CausalSmith · seminar slides

Exact Randomized Designs for Two-Block Interference Experiments

We characterize when a covariance relaxation for network-interference design is exactly implementable by a finite randomized $\{\pm1\}$ assignment law, and we compute the exact loss when implementation binds.

slides for Exact Randomized Designs for Two-Block Interference Experiments

Overview

The design problem chooses a randomized treatment assignment under network interference.
The relaxed problem chooses a covariance matrix.
The implementable problem requires that covariance to come from an actual finite assignment law.
In a two-block homophilous graph, the whole question reduces to two covariance parameters.
We give exact regions with zero loss, an odd-block region with strictly positive loss, and a finite formula for the loss everywhere.

Motivation

Interference-aware design often optimizes a covariance $X=\mathbb E[ZZ^\top]$ , where $Z$ is the treatment-sign vector.
A covariance optimum is useful operationally when some randomized $\{\pm1\}$ design generates it exactly.
In finite samples, the elliptope relaxation can contain covariance matrices outside the assignment-induced moment set.
Our question is: when can the relaxed optimum be implemented as a randomized design?

Running Example

Think of $2m$ villages, schools, or online neighborhoods split into two equal communities.
Within-community exposure has weight $a$ , and across-community exposure has weight $b$ .
Homophily means $a>b>0$ : spillovers are stronger inside each community.
The assignment law may correlate treatment signs across units to manage interference and balance.

illustrative Box-and-arrow schematic with two community boxes, within-community exposure

a/m

, across-community exposure

b/m

, and randomized treatment signs feeding the design objective.

Setup

We work in a design-based finite-population model: units and graph weights are fixed, and only assignment is random.
The sample size is $n=2m$ , with $m$ units in each community.
A design is a law $P$ on $Z\in\{-1,1\}^n$ .
Sign symmetry means treatment and control labels are balanced at the law level.

Assumption A-1 (Two-Block Homophily)

The block size and edge intensities satisfy $m\ge 2 \qquad\text{and}\qquad a>b>0.$

Assumption A-2 (Balanced Sign Symmetry)

The assignment law is sign-symmetric: for every $z\in\{-1,1\}^n$ , $P(Z=z)=P(Z=-z).$

Design Objective

The objective combines four design pressures.
$\operatorname{tr}(L_mX)$ rewards agreement across high-weight graph edges.
$r$ , the pseudoinverse weight, controls the Laplacian-pseudoinverse term.
$\kappa$ , the robustness weight, controls the Schatten--2 penalty.
$\operatorname{tr}(J_nX)$ penalizes aggregate treatment imbalance.

$F_{r,\kappa}(X) = \operatorname{tr}(L_m X) + r\,\operatorname{tr}(L_m^\dagger X) + \kappa \|X\|_{S_2} + \operatorname{tr}(J_n X).$

Relaxed Versus Implementable

The relaxed feasible set is the block elliptope $\mathcal E_m^{\mathrm{blk}}$ , the block-symmetric positive-semidefinite covariance slice.
The implementable feasible set is $\mathcal C_m^{\pm}$ , the covariances generated by sign-symmetric block-exchangeable assignment laws.
The implementability loss $\Delta_m^{\pm}(r,\kappa)$ is the objective-value price of returning from the relaxation to an actual randomized design.

$\Delta_m^{\pm}(r,\kappa) = \inf_{P\in \mathcal P_m^{\mathrm{sym}}} F_{r,\kappa}(X(P)) - \inf_{X\in \mathcal E_m^{\mathrm{blk}}} F_{r,\kappa}(X).$

Symmetry Reduction

Averaging over the two-block automorphism group preserves the relevant design problem.
For relaxed matrices, only the average within-block covariance $u$ and across-block covariance $v$ matter.
For assignment laws, block-exchangeable laws attain the same implementable infimum.
This turns the original matrix problem into a two-parameter geometry.

informal · Lemma L-1 Under two-block homophily, optimizing the original relaxed and implementable problems is equivalent to optimizing over the block elliptope and block-exchangeable assignment laws.

Related Literature

Fisher (1935), Horvitz and Thompson (1952), and Rubin (1974) provide the design-based foundation.
Hudgens and Halloran (2008), Aronow and Samii (2013), Athey et al. (2015), Sävje et al. (2017), Leung (2019), and Li and Wager (2020) develop causal inference under interference.
Ugander et al. (2013), Eckles et al. (2014), Baird et al. (2018), Ugander and Yin (2020), Viviano et al. (2023), Chen et al. (2023), and Cai et al. (2023) study network-aware designs.
Thiyageswaran et al. (2026) is the immediate covariance-design predecessor.
Goemans and Williamson (1995), Deza and Laurent (1997), and Boyd and Vandenberghe (2004) provide the elliptope and convex-optimization backdrop.

Key Idea

The two-block covariance has three spectral coordinates.
$x$ tracks within-block spread, $y$ tracks the between-block contrast, and $z_{\mathrm{sp}}$ tracks aggregate balance.
The relaxed problem is a weighted simplex problem in $(x,y,z_{\mathrm{sp}})$ .
Implementability adds one finite parity condition.
When $m$ is odd, each block sum is odd, so the relaxed optimum can fall just outside the implementable slice.

illustrative Box-and-arrow schematic from assignment law P to block sums S_A and S_B to covariance X(u,v) to spectral coordinates x, y, z_sp to implementability check.

Cut Exactness

The cut design assigns opposite signs to the two communities, up to global sign reversal.
In the strict cut region, the relaxed optimum is exactly this implementable covariance.
For the running example, this is the regime where separating communities is optimal for the criterion.

informal · Theorem T-1 Under two-block homophily and the stated strict cut inequality for $r$ , $X_{\mathrm{cut}}$ is the unique relaxed and implementable minimizer, and the implementability loss is zero.

Independent Assignment

The iid design assigns independent fair signs and generates $I_n$ .
It is always an implementable block-exchangeable design.
Its exact finite-robustness optimality occurs on the affine-balanced locus.
Away from that locus, increasing robustness drives relaxed minimizers entrywise toward $I_n$ .

informal · Theorem T-2 Under two-block homophily and $r\ge0$ , $I_n$ is a finite-robustness relaxed minimizer exactly on $a+3b=2m$ and $r=2b(a+b)$ , and relaxed minimizers converge entrywise to $I_n$ as $\kappa\to\infty$ elsewhere.

Positive Loss

The spread covariance concentrates on the within-block spectral direction.
Under odd block size, low scale, and the theorem’s robustness range, the spread covariance is the unique relaxed optimum over the block elliptope.
The same parity condition places it outside the implementable covariance class.
The implementability loss is then strictly positive.

informal · Theorem T-3 Under two-block homophily, the low-scale condition, odd $m$ , and the stated interval for $\kappa$ and $r$ , $X_{\mathrm{spread}}$ is the unique relaxed minimizer, $X_{\mathrm{spread}}\notin\mathcal C_m^{\pm}$ , and $\Delta_m^{\pm}(r,\kappa)>0$ .

Parity Contrast

Odd $m$ : every realized block sum has absolute value at least one.
That arithmetic restriction becomes $y+z_{\mathrm{sp}}\ge d_m$ with $d_m=2/m$ .
Even $m$ : each block can have zero sign sum.
In the maintained symmetric two-block model, every relaxed block covariance is implementable when $m$ is even.

informal · Theorem T-3 If $m$ is even, then every covariance in the block elliptope is implementable.

Sharp Loss Formula

The active-set formula computes the exact finite implementability loss for any admissible $m,a,b,r,\kappa$ .
First solve the relaxed weighted-simplex problem.
Then check whether its minimizer satisfies $d_m\le y+z_{\mathrm{sp}}$ .
If the check passes, the loss is zero.
If it binds, the implementable value is found on the one-dimensional truncation segment.

informal · Theorem T-4 Under two-block homophily, the sharp active-set value $\rho_\star(m,a,b,r,\kappa)$ equals $\Delta_m^{\pm}(r,\kappa)$ , is nonnegative, gives the exact zero-loss condition, and is zero for even $m$ .

Proof Sketch

Symmetry does the first reduction: averaging makes all relevant matrices block-symmetric and all relevant laws block-exchangeable.
Spectral coordinates do the second reduction: the matrix objective becomes a convex weighted-simplex objective.
Block sums characterize implementability: finite sign assignments translate into a parity truncation.
Vertex certificates identify the cut and spread regions.
The active-set calculation solves the relaxed simplex problem and the truncated implementable problem exactly.

Takeaways

We give an exact finite-sample implementability theory for covariance-based design in a homophilous two-block network.
In the strict cut region, the relaxed optimum is generated by a two-point randomized design.
On the affine-balanced locus, iid assignment is the exact finite-robustness optimum.
For odd block size in the stated low-scale window, the unique relaxed optimum has strictly positive implementability loss.
The active-set formula computes that loss exactly for all admissible parameters.