Epsilon-Minimax Solutions of Statistical Decision Problems

TL;DR

The paper tackles the computational challenge of epsilon-minimax statistical decision problems when a decision maker randomizes over a finite menu of rules. It recasts the minimax objective as a convex program over the $(I-1)$-simplex and proposes a Hedge/mirror-descent algorithm with an explicit iteration bound $T=\lceil 2 M^2 \ln(I)/\epsilon^2 \rceil$ to obtain an $\epsilon$-minimax rule, with guarantees that the averaged iterate achieves the target accuracy. The authors connect the method to online learning and two-player zero-sum games, show optimality up to a logarithmic factor, and demonstrate its utility through illustrative examples with partial identification and a concrete site-selection application to maximize external validity. The empirical application reveals that optimal randomization over site selections systematically weighs sites by covariate-based representativeness rather than choosing uniformly at random, offering practical guidance for policy evaluation campaigns. Overall, the work provides a scalable, theoretically grounded approach to near-optimal randomized decision rules in econometric minimax problems.

Abstract

A decision rule is epsilon-minimax if it is minimax up to an additive factor epsilon. We present an algorithm for provably obtaining epsilon-minimax solutions for a class of statistical decision problems. In particular, we are interested in problems where the statistician chooses randomly among I decision rules. The minimax solution of these problems admits a convex programming representation over the (I-1)-simplex. Our suggested algorithm is a well-known mirror subgradient descent routine, designed to approximately solve the convex optimization problem that defines the minimax decision rule. This iterative routine is known in the computer science literature as the hedge algorithm and is used in algorithmic game theory as a practical tool to find approximate solutions of two-person zero-sum games. We apply the suggested algorithm to different minimax problems in the econometrics literature. An empirical application to the problem of optimally selecting sites to maximize the external validity of an experimental policy evaluation illustrates the usefulness of the suggested procedure.

Figures (4)

• Figure 1: $\epsilon$-Minimax Decision Rule via the Hedge algorithm. The graph is generated using $\sigma=1$, $k=2$. The value of $\rho^*$ in Equation \ref{['eq:rho.star.main.proof']} is 1.8797.
• Figure 2: $\epsilon$-Minimax Decision Rule for the 2-point Robust Bayes problem via the Hedge algorithm. The graph is generated using $\sigma=1$, $k=2$. The $\rho^\star$-adjusted value is about $1.8486$.
• Figure 3: Distances from each of the experimental sites to each of the policy-relevant sites.
• Figure 4: $\epsilon$-Minimax decision rule for the Site Selection Problem via the Hedge algorithm. The graph is generated using $C=1.1834$, $\sigma = 4.5$, and $\epsilon=.1$.

Theorems & Definitions (15)

• Lemma 1
• proof
• Definition 1
• Theorem 1
• proof
• Remark 1: Optimality of Algorithm \ref{['alg:mirror_descent']}
• Remark 2: Approximate Least-favorable distribution
• Remark 3: Approximate evaluation of $f(p)$
• Remark 4: Minimax problems with infinitely many decision rules
• Remark 5: Stopping the algorithm before our suggested $T$ epochs