CausalSmith · seminar slides

Uniform Expected Risk for Boundary Distance Designs

We characterize how distance compression sets the expected uniform-risk scale for boundary regression discontinuity designs: unsigned distance forces the $(\log n/n)^{1/4}$ lower bound, and signed known-geometry local polynomials match that scale under the stated analytic inputs.

slides for Uniform Expected Risk for Distance-Based Boundary Regression Designs

Overview

- The frontier normalization $a_n$ , the benchmark for expected supremum loss, is $a_n=\left(\frac{\log n}{n}\right)^{1/4}.$

Unsigned distance rules observe outcomes and scalar distances from each boundary query point.

For those rules, expected boundary sup-loss is bounded below on the $a_n$ scale.

Signed-distance rules also know the treatment side and boundary geometry.

Under $\mathsf{AI}_{p,\nu,L}$ , the signed known-geometry minimax risk is characterized up to constants on the same scale.

Motivation

In a geographic regression discontinuity design, treatment changes at a border: school districts, precincts, jurisdictions, or policy regions.

Distance to the border is a natural one-dimensional running coordinate.

For one boundary location, distance looks like the familiar scalar regression discontinuity coordinate.

For the entire boundary, the same sample is re-expressed at many query points.

Uniform expected risk asks how accurately a distance-based rule can recover the whole boundary curve.

The central question is how much uniform accuracy survives the distance reduction.

Distance Experiments

illustrative Box-and-arrow schematic comparing a boundary point, nearby observations, unsigned distance, signed distance, side labels, known geometry, and supremum loss.

Unsigned distance keeps how far an observation is from the query point.

Signed distance keeps how far and which side of the treatment boundary the observation lies on.

The unsigned target is the boundary regression function $\mu_P$ , the conditional mean at the support boundary.

The signed target is $\tau_P(x)$ , the boundary treatment-effect curve.

Both risks evaluate the expected supremum error along the boundary.

Unsigned Setup

The class $\mathcal P_{\mathrm{NP}}(L,q)$ , the compact bivariate nonparametric law class, has bounded density, Hölder regression, continuous bounded variance, and a Lipschitz support boundary.

At boundary point $x$ , the rule sees $(Y_i,\lVert X_i-x\rVert_2)$ .

The common-map class uses one law-independent distance rule across all boundary points.

The point-indexed class permits a separate law-independent distance rule at each query point.

The loss is expected sup-error for estimating $\mu_P(x)$ over $\operatorname{bd}(\mathcal X_P)$ .

Signed Setup

The known geometry $G_P$ , the bundled design object, gives the support, assignment regions, interface, Euclidean metric, and uniform kernel.

Signed distance assigns positive and negative signs using the treatment side.

The class $\mathcal P_{12}(p,\nu,L)$ , the A1/A2 law class, supplies rectangular support, bounded density, smooth arm regressions, moments, variances, rectifiable interface, local mass, and stable Gram matrices.

The target $\tau_P(x)$ , the treatment-effect curve, is the arm-specific regression jump at boundary point $x$ .

In the geographic example, this is the effect curve traced along the policy border.

Key Idea

illustrative Box-and-arrow schematic showing separated boundary cells, hidden binary perturbations, unsigned distance observations, supremum loss, the scale calculation, balancing, and decoding difficulty.

The lower-bound construction plants many separated cells along the boundary.

Each cell carries a hidden binary perturbation and a boundary signal $\Delta$ .

Smoothness permits cells with radius at the $\Delta^{1/q}$ scale and count at the $\Delta^{-1/q}$ scale.

Unsigned distances preserve radius and mix angular directions, so each bit carries weak information $n\Delta^4$ .

Uniform loss forces simultaneous recovery across cells.

Balancing $n\Delta^4$ with $\log M$ yields the $a_n$ scale.

Main Result I

informal · Theorem T-1 For every $q\geq1$ and $L\geq4$ , common-map unsigned rules have minimax expected boundary sup-loss at least a constant multiple of $a_n$ .

This is the Cattaneo et al. (2026) common-map architecture.

The result keeps the law class, information set, target, decision class, and expected supremum loss fixed.

Theorem T-1 (Same-Class Log Converse)

For every integer $q\geq 1$ and every $L\geq 4$ , there exists a constant $c_{\mathrm{CTY}}=c_{\mathrm{CTY}}(q,L)>0$ such that, for the common-map minimax risk $R_n^{\mathrm{NP}}$ in Definition P-5 and $a_n=(\log n/n)^{1/4}$ , $\liminf_{n\to\infty} \left(\frac{n}{\log n}\right)^{1/4} R_n^{\mathrm{NP}} = \liminf_{n\to\infty}\frac{R_n^{\mathrm{NP}}}{a_n} \geq c_{\mathrm{CTY}} .$

Main Result II

informal · Lemma L-3 For $n\geq1$ , $q\geq1$ , and $L\geq4$ , point-indexed unsigned rules strictly contain common-map rules.

informal · Theorem T-2 For $q\geq1$ and $L\geq4$ , point-indexed unsigned rules also have minimax outer-expected boundary sup-loss at least a constant multiple of $a_n$ .

The rule may use a different law-independent distance rule at each boundary point.

The lower scale is tied to outcomes plus unsigned scalar distances.

Theorem T-2 (Logarithmic Distance Converse)

For every $q\in\mathbb N$ and $L\in\mathbb R$ satisfying $q\geq 1$ and $L\geq 4$ , there exists a constant $c_{\mathrm{PI}}=c_{\mathrm{PI}}(q,L)>0$ such that, with $R_n^{\mathrm{PI}}$ the point-indexed outer-expectation minimax risk in Definition P-7 and $a_n=(\log n/n)^{1/4}$ , $\liminf_{n\to\infty} \left(\frac{n}{\log n}\right)^{1/4} R_n^{\mathrm{PI}} = \liminf_{n\to\infty} \frac{R_n^{\mathrm{PI}}}{a_n} \geq c_{\mathrm{PI}}.$

Signed Lower Bound

informal · Lemma L-7 For each $p$ , the fixed rectangle supports a signed-distance hypercube with exact $\Delta$ target separation and local KL bounded by the $\Delta^4/w^2$ scale.

informal · Theorem T-3 For every fixed $p$ , $\nu\geq2$ , and $L\geq L_0(p)$ , signed-distance known-geometry rules have minimax outer risk at least a constant multiple of $a_n$ .

The hard experiment is a fixed treated rectangle inside a fixed square, including corner points.

Signed-distance rules know the geometry and side labels; the construction carries many weak boundary bits under that information set.

Theorem T-3 (Point-Indexed Log Converse)

For every nonnegative integer $p$ , there exists a real number $L_0=L_0(p)$ such that:

(Envelope threshold.) $L_0\ge 48$ .
(Moment exponent.) $\nu\ge 2$ .
(Uniform envelope.) $L\ge L_0$ .
(Risk definition.) $R^{12,\pm}_n(p,\nu,L)$ is the signed-distance minimax risk in Definition P-13.
(Fixed-geometry risk.) $R^{12,\pm,\mathrm{fix}}_n(p,\nu,L)$ is the infimum over the same known-geometry point-indexed decision rules of the same worst-case outer-expected interface sup-loss, with the law supremum further restricted to laws in $\mathcal P_{12}(p,\nu,L)$ satisfying the fixed hard-geometry condition.

For every $\nu$ and $L$ satisfying these conditions, there exists a constant $c=c(p,\nu,L)>0$ such that, writing $a_n=(\log n/n)^{1/4}$ , $\liminf_{n\to\infty} a_n^{-1}R^{12,\pm}_n(p,\nu,L) = \liminf_{n\to\infty}{R^{12,\pm}_n(p,\nu,L)\over a_n},$ and $c \le \liminf_{n\to\infty}{R^{12,\pm}_n(p,\nu,L)\over a_n}, \qquad c \le \liminf_{n\to\infty}{R^{12,\pm,\mathrm{fix}}_n(p,\nu,L)\over a_n}.$

Analytic Inputs

The signed upper bound is conditional on $\mathsf{AI}_{p,\nu,L}$ , the three assumed signed-distance inputs.

Identification: one-sided signed-distance conditional means identify $\tau_P(x)$ .

Approximation: population local-polynomial intercept bias is uniformly first order in bandwidth.

Stochastic control: expected uniform Gram and raw-score deviations obey the maximal bounds, including the heavy-tail exponent $n^{(1+\nu)/(2+\nu)}$ .

informal · Lemma L-4 Closed Euclidean balls in the plane shatter at most three points in this finite-set certificate.

informal · Lemma L-6 For clipped signed-distance local-polynomial scores, the expected outer supremum is at most the sum of a square-root empirical term and a bounded-envelope term.

Signed Upper Bound

informal · Theorem T-4 Under the three signed-distance analytic inputs, the stabilized signed-distance local-polynomial estimator has expected outer interface sup-loss at most $C a_n$ for all sufficiently large $n$ .

Mechanism: at bandwidth $h_n=a_n$ , winsorize outcomes at $B_n=a_n^{-1/3}$ .

Fit separate degree- $p$ signed-distance local polynomials on each side.

Stabilize the Gram inverse and clip the final jump estimate.

The bandwidth balances first-order bias with the uniform stochastic scale.

Theorem T-4 (Expected Outer Upper Bound)

Let $p\geq 0$ be an integer, let $\nu\geq 2$ , and let $L\geq 4$ . Assume:

(Distance identification.) For every A1/A2 law $P$ , every distance map $d$ , and every $x\in\mathcal B_P$ satisfying the CTY identification assumptions at order $p$ , there are signed-distance conditional-mean versions $\theta_1,\theta_0:\mathbb R\to\mathbb R$ and one-sided limits $\theta_1(0+)$ and $\theta_0(0-)$ such that $\tau_P(x)=\theta_1(0+)-\theta_0(0-).$
(Uniform first-order bias.) There is a constant $C_b>0$ , depending on $p$ and $L$ , such that for every $\nu'\geq 2$ and every positive antitone deterministic bandwidth sequence $h_n$ with $h_n\to0$ and $n h_n^2\to\infty$ , $\limsup_{n\to\infty} \mathrm{BiasRatio}_{p,\nu',L}(h_n)\leq C_b .$
(Expected maximal bounds.) There is a constant $C_m>0$ such that, for every positive deterministic bandwidth sequence $h_n$ with $h_n\to0$ and $\frac{n^{(1+\nu)/(2+\nu)}h_n^2}{\log(h_n^{-1})}\to\infty,$ the following bounds hold for all sufficiently large $n$ , uniformly over $P\in\mathcal P_{12}(p,\nu,L)$ : $\mathbb E_P^*\!\left[\mathrm{GramDev}_{n,p,P}(h_n)\right] \leq C_m \sqrt{\frac{\log(h_n^{-1})}{n h_n^2}},$ and $\mathbb E_P^*\!\left[\mathrm{ScoreDev}_{n,p,P}(h_n)\right] \leq C_m\left\{ \sqrt{\frac{\log(h_n^{-1})}{n h_n^2}} + \frac{\log(h_n^{-1})}{n^{(1+\nu)/(2+\nu)}h_n^2} \right\}.$

With $a_n=(\log n/n)^{1/4}$ as in Definition P-1, take the bandwidth $h_n=a_n$ and the winsorization level $B_n=a_n^{-1/3}$ in the winsorized, Gram-stabilized signed-distance local-polynomial estimator clipped to $[-2L,2L]$ . Then there are constants $C>0$ and $N$ such that, for every $n\geq N$ , $\sup_{P\in\mathcal P_{12}(p,\nu,L)} \mathbb E_P^*\!\left[ \sup_{x\in\mathcal B_P} \left|\widehat\tau^{12,\mathrm{LP}}_{n,h_n}(x)-\tau_P(x)\right| \right] \leq C a_n .$

Matched Frontier

informal · Theorem T-5 Under $\mathsf{AI}_{p,\nu,L}$ , for every fixed $p$ , $\nu\geq2$ , and $L\geq L_0$ , the signed-distance minimax risk is bounded below and above by constant multiples of $a_n$ .

The lower inequality comes from the fixed-geometry signed hypercube.

The upper inequality is attained by the explicit winsorized, Gram-stabilized degree- $p$ estimator.

In the geographic example, known side labels and boundary geometry support a two-sided risk characterization for the treatment-effect curve.

Theorem T-5 (Matched Frontier Rate)

For every integer $p\geq 0$ , there exists a constant $L_0\geq 48$ such that, for every $\nu\geq 2$ and every $L\geq L_0$ , the following conditions imply the frontier conclusion below:

(Distance identification.) For every A1/A2 law $P$ , every map $d$ satisfying the CTY identification assumptions for $p$ and $P$ , and every $x\in\mathcal B_P$ , there are selected source-coherent signed-distance conditional-mean versions $\theta_1,\theta_0:\mathbb R\to\mathbb R$ whose right and left limits at zero exist and identify $\tau_P(x)=\lim_{u\downarrow 0}\theta_1(u)-\lim_{u\uparrow 0}\theta_0(u).$
(Uniform first-order bias.) There is a constant depending only on $p$ and $L$ such that, uniformly over every $\nu'\geq 2$ and every positive antitone deterministic bandwidth sequence $h_n\to0$ with $n h_n^2\to\infty$ , the finite limsup of the normalized uniform first-order bias ratio is bounded by that constant.
(Expected maximal bounds.) For this $p,\nu,L$ , there is a constant $C_{\mathrm{max}}>0$ such that, along every positive bandwidth sequence $h_n\to0$ with $\frac{n^{(1+\nu)/(2+\nu)}h_n^2}{\log(h_n^{-1})}\to\infty,$ for all sufficiently large $n$ and every $P\in\mathcal P_{12}(p,\nu,L)$ , the expected outer Gram-deviation supremum satisfies $\mathbb E_P^*\!\left[ \sup_{t\in\{0,1\}}\sup_{x\in\mathcal B_P}\sup_{j,k} \left|\widehat\Psi_{t,x}(h_n)_{jk}-\Psi_{t,P,x}(h_n)_{jk}\right| \right] \leq C_{\mathrm{max}} \sqrt{\frac{\log(h_n^{-1})}{n h_n^2}},$ and the expected outer centered raw-score supremum satisfies $\mathbb E_P^*\!\left[ \sup_{t\in\{0,1\}}\sup_{x\in\mathcal B_P} \left\|\widehat m_{t,x}(h_n)-m_{t,P,x}(h_n)\right\| \right] \leq C_{\mathrm{max}}\left( \sqrt{\frac{\log(h_n^{-1})}{n h_n^2}} + \frac{\log(h_n^{-1})}{n^{(1+\nu)/(2+\nu)}h_n^2} \right).$

Then there exist constants $c$ and $C$ with $0<c\leq C$ such that, for the signed-distance minimax risk $R^{12,\pm}_n(p,\nu,L)$ of Definition P-13 and $a_n=(\log n/n)^{1/4}$ , $c \leq \liminf_{n\to\infty}\frac{R^{12,\pm}_n(p,\nu,L)}{a_n} \leq \limsup_{n\to\infty}\frac{R^{12,\pm}_n(p,\nu,L)}{a_n} \leq C .$ Moreover, the explicit winsorized, Gram-stabilized degree- $p$ signed-distance local-polynomial estimator with bandwidth $h_n=a_n$ and winsorization level $B_n=a_n^{-1/3}$ , clipped to $[-2L,2L]$ , has uniform outer risk over $P\in\mathcal P_{12}(p,\nu,L)$ satisfying $\limsup_{n\to\infty} \frac{ \sup_{P\in\mathcal P_{12}(p,\nu,L)} \mathbb E_P^*\!\left[ \sup_{x\in\mathcal B_P} \left|\widehat\tau^{12,\mathrm{LP}}_{n,a_n}(x)-\tau_P(x)\right| \right] }{a_n} \leq C .$

Lower-bound Mechanics

informal · Lemma L-1 If each compressed coordinate has KL at most a fixed fraction of $\log M$ , the probability of at least one binary decoding error is at least the stated overlap lower bound.

informal · Lemma L-2 For $q\geq1$ and $L\geq4$ , the support-boundary construction supplies separated cells with signal separation at least $c_1a_n$ and compressed-distance KL at most $\alpha\log M_n$ .

A rule with small uniform error would decode every hidden bit.

The direct-product step converts weak per-cell information into a high probability of at least one decoding error.

One decoding error forces supremum loss at the boundary signal scale.

Upper-bound Mechanics

Identification turns the signed-distance conditional mean jump into the treatment-effect target.

The population local polynomial tracks that jump to first order in the bandwidth.

The empirical Gram condition keeps the two side-specific fits stable.

Winsorization converts the moment envelope into a bounded-score empirical process.

The expected maximal bounds control the supremum over arms, interface points, and laws.

Background

Classical regression discontinuity starts with threshold designs: Thistlethwaite and Campbell (1960) and Hahn et al. (2001).

Modern RD inference emphasizes local polynomial estimation, bandwidths, and robust correction: Imbens and Lemieux (2008), Lee and Lemieux (2010), Fan and Gijbels (1996), Calonico et al. (2014), and Calonico et al. (2020).

Boundary and geographic RD motivate vector scores and interface-indexed targets: Keele and Titiunik (2015), Keele et al. (2015), Keele and Titiunik (2016), and Cattaneo et al. (2026).

Our contribution uses the minimax and empirical-process vocabulary of Stone (1982), Tsybakov (2009), Vapnik and Chervonenkis (1971), Pollard (1984), and van der Vaart and Wellner (1996).

Open Questions

We establish unconditional logarithmic lower bounds for unsigned common-map and point-indexed distance rules over $\mathcal P_{\mathrm{NP}}(L,q)$ .

We establish an unconditional signed-distance lower bound on a fixed rectangular subexperiment.

Under $\mathsf{AI}_{p,\nu,L}$ , we obtain the signed known-geometry two-sided expected outer-risk frontier.

The shared $a_n$ scale compares delivered lower scales; the signed two-sided frontier is the conditional characterization.

Open directions: a matching unsigned upper bound, a self-contained derivation of the signed analytic inputs on the displayed law class, and extensions to non-Euclidean metrics, kernels, and geometry envelopes.