experiment and would force a measure-theoretic re-encoding of the finite design.
Setup S-4 — least-favourable block world: complete directed d-blocks with loops, the block Bernoulli score/energy/representer, the continuous cosine-squared block baseline prior, and the two prior-predictive laws on ({0,1}^V) × ℝ^m (counting ⊗ Lebesgue) used for the Le Cam converse. — REVISED (round 3, FLAG-finite-mixture) on top of round 2 (FLAG-reuse-precondition). GENERIC Le Cam reuse is RETAINED and unchanged: Causalean.Stat.tvDist, one_sub_tvDist_le_test, one_sub_tvDist_le_error_sum, real_two_point_lower_bound, two_point_lower_bound_of_tvDist_le (TotalVariation.lean / LeCam.lean / MinimaxRisk.lean) — none of these has an absolute-continuity premise, and none of them requires a finite hypothesis index. DROPPED in round 3: Causalean.Stat.mixtureReal_le (Minimax/Mixture.lean), which the round-2 plan cited for the PRIOR REALIZABILITY step. Verified against the source (Grep 'mixtureReal' ../Causalean/Stat/Minimax/ → Mixture.lean:53 theorem mixtureReal_le (w : ι → ℝ≥0∞) (hw : ∑ i, w i = 1) … with [Fintype ι]): it is a .real-MASS domination bound for a FINITE Fintype-indexed convex combination of measures, whereas the baseline prior here is the CONTINUOUS product ⊗_{b ≤ m} f_s(u)du over U : Fin m → ℝ (a compactly supported cosine-squared density, not a finite weight vector), and the quantity to be dominated is an integrated RISK, not a set mass. It cannot discharge the realizability step and the whole Causalean.Stat.Minimax.Mixture module is therefore removed from this S-block's modules. REPLACEMENT (local, small, built in Helpers/LeastFavourable.lean — see def:block-family for the same text): (R0) prior_integrated_risk_le_worstRisk — for an admissible estimator est (the AdmissibleEstimator predicate of def:minimax-risk already carries BddAbove {riskAt p M est | M} and outcome-measurability), ∫ riskAt p (blockSchedule σ U) est ∂(blockBaselinePrior s m) ≤ worstRisk p est. Proof is the ordinary continuous-prior ∫ ≤ sup: the admissibility lemma of def:block-family puts blockSchedule σ U in ModelClass V d β B for EVERY U in the support, so riskAt p (blockSchedule σ U) est ≤ worstRisk p est pointwise by the reused API lemma le_worstRisk; then MeasureTheory.integral_mono_ae against the constant plus integral_const and IsProbabilityMeasure (∫ f_s = 1 is the closed-form intervalIntegral computation already needed for (C2), lifted to the product prior by MeasureTheory.Measure.pi + MeasureTheory.isProbabilityMeasure_pi). Integrability is NOT assumed away: 0 ≤ riskAt ≤ worstRisk (from mse_nonneg and le_worstRisk) gives a two-sided bound, and measurability in U holds because riskAt is a FINITE sum over the finite assignment space of terms polynomial in U composed with the outcome-measurable est, so Integrable follows from Measure.integrable_of_bounded-style reasoning on a probability measure. Search trace for a reusable form: lean_local_search 'mixtureReal_le' → EMPTY in the index; lean_local_search 'bayesRisk' → only ProbabilityTheory.bayesRisk* (Mathlib Decision/Risk), which is Kernel-shaped over an abstract experiment and would force a measure-theoretic re-encoding of the finite design (same rejection as in def:minimax-risk); Grep 'integral_mono|Bayes' ../Causalean/Stat/Minimax/ → no integrated-risk-vs-worst-case-risk lemma. Not found ⇒ local, build-inline (three lines of Mathlib integral_mono_ae). DROPPED in round 2, and why: the round-1 plan routed the converse through Causalean.Stat.sq_bhattacharyya_le_two_mul_integral_min + Causalean.Stat.integral_min_le_one_sub_tvDist (BretagnolleHuber.lean), but BOTH are stated for P₊ ≪ P₋ (they are RN-derivative statements, ∫ √(dP₊/dP₋) dP₋). The two prior-predictive laws here are mixtures of COMPACTLY SUPPORTED translated cosine-square densities whose supports are non-nested, so P₊ ≪ P₋ FAILS and those lemmas are unusable; Pinsker.lean / ChiSquared.lean are excluded for the same reason (KL and χ² are infinite here), as the core's technical_internal_limitation records. REPLACEMENT (built locally in Helpers/HellingerAffinity.lean, all statements phrased against a COMMON DOMINATING measure λ = count(V → Bool) ⊗ volume^m, never against a mutual RN derivative): (C0) tvDist_le_sqrt_two_mul_one_sub_affinity — for P = λ.withDensity f, Q = λ.withDensity g with f, g ≥ 0 integrable densities of total mass one, tvDist P Q ≤ √(2·(1 − ∫ √(f·g) ∂λ)). Proof, both halves elementary and premise-free: |P.real A − Q.real A| = |∫_A (f − g) ∂λ| ≤ ½∫|f − g| ∂λ by REUSING Causalean.Stat.abs_setIntegral_le_half_integral_abs_of_integral_eq_zero (Scheffe.lean — verified generic: it takes only Integrable f ν and ∫ f = 0, NO ≪), pushed through MeasureTheory.withDensity_apply and sSup to bound tvDist; then the Cauchy–Schwarz factorization ∫|f−g| = ∫|√f−√g|·(√f+√g) ≤ √(∫(√f−√g)²)·√(∫(√f+√g)²) ≤ 2√(∫(√f−√g)²) with ∫(√f−√g)² = 2(1 − ∫√(fg)). This is exactly the paper's own step (‖P₊−P₋‖_TV ≤ H, 'Cauchy–Schwarz applied to the densities'), so no fidelity is lost: H² ≤ 1/4 ⇒ tvDist ≤ 1/2. (C1) affinity TENSORIZATION for the explicit product density: ∫ √(∏_b f_b · ∏_b g_b) ∂(λ₀ ⊗ volume^m) = ∏_b ∫ √(f_b g_b), by Real.sqrt_mul on the pointwise product followed by Fubini on MeasureTheory.Measure.pi (integral_finset_prod-style), with the shared DISCRETE assignment coordinate handled by summing over z first (blocks are conditionally independent given Z, and the law of Z is itself a product over blocks); combined with 1 − ∏ a_b ≤ ∑ (1 − a_b) for a_b ∈ [0,1]. (C2) unchanged: the cosine-squared translation affinity bound 1 − ∫ √(f_s(u−a) f_s(u−b)) du ≤ C(a−b)²/s², a closed-form intervalIntegral computation (product-to-sum on cos·cos over the overlap). NOTE the direction of travel: nothing is now needed ABOUT a Hellinger distance object — only the affinity number ∫√(fg), so no general Hellinger theory is built and no ≪ ever appears.
Setup S-5 — block-local Riesz world: the polynomial subspace P_d ⊆ L²(P_Z) spanned by raw monomials of degree ≤ β̄_d on one d-block, the all-one-versus-all-zero contrast functional L_d, and the fixed-complete-block sequence (n_t, d_t, G_t) carrying the block-local linear design-unbiased estimator class. — Search trace: lean_local_search for 'Riesz', 'representer', 'L2Space', 'inner product design', 'polynomial space'; read ../doc/API.md §18 (DesignBased substrate tables) and §10g Mathlib/SemiInnerProjection.lean, §10f Mathlib/FWL.lean. Result: not found for a design-based L²(P_Z) inner-product layer. Bypass-justified and DELIBERATE: L²(P_Z) here is the FINITE-dimensional space of functions (V_b → Bool) → ℝ with the design inner product ⟪f,g⟫ = D.E (f·g), so the whole Riesz story is Finset algebra over FiniteDesign.E/Cov (no Mathlib L2Space, no MeasureTheory needed); P_d is a Submodule ℝ ((V_b → Bool) → ℝ) spanned by raw monomials, and orthogonal decomposition w = g_d + u, u ⊥ P_d, is finite-dimensional projection. The programs PerturbProg/WeightProg are sInf over the feasible sets, with their minimizers identified in lem:block-energy-representer. The only genuinely non-elementary piece is the non-necessity witness, which needs dim P_{d_t} = ∑_{r ≤ β} C(d_t,r) < 2^{d_t} for large d_t — the centered-monomial family is orthogonal (hence independent) under the product Bernoulli design, so this is a Finset.card computation plus Module.finrank of the full function space; Mathlib.LinearAlgebra.Dimension + Submodule.exists_mem_ne_zero_of_finrank_lt style reasoning covers it.

Definitions (20)

Definition P-1 verified CausalSmith.Experimentation.SnipeDegreeFrontier.kStar Set $\bar\beta_d:=\min\{\beta,d\}$ for every $d\ge0$ ; in particular $\bar\beta_0=0$ . If $\bar\beta_d\ge1$ , set $k_\star(d,\beta,p):=\max\{r\in\{1,\ldots,\bar\beta_d\}:\Delta_r(p)\ne0\}$ ; since $\Delta_1(p)=1$ , this index set contains $r=1$ , so its maximum is well defined. When $\bar\beta_d=0$ , equivalently $d=0$ or $\beta=0$ , set $k_\star(d,\beta,p):=0$ as a totality convention only, never as an attained interaction order. Thus the maximum case applies exactly when both $d\ge1$ and $\beta\ge1$ ; every statement below fixes $\beta\ge1$ .
Definition P-2 verified CausalSmith.Experimentation.SnipeDegreeFrontier.ZeroDegreeConventions Set $\binom{0}{0}:=1$ , $g_0:=0$ , and $A_0:=0$ . Interpret sums over nonempty subsets of $\{1,\ldots,0\}$ as zero; its assignment space has one empty vector and the empty product equals one. Set $h_0:=0$ only as an empty-object convention, and never interpret it as $g_0/A_0$ .
Definition P-3 verified CausalSmith.Experimentation.SnipeDegreeFrontier.GraphClass $\mathcal G_{n,d}:=\{G_n\in2^{V_n\times V_n}:G_n\text{ satisfies }\mathrm{ass:bounded-degree}\}$ .
Definition P-4 verified CausalSmith.Experimentation.SnipeDegreeFrontier.CoeffClass $\mathcal C_{n,d,\beta}(B):=\{c:c\text{ satisfies }\mathrm{ass:low-order}\text{ and }\mathrm{ass:bounded-coefficient-mass}\}$ .
Definition P-3b verified CausalSmith.Experimentation.SnipeDegreeFrontier.potentialOutcome $Y_i(z):=\sum_{S\subseteq N_i}c_{i,S}\prod_{j\in S}z_j$ is the potential outcome of unit $i$ at assignment $z$ , and $Y_i^{\mathrm{obs}}:=Y_i(Z)$ is the observed outcome at the realized assignment.
Definition P-5 verified CausalSmith.Experimentation.SnipeDegreeFrontier.ModelClass $\mathcal M_{n,d,\beta}(B):=\{(G_n,c):G_n\in\mathcal G_{n,d},\ c\in\mathcal C_{n,d,\beta}(B)\}$ , where $\mathcal G_{n,d}$ and $\mathcal C_{n,d,\beta}(B)$ are the classes in ${\rm def:graph-class}$ and ${\rm def:coefficient-class}$ , respectively.
Definition P-6 verified CausalSmith.Experimentation.SnipeDegreeFrontier.snipeScore $g_{i,\beta,p}(z):=\sum_{r=1}^{\bar\beta_d}\frac{\Delta_r(p)}{[p(1-p)]^r}\sum_{\substack{S\subseteq N_i\\|S|=r}}\prod_{j\in S}(z_j-p)$ .
Definition P-7 verified CausalSmith.Experimentation.SnipeDegreeFrontier.snipeEstimatorBundle $\widehat\tau_n^{\mathrm{SNIPE}}:=n^{-1}\sum_{i\in V_n}Y_i^{\mathrm{obs}}g_{i,\beta,p}(Z)$ and $\widehat\tau_n^{\mathrm{up}}:=\Pi_{[-B,B]}(\widehat\tau_n^{\mathrm{SNIPE}})$ , where $\Pi_{[-B,B]}$ is Euclidean projection.
Definition P-8 verified CausalSmith.Experimentation.SnipeDegreeFrontier.minimaxRisk $R_n^\star(d,\beta,p,B):=\inf_{\widehat\tau_n}\sup_{(G_n,c)\in\mathcal M_{n,d,\beta}(B)}\mathbb E_Z[(\widehat\tau_n-\tau_n)^2]$ , where $\widehat\tau_n$ is measurable in $(G_n,Z,\{Y_i^{\mathrm{obs}}:i\in V_n\})$ .
Definition P-9 verified CausalSmith.Experimentation.SnipeDegreeFrontier.blockScoreEnergyBundle For $d\ge1$ and a $d$ -coordinate Bernoulli vector, set $g_d(z):=\sum_{r=1}^{\bar\beta_d}\frac{\Delta_r(p)}{[p(1-p)]^r}\sum_{\substack{S\subseteq\{1,\ldots,d\}\\|S|=r}}\prod_{j\in S}(z_j-p)$ , $A_d:=\sum_{r=1}^{\bar\beta_d}\binom{d}{r}\Delta_r(p)^2/[p(1-p)]^r$ , and $h_d:=g_d/A_d=\sum_{T\subseteq\{1,\ldots,d\}}h_{d,T}\prod_{j\in T}z_j$ , the last equality being the raw-monomial expansion.
Definition P-10 verified CausalSmith.Experimentation.SnipeDegreeFrontier.blockFamilyBundle For $B>0$ and $1\le d\le n$ , set $m:=\lfloor n/d\rfloor$ , $q:=md$ , and $\rho:=q/n$ . Partition $q$ units into $m$ blocks $V_b$ of size $d$ ; include every within-block directed arrow, including loops, and leave the remaining $n-q$ units isolated. Set $s:=B/2$ , $f_s(u):=s^{-1}\cos^2(\pi u/(2s))\mathbf1\{|u|\le s\}$ , and draw $U_1,\ldots,U_m$ independently from $f_s$ . Let $H_{\beta,p}:=\sup_{d\ge1}\sum_T|h_{d,T}|$ and choose $\delta:=B\min\{(2H_{\beta,p})^{-1},(4\pi)^{-1}\}\min\{1,\sqrt{A_d/m}\}$ . Under $\Pi_\sigma$ , active $i\in V_b$ has $c_{i,T}:=U_b\mathbf1\{T=\varnothing\}+\sigma\delta h_{d,T}$ for $|T|\le k_\star$ , and all other coefficients, including those of inactive units, are zero. The $U_b$ randomize fixed potential-outcome schedules only; they are not observation noise.
Definition P-11 verified CausalSmith.Experimentation.SnipeDegreeFrontier.perturbProg For $d\ge1$ , $\mathsf{PerturbProg}_{d,\beta,p}:=\min_{h\in\mathcal P_d}\mathbb E_Z[h(Z)^2]$ subject to $L_d(h)=1$ .
Definition P-12 verified CausalSmith.Experimentation.SnipeDegreeFrontier.weightProg For $d\ge1$ , $\mathsf{WeightProg}_{d,\beta,p}:=\min_{w\in L^2(P_Z)}\mathbb E_Z[w(Z)^2]$ subject to $\mathbb E_Z[w(Z)]=0$ and $\mathbb E_Z[w(Z)\prod_{j\in S}Z_j]=1$ for every nonempty $S\subseteq\{1,\ldots,d\}$ with $|S|\le\bar\beta_d$ .
Definition P-13 verified CausalSmith.Experimentation.SnipeDegreeFrontier.LocLinSchedClass On the fixed graph $G_t$ , set $\mathcal M_t(G_t):=\{c:c_{i,S}=0\text{ unless }S\subseteq N_i^{G_t}\text{ and }|S|\le\bar\beta_{d_t},\ \sum_{S\subseteq N_i^{G_t}}|c_{i,S}|\le B\text{ for every }i\in V_{n_t}\}$ .
Definition P-14 verified CausalSmith.Experimentation.SnipeDegreeFrontier.locLinEstimatorClass Set $\mathcal E_t^{\mathrm{loc,lin}}:=\{\widehat\tau_{w,t}=n_t^{-1}\sum_{i\in V_{n_t}}w_{i,t}(Z_{b(i)})Y_i(Z):w_{i,t}\in L^2(P_Z)\text{ is measurable with respect to }Z_{b(i)},\ \mathbb E_Z[w_{i,t}]=0,\text{ and }\mathbb E_Z[w_{i,t}\prod_{j\in S}Z_j]=1\text{ for every nonempty }S\subseteq N_i^{G_t}\text{ with }|S|\le\bar\beta_{d_t}\}$ .
Definition P-15 verified CausalSmith.Experimentation.SnipeDegreeFrontier.locLinMinimaxRisk Define $R_t^{\mathrm{loc,lin}}:=\inf_{\widehat\tau_{w,t}\in\mathcal E_t^{\mathrm{loc,lin}}}\sup_{c\in\mathcal M_t(G_t)}\mathbb E_Z[(\widehat\tau_{w,t}-\tau_{n_t})^2]$ ; the graph is fixed and is not varied in this supremum.
Definition P-16 verified CausalSmith.Experimentation.SnipeDegreeFrontier.BddOutcomeCoeffClass $\mathcal C_{n,d,\beta}^{\infty}(B):=\{c:c_{i,S}=0\text{ whenever }|S|>\beta,\ \sup_{z\in\{0,1\}^{V_n}}|Y_i(z)|\le B\text{ for every }i\in V_n\}$ .
Definition P-17 verified CausalSmith.Experimentation.SnipeDegreeFrontier.BddOutcomeModelClass $\mathcal M_{n,d,\beta}^{\infty}(B):=\{(G_n,c):G_n\in\mathcal G_{n,d},\ c\in\mathcal C_{n,d,\beta}^{\infty}(B)\}$ , where the two component classes are those in ${\rm def:graph-class}$ and ${\rm def:bounded-outcome-coefficient-class}$ .
Definition P-18 verified CausalSmith.Experimentation.SnipeDegreeFrontier.twoClassMinimaxRisks $R_{n,\ell_1}^{\star}(d,\beta,p,B):=R_n^\star(d,\beta,p,B)$ and $R_{n,\infty}^{\star}(d,\beta,p,B):=\inf_{\widehat\tau_n}\sup_{(G_n,c)\in\mathcal M_{n,d,\beta}^{\infty}(B)}\mathbb E_Z[(\widehat\tau_n-\tau_n)^2]$ , where $\widehat\tau_n$ is measurable in $(G_n,Z,\{Y_i^{\mathrm{obs}}:i\in V_n\})$ .
Definition P-19 verified CausalSmith.Experimentation.SnipeDegreeFrontier.snipeClippedBdd $\widehat\tau_{n,\infty}^{\mathrm{up}}:=\Pi_{[-2B,2B]}(\widehat\tau_n^{\mathrm{SNIPE}})$ , where $\Pi_{[-2B,2B]}$ is Euclidean projection.

Assumptions (4)

Assumption A-1 verified CausalSmith.Experimentation.SnipeDegreeFrontier.IsProductBernoulli $Z_j$ are mutually independent and $\Pr(Z_j=1)=p$ for every $j\in V_n$ .
Assumption A-2 verified CausalSmith.Experimentation.SnipeDegreeFrontier.BoundedDegree Self-loops are allowed and count in both degrees, with $\max_{i\in V_n}|N_i|\le d$ and $\max_{j\in V_n}|\{i\in V_n:j\in N_i\}|\le d$ .
Assumption A-3 verified CausalSmith.Experimentation.SnipeDegreeFrontier.LowOrder $c_{i,S}=0$ whenever $|S|>\beta$ .
Assumption A-4 verified CausalSmith.Experimentation.SnipeDegreeFrontier.BoundedCoeffMass $\sum_{S\subseteq N_i}|c_{i,S}|\le B$ for every $i\in V_n$ .

Lemmas (4)

Lemma L-1 verified CausalSmith.Experimentation.SnipeDegreeFrontier.blockEnergy_representer For fixed $\beta\ge1$ , $p\in(0,1)$ , and $d\ge1$ , $A_d\asymp_{\beta,p}\binom{d}{k_\star}$ , $\mathbb E_Z[g_d(Z)f(Z)]=L_d(f)$ for every $f\in\mathcal P_d$ , $h_d(\mathbf1_d)-h_d(\mathbf0_d)=1$ , and $\mathbb E_Z[h_d(Z)^2]=A_d^{-1}$ . The unique minimizers of ${\rm def:perturbation-program}$ and ${\rm def:weight-program}$ are respectively $h_d=g_d/A_d$ , with value $A_d^{-1}$ , and $w=g_d$ , with value $A_d$ . Moreover $H_{\beta,p}=\sup_{d\ge1}\sum_T|h_{d,T}|<\infty$ .
Lemma L-2 verified CausalSmith.Experimentation.SnipeDegreeFrontier.overlap_count_le For every $r\ge1$ , including $d=0$ , and every $i\in V_n$ , $\sum_{l\in V_n}\binom{|N_i\cap N_l|}{r}=\sum_{\substack{S\subseteq N_i\\|S|=r}}\#\{l\in V_n:S\subseteq N_l\}\le d\binom{|N_i|}{r}\le d\binom{d}{r}.$
Lemma lem:block-prior-dominating-measure verified CausalSmith.Experimentation.SnipeDegreeFrontier.blockPriorDensity_integral_one Fix a population size $n\ge1$ , a block size $d$ with $1\le d\le n$ , an interaction order $\beta$ , a radius $B>0$ , a treatment probability $p\in(0,1)$ , and a sign $\sigma\in\{-1,+1\}$ . Let $m=\lfloor n/d\rfloor$ , let $s=B/2$ , let $f_s$ , $\delta$ , and $V_1,\ldots,V_m$ be the cosine-squared baseline density, the perturbation amplitude, and the complete $d$ -blocks of def:block-family, and let $h_d$ be the normalized block representer of def:block-score-energy. Let $\mu_{n,d}$ denote the block dominating measure on $\{0,1\}^{V_n}\times\mathbb R^m$ , that is, counting measure on the finite assignment set times $m$ -fold Lebesgue measure, and let $\pi_\sigma$ denote the sign- $\sigma$ block prior density $\pi_\sigma(z,u)=\Bigl(\prod_{j\in V_n}p^{z_j}(1-p)^{1-z_j}\Bigr)\prod_{b=1}^{m}f_s\bigl(u_b-\sigma\delta\,h_d(z_{V_b})\bigr),$ where $z_{V_b}$ is the restriction of the assignment $z$ to the $b$ -th block. Then $\int\pi_\sigma\,d\mu_{n,d}=1$ ; that is, $\pi_\sigma$ is a probability density with respect to $\mu_{n,d}$ , which is therefore a common dominating measure for the two signed block priors.
Lemma lem:block-prior-hellinger-bound verified CausalSmith.Experimentation.SnipeDegreeFrontier.blockPrior_hellinger_le Fix a population size $n\ge1$ , a block size $d$ with $1\le d\le n$ , an interaction order $\beta\ge1$ , a radius $B>0$ , and a treatment probability $p\in(0,1)$ . Let $\mu_{n,d}$ , $\pi_{+1}$ , and $\pi_{-1}$ be the block dominating measure and the two signed block prior densities of lem:block-prior-dominating-measure, let $m=\lfloor n/d\rfloor$ be the number of complete $d$ -blocks, and let $\delta$ be the perturbation amplitude of def:block-family. Then the unhalved squared Hellinger distance between the two densities against their common dominating measure, $H^2(\pi_{+1},\pi_{-1})=\int\bigl(\sqrt{\pi_{+1}}-\sqrt{\pi_{-1}}\bigr)^2\,d\mu_{n,d},$ satisfies $H^2(\pi_{+1},\pi_{-1})\le\frac{4\pi^2m\delta^2}{B^2A_d},$ where $A_d$ is the complete-block score energy of def:block-score-energy.

Theorems (4)

Theorem T-1 verified CausalSmith.Experimentation.SnipeDegreeFrontier.degree_frontier Fix $\beta\ge1$ , $p\in(0,1)$ , $B\ge0$ , integers $n\ge1$ , and $0\le d\le n$ . There are $0<c_{\beta,p}\le C_{\beta,p}<\infty$ such that $c_{\beta,p}B^2\min\!\left\{1,\frac{d\binom{d}{k_\star}}{n}\right\}\le R_n^\star(d,\beta,p,B)\le\sup_{(G_n,c)\in\mathcal M_{n,d,\beta}(B)}\mathbb E_Z[(\widehat\tau_n^{\mathrm{up}}-\tau_n)^2]\le C_{\beta,p}B^2\min\!\left\{1,\frac{d\binom{d}{k_\star}}{n}\right\}.$ In the nonsaturated regime $d\binom{d}{k_\star}/n\lesssim_{\beta,p}1$ , unclipped SNIPE attains the same rate. The upper bound uses $A_i=\sum_{r=1}^{\min\{\beta,|N_i|\}}\binom{|N_i|}{r}\Delta_r(p)^2/[p(1-p)]^r\le A_d$ and the single-out-degree overlap count. For $B>0$ and $d\ge1$ , the block mixtures have TTE separation $2\rho\delta$ and squared Hellinger distance at most $4\pi^2m\delta^2/(B^2A_d)$ ; since $1/2\le\rho\le1$ and $A_d/m=(dA_d/n)/\rho$ , this yields the stated lower branch without a divisibility condition.
Theorem T-2 verified CausalSmith.Experimentation.SnipeDegreeFrontier.sharp_local_linear_constant_and_representers Fix $\beta\ge1$ , $p\in(0,1)$ , and $B>0$ . For every index $t$ , let $n_t$ and $d_t$ be positive integers such that $d_t\mid n_t$ , write $m_t:=n_t/d_t$ , and let $G_t$ be the fixed directed graph that is the disjoint union of $m_t$ complete directed blocks $V_b$ of size $d_t$ , including all self-loops. Then, for every $t$ , $R_t^{\mathrm{loc,lin}}=\frac{B^2A_{d_t}}{m_t}=\frac{B^2d_tA_{d_t}}{n_t}.$ In particular, $R_t^{\mathrm{loc,lin}}/(B^2d_tA_{d_t}/n_t)=1$ for every $t$ . More precisely, put $Z_S:=\prod_{j\in S}Z_j$ and $Z_b:=(Z_j)_{j\in V_b}$ , and for every block $V_b$ define $\Psi_{b,t}(w):=\max_{\substack{(\varepsilon_i,S_i)_{i\in V_b}\\\varepsilon_i\in\{-1,1\},\ S_i\subseteq V_b,\ |S_i|\le\bar\beta_{d_t}}} \mathbb E_Z\!\left[\left(\sum_{i\in V_b}\varepsilon_i\{w_{i,t}(Z_b)Z_{S_i}-\mathbf1(S_i\ne\varnothing)\}\right)^2\right].$ Then, for every $t$ , $\sup_{c\in\mathcal M_t(G_t)}\mathbb E_Z[(\widehat\tau_{w,t}-\tau_{n_t})^2] =\frac{B^2}{n_t^2}\sum_{b=1}^{m_t}\Psi_{b,t}(w), \qquad \Psi_{b,t}(w)\ge d_t^2A_{d_t}.$ A sequence in $\mathcal E_t^{\mathrm{loc,lin}}$ is asymptotically minimax if and only if $\frac{1}{m_td_t^2A_{d_t}}\sum_{b=1}^{m_t}\{\Psi_{b,t}(w)-d_t^2A_{d_t}\}\longrightarrow0.$ The condition $\frac{1}{n_tA_{d_t}}\sum_{i\in V_{n_t}}\mathbb E_Z[(w_{i,t}(Z_{b(i)})-g_{d_t}(Z_{b(i)}))^2]\longrightarrow0$ is sufficient for asymptotic minimaxity. Suppose in addition that $d_t\to\infty$ . This sufficient condition is not necessary, even up to relabeling coordinates within blocks. There are asymptotically minimax weights for which the last display equals $2$ for every sufficiently large $t$ .
Theorem T-3 verified CausalSmith.Experimentation.SnipeDegreeFrontier.bounded_outcome_degree_frontier Fix $\beta\ge1$ , $p\in(0,1)$ , $B\ge0$ , integers $n\ge1$ , and $0\le d\le n$ . The coefficient-mass model is contained in the uniformly bounded-outcome model, and the inclusion is strict whenever $B>0$ and $d\ge1$ . There are $0<c_{\beta,p}\le C_{\beta,p}<\infty$ such that $c_{\beta,p}B^2\min\!\left\{1,\frac{dA_d}{n}\right\} \le R_{n,\ell_1}^\star(d,\beta,p,B)=R_n^\star(d,\beta,p,B) \le R_{n,\infty}^\star(d,\beta,p,B) \le \sup_{(G_n,c)\in\mathcal M_{n,d,\beta}^{\infty}(B)} \mathbb E_Z[(\widehat\tau_{n,\infty}^{\mathrm{up}}-\tau_n)^2] \le C_{\beta,p}B^2\min\!\left\{1,\frac{dA_d}{n}\right\}.$ Moreover, $\sup_{(G_n,c)\in\mathcal M_{n,d,\beta}(B)} \mathbb E_Z[(\widehat\tau_n^{\mathrm{up}}-\tau_n)^2] \le C_{\beta,p}B^2\min\!\left\{1,\frac{dA_d}{n}\right\},$ and on either model class unclipped SNIPE is design-unbiased and has worst-case mean squared error at most $B^2dA_d/n$ . For every $0\le d\le n$ , $dA_d\asymp_{\beta,p}d\binom d{k_\star}$ , so both minimax risks are also $\asymp_{\beta,p}B^2\min\{1,d\binom d{k_\star}/n\}$ , and unclipped SNIPE attains the nonsaturated branch. If $1\le d\le n$ and $d\mid n$ , put $m=n/d$ and take the disjoint union of $m$ complete directed $d$ -blocks, including loops. Then canonical unclipped SNIPE has the exact global worst-case risk on both classes, $\sup_{(G_n,c)\in\mathcal M_{n,d,\beta}(B)}\mathbb E_Z[(\widehat\tau_n^{\mathrm{SNIPE}}-\tau_n)^2] =\sup_{(G_n,c)\in\mathcal M_{n,d,\beta}^{\infty}(B)}\mathbb E_Z[(\widehat\tau_n^{\mathrm{SNIPE}}-\tau_n)^2] =\frac{B^2A_d}{m}=\frac{B^2dA_d}{n},$ so the constant one in the global unclipped variance bound is sharp. The upper-bound calculation uses $A_i=\sum_{r=1}^{\min\{\beta,|N_i|\}}\binom{|N_i|}{r}\frac{\Delta_r(p)^2}{[p(1-p)]^r}\le A_d$ and charges out-degree only once. For $B>0$ and $d\ge1$ , the block mixtures in ${\rm def:block-family}$ lie in the smaller class, have TTE separation $2\rho\delta$ , and have squared Hellinger distance at most $4\pi^2m\delta^2/(B^2A_d)$ ; furthermore $1/2\le\rho\le1$ and $A_d/m=(dA_d/n)/\rho$ .
Theorem T-4 verified CausalSmith.Experimentation.SnipeDegreeFrontier.fair_coin_energy_frontier Fix $n\ge1$ , $1\le d\le n$ , $\beta\ge1$ , $B\ge0$ , and $p=1/2$ . Then $\Delta_r(1/2)=\begin{cases}2^{1-r},&r\text{ odd},\\0,&r\text{ even},\end{cases} \qquad A_d=4\sum_{\substack{1\le r\le\min\{\beta,d\}\\r\text{ odd}}}\binom dr.$ Consequently $k_\star$ is the largest odd integer no greater than $\min\{\beta,d\}$ . For $\beta=1$ , $A_d=4d$ and $R_n^\star(d,1,1/2,B)=R_{n,\ell_1}^\star(d,1,1/2,B) \asymp R_{n,\infty}^\star(d,1,1/2,B) \asymp B^2\min\!\left\{1,\frac{d^2}{n}\right\}.$ If additionally $d\mid n$ , canonical unclipped SNIPE has exact worst-case mean squared error $4B^2d^2/n$ on both model classes.

Auxiliary Lean lemmas (machine-checked) (83)

Definition activeShare —
Definition bernoulliContrast —
Definition blockCount —
Definition blockDominatingMeasure —
Definition blockEnergy —
Definition blockExtremal —
Definition blockGraph —
Definition blockPriorDensity —
Definition blockSchedule —
Definition blockUnits —
Definition edgeFn —
Definition edgeFnBdd —
Definition effBeta —
Definition Hbound —
Definition hellingerSqDensity —
Definition largestOddLE —
Definition localEnergy —
Definition locLinMinimaxRisk —
Definition locLinRiskRatio —
Definition LocLinWeights —
Definition locLinWorstRisk —
Definition minimaxRiskL1 —
Definition ModelClassIncluded —
Definition ModelClassStrict —
Definition normalizedBlockExcess —
Definition normalizedWeightDistance —
Definition normalizedWeightDistanceRelabeled —
Definition obsOutcome —
Definition OutcomeMeasurable —
Definition snipeClipped —
Definition tiltAmplitude —
Definition tte —
Definition worstRisk —
Definition worstRiskBdd —
Definition worstRiskBddFixedGraph —
Definition worstRiskFixedGraph —
Definition AdmissibleEstimator —
Definition AdmissibleEstimatorBdd —
Definition containingNeighborhoods —
Definition DegreeIndex —
Definition minimaxRiskBddOutcome —
Definition nbhd —
Definition snipeEstimator —
Lemma CausalSmith.Experimentation.SnipeDegreeFrontier.fairCoinContrast —
Lemma CausalSmith.Experimentation.SnipeDegreeFrontier.fairCoinEnergy —
Lemma CausalSmith.Experimentation.SnipeDegreeFrontier.fairCoinKStar —
Lemma CausalSmith.Experimentation.SnipeDegreeFrontier.E_coordinate_prod —
Lemma CausalSmith.Experimentation.SnipeDegreeFrontier.one_sub_prod_le_sum —
Lemma CausalSmith.Experimentation.SnipeDegreeFrontier.potentialOutcome_abs_le_of_mem_modelClass —
Lemma CausalSmith.Experimentation.SnipeDegreeFrontier.tte_abs_le_two_mul_of_mem_modelClass —
Lemma CausalSmith.Experimentation.SnipeDegreeFrontier.snipeClipped_admissible —
Lemma CausalSmith.Experimentation.SnipeDegreeFrontier.minimaxRisk_le_worstRisk_of_admissible —
Lemma CausalSmith.Experimentation.SnipeDegreeFrontier.blockScore_contrast —
Lemma CausalSmith.Experimentation.SnipeDegreeFrontier.blockEnergy_pos —
Lemma CausalSmith.Experimentation.SnipeDegreeFrontier.localEnergy_le_blockEnergy —
Lemma CausalSmith.Experimentation.SnipeDegreeFrontier.blockContrast_binomial —
Lemma CausalSmith.Experimentation.SnipeDegreeFrontier.blockScore_raw_moment —
Lemma CausalSmith.Experimentation.SnipeDegreeFrontier.blockScore_mean_zero —
Lemma CausalSmith.Experimentation.SnipeDegreeFrontier.centeredMonomial_raw_expansion —
Lemma CausalSmith.Experimentation.SnipeDegreeFrontier.sum_powerset_subset_exchange —
Lemma CausalSmith.Experimentation.SnipeDegreeFrontier.nbhdB_edgeFn_eq_nbhd —
Lemma CausalSmith.Experimentation.SnipeDegreeFrontier.nbhdB_edgeFnBdd_eq_nbhd —
Lemma CausalSmith.Experimentation.SnipeDegreeFrontier.E_global_coordinate_prod —
Lemma CausalSmith.Experimentation.SnipeDegreeFrontier.snipeScore_raw_moment —
Lemma CausalSmith.Experimentation.SnipeDegreeFrontier.snipeScore_mean_zero —
Lemma CausalSmith.Experimentation.SnipeDegreeFrontier.snipeScore_potentialOutcome_moment —
Lemma CausalSmith.Experimentation.SnipeDegreeFrontier.blockScore_weightFeasibleAt —
Lemma CausalSmith.Experimentation.SnipeDegreeFrontier.weightProg_le_blockEnergy —
Lemma CausalSmith.Experimentation.SnipeDegreeFrontier.kStar_mem_exposedOrder —
Lemma CausalSmith.Experimentation.SnipeDegreeFrontier.blockEnergy_topExposed_le —
Lemma CausalSmith.Experimentation.SnipeDegreeFrontier.blockDesign_sq_eq_zero_iff —
Lemma CausalSmith.Experimentation.SnipeDegreeFrontier.blockScore_mem_polySpace —
Lemma CausalSmith.Experimentation.SnipeDegreeFrontier.weightFeasibleAt_represents —
Lemma CausalSmith.Experimentation.SnipeDegreeFrontier.perturbFeasible_energy_unique —
Lemma CausalSmith.Experimentation.SnipeDegreeFrontier.weightFeasibleAt_energy_unique —
Lemma CausalSmith.Experimentation.SnipeDegreeFrontier.blockPrograms_exact —
Lemma CausalSmith.Experimentation.SnipeDegreeFrontier.choose_le_pow_mul_choose —
Lemma CausalSmith.Experimentation.SnipeDegreeFrontier.eligibleOrder_le_kStar —
Lemma CausalSmith.Experimentation.SnipeDegreeFrontier.blockLowerConst_spec —
Lemma CausalSmith.Experimentation.SnipeDegreeFrontier.blockEnergy_uniform_compare —
Lemma CausalSmith.Experimentation.SnipeDegreeFrontier.blockRawCoef_mass_estimate —
Lemma CausalSmith.Experimentation.SnipeDegreeFrontier.blockRawCoef_mass_uniform —
Lemma CausalSmith.Experimentation.SnipeDegreeFrontier.modelClassIncluded_of_coeffMass —

Symbol realizations (Lean @realizes) (80)

V_n verified realized by variable@27
n verified realized by variable@27
i verified realized by variable@27
j verified realized by variable@27
l verified realized by variable@27
G_n verified realized by nbhd, GraphClass
N_i verified realized by nbhd
$\Delta_r(p)$ verified realized by bernoulliContrast
$\beta$ verified realized by effBeta
$\bar\beta_d$ verified realized by effBeta
d verified realized by DegreeIndex
r verified realized by OrderIndex
S verified realized by NeighborhoodSubset
$k_\star$ verified realized by kStar
p verified realized by IsProductBernoulli
P_Z verified realized by IsProductBernoulli
$\mathbb E_Z$ verified realized by IsProductBernoulli
c verified realized by LowOrder
c_{i,S} verified realized by LowOrder
B verified realized by BoundedCoeffMass
z verified realized by potentialOutcome
Y_i(z) verified realized by potentialOutcome
Z verified realized by obsOutcome
$Y_i^{\mathrm{obs}}$ verified realized by obsOutcome
$\tau_n$ verified realized by tte
$\mathcal G_{n,d}$ verified realized by GraphClass
$\mathcal C_{n,d,\beta}(B)$ verified realized by CoeffClass
$\mathcal M_{n,d,\beta}(B)$ verified realized by ModelClass
$\mathcal C_{n,d,\beta}^{\infty}(B)$ verified realized by BddOutcomeCoeffClass
$\mathcal M_{n,d,\beta}^{\infty}(B)$ verified realized by BddOutcomeModelClass
$\widehat\tau_n$ verified realized by Estimator
$g_{i,\beta,p}$ verified realized by snipeScore
$\widehat\tau_n^{\mathrm{SNIPE}}$ verified realized by snipeEstimator
$\widehat\tau_n^{\mathrm{up}}$ verified realized by snipeClipped
$R_n^\star(d,\beta,p,B)$ verified realized by minimaxRisk
$R_{n,\ell_1}^{\star}(d,\beta,p,B)$ verified realized by minimaxRiskL1
$R_{n,\infty}^{\star}(d,\beta,p,B)$ verified realized by minimaxRiskBddOutcome
$\widehat\tau_{n,\infty}^{\mathrm{up}}$ verified realized by snipeClippedBdd
$H_{\beta,p}$ verified realized by blockEnergy_representer, representerMassSup
$c_{\beta,p}$ verified realized by blockEnergy_representer, bounded_outcome_degree_frontier
$C_{\beta,p}$ verified realized by blockEnergy_representer, bounded_outcome_degree_frontier
g_d verified realized by blockScore
A_d verified realized by blockEnergy
h_d verified realized by blockRepresenter
T verified realized by blockRawCoef
h_{d,T} verified realized by blockRawCoef
$\mathcal P_d$ verified realized by polySpace
f verified realized by polySpace
h verified realized by polySpace
L_d verified realized by contrastFunctional
$\mathsf{PerturbProg}_{d,\beta,p}$ verified realized by perturbProg
w verified realized by WeightFeasible
$\mathsf{WeightProg}_{d,\beta,p}$ verified realized by weightProg
s verified realized by baselineProductDensity, cosSqDensity, blockPriorDensity
f_s verified realized by baselineProductDensity, cosSqDensity
m verified realized by blockCount
q verified realized by activeCount
$\rho$ verified realized by activeShare
$G_{n,d}^{\mathrm{blk}}$ verified realized by blockGraph
V_b verified realized by blockGraph
b verified realized by blockGraph
U_b verified realized by baselineAt
$\delta$ verified realized by tiltAmplitude
$\sigma$ verified realized by blockSchedule
$\Pi_\sigma$ verified realized by blockSchedulePrior
$P_\sigma$ verified realized by priorPredictiveLaw
$\mathcal M_t(G_t)$ verified realized by LocLinSchedClass
G_t verified realized by LocLinSchedClass
N_i^{G_t} verified realized by LocLinSchedClass
n_t verified realized by LocLinSchedClass
V_{n_t} verified realized by LocLinSchedClass
d_t verified realized by LocLinSchedClass
w_{i,t} verified realized by LocLinWeights
b(i) verified realized by LocLinWeights
Z_{b(i)} verified realized by LocLinWeights
$\widehat\tau_{w,t}$ verified realized by locLinEstimator
$\mathcal E_t^{\mathrm{loc,lin}}$ verified realized by locLinEstimatorClass
$R_t^{\mathrm{loc,lin}}$ verified realized by locLinMinimaxRisk
A_i verified realized by localEnergy
$\Psi_{b,t}$ verified realized by sharp_local_linear_constant_and_representers

Proof map 8 statements · 10 edges

Minimax Mean Squared Error for Low-order Network Interference under Bernoulli Assignment

Abstract

Introduction

Related work

Setup and assumptions

Discussion of the assumptions

Main results

Discussion and extensions

Limitations and future work

Appendices

Proofs and auxiliary lemmas

Verification note

Proofs of the main results

References