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Compound Selection Decisions: An Almost SURE Approach

Jiafeng Chen, Lihua Lei, Timothy Sudijono, Liyang Sun, Tian Xie

TL;DR

This work advances compound decision making under parallel noisy estimates by introducing ASSURE, a SURE-inspired estimator for the welfare of threshold-based selections in a Gaussian setting. By constructing a near-unbiased welfare estimator and optimizing it over a controlled class of decision rules, ASSURE yields welfare-improving selections while borrowing strength across units. The authors establish minimax-type regret bounds, identify fast-rate scenarios under margin-like conditions, and extend the framework to Poisson cases and complex decisions. Through calibrated simulations and empirical applications in economic mobility, experimentation programs, and discrimination analyses, ASSURE demonstrates robustness to misspecification and practical value for data-driven policy and business decisions.

Abstract

This paper proposes methods for producing compound selection decisions in a Gaussian sequence model. Given unknown, fixed parameters $μ_ {1:n}$ and known $σ_{1:n}$ with observations $Y_i \sim \textsf{N}(μ_i, σ_i^2)$, the decision maker would like to select a subset of indices $S$ so as to maximize utility $\frac{1}{n}\sum_{i\in S} (μ_i - K_i)$, for known costs $K_i$. Inspired by Stein's unbiased risk estimate (SURE), we introduce an almost unbiased estimator, called ASSURE, for the expected utility of a proposed decision rule. ASSURE allows a user to choose a welfare-maximizing rule from a pre-specified class by optimizing the estimated welfare, thereby producing selection decisions that borrow strength across noisy estimates. We show that ASSURE produces decision rules that are asymptotically no worse than the optimal but infeasible decision rule in the pre-specified class. We apply ASSURE to the selection of Census tracts for economic opportunity, the identification of discriminating firms, and the analysis of $p$-value decision procedures in A/B testing.

Compound Selection Decisions: An Almost SURE Approach

TL;DR

This work advances compound decision making under parallel noisy estimates by introducing ASSURE, a SURE-inspired estimator for the welfare of threshold-based selections in a Gaussian setting. By constructing a near-unbiased welfare estimator and optimizing it over a controlled class of decision rules, ASSURE yields welfare-improving selections while borrowing strength across units. The authors establish minimax-type regret bounds, identify fast-rate scenarios under margin-like conditions, and extend the framework to Poisson cases and complex decisions. Through calibrated simulations and empirical applications in economic mobility, experimentation programs, and discrimination analyses, ASSURE demonstrates robustness to misspecification and practical value for data-driven policy and business decisions.

Abstract

This paper proposes methods for producing compound selection decisions in a Gaussian sequence model. Given unknown, fixed parameters and known with observations , the decision maker would like to select a subset of indices so as to maximize utility , for known costs . Inspired by Stein's unbiased risk estimate (SURE), we introduce an almost unbiased estimator, called ASSURE, for the expected utility of a proposed decision rule. ASSURE allows a user to choose a welfare-maximizing rule from a pre-specified class by optimizing the estimated welfare, thereby producing selection decisions that borrow strength across noisy estimates. We show that ASSURE produces decision rules that are asymptotically no worse than the optimal but infeasible decision rule in the pre-specified class. We apply ASSURE to the selection of Census tracts for economic opportunity, the identification of discriminating firms, and the analysis of -value decision procedures in A/B testing.

Paper Structure

This paper contains 42 sections, 41 theorems, 321 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Setup & Methodology
  3. The ASSURE estimator for welfare
  4. Intuition for ASSURE
  5. Discussion
  6. Theoretical assurances
  7. Main regret bound
  8. Examples
  9. Lower bound
  10. Fast Rates for ASSURE
  11. Extensions
  12. Other observation distributions
  13. Complex Decisions
  14. Simulation Results
  15. Calibrated Opportunity Atlas Simulation
  16. ...and 27 more sections

Key Result

Proposition 1

Fix $\mu, K \in \mathbb{R}$. Let $Y \sim \textsf{N}(\mu, \sigma^2)$ and $\delta(Z_i;\beta) = \beta$, then $\lim_{h\to 0}\mathbb{E}_\mu w_h(Y; (\sigma, K), \beta) = (\mu - K) \Phi\left({\frac{\mu - \beta}{\sigma}}\right).$ The bias attains the following bound: For all $h > 0$, Thus, with $h := h_n := \lambda_n^{-1} := 1/\sqrt{2\log n}$,We upper bound bias with $h^2 e^{-Ch^2}$ and variance with $n^

Figures (12)

  • Figure 1: A plot of the function $\frac{y}{2} + \frac{y}{\pi} \mathop{\mathrm{Si}}\limits((y-C)/h) - \frac{1}{h} \mathop{\mathrm{sinc}}\limits((y-C)/h)$, for $C = 1$ and various values of the bandwidth $h$.
  • Figure 2: Semisynthetic simulation comparison on Opportunity Atlas dataset. Box and whisker plots summarize 40 Monte Carlo runs where only $Y_i$ are redrawn. Constants costs are taken with $K = 0.361;$ the rationale behind this choice is discussed in \ref{['sec:oa_application']}.
  • Figure 3: Experimentation program semisynthetic simulation, with $X_i = \mu_i + \sigma_i t_{10}, (\rho = 0.7)$. Boxplots show 100 Monte Carlo comparisons where $Y_i$ is rerandomized.
  • Figure 4: Visualization of Opportunity Atlas estimates from chetty2018opportunitybergman2024creating. Each point represents a census tract within the largest 20 Commuting Zones, measuring the household income rank in adulthood for Black children, with genders pooled, whose parents were at the 25th percentile of income.
  • Figure 5: ASSURE estimate curves for linear shrinkage class with constant costs on the OA dataset of Figure \ref{['fig:oa_viz']}. See text for an explanation of the different cost regimes. Grey lines indicate 1.96 times the standard errors of the ASSURE estimate, calculated pointwise.
  • ...and 7 more figures

Theorems & Definitions (84)

  • Definition 1: ASSURE
  • Proposition 1: ASSURE has low bias
  • Remark 1
  • Theorem 1: $\tilde O(1/\sqrt{n})$-regret for ASSURE
  • Corollary 1
  • Example 1: Simple truncation rules
  • Example 2: Thresholding $t$-statistics
  • Example 3: Finite Classes of Decision Rules
  • Example 4: Gaussian Empirical Bayes Models
  • Theorem 2: Matching Regret Lower Bounds
  • ...and 74 more