How to sample and when to stop sampling: The generalized Wald problem and minimax policies

TL;DR

This paper analyzes optimal sampling and stopping strategies in sequential experiments with costly sampling, establishing minimax policies that maximize expected welfare and are robust across different outcome distribution assumptions.

Contribution

It introduces minimax optimal policies for adaptive sampling and stopping in sequential experiments, applicable under both parametric and non-parametric settings, with a focus on the continuous time limit.

Findings

Neyman allocation is the minimax optimal sampling rule.

Optimal stopping occurs when the product of treatment difference and sample size exceeds a threshold.

Policies are robust to outcome distribution assumptions and applicable to fixed-sample scenarios.

Abstract

We study sequential experiments where sampling is costly and a decision-maker aims to determine the best treatment for full scale implementation by (1) adaptively allocating units between two possible treatments, and (2) stopping the experiment when the expected welfare (inclusive of sampling costs) from implementing the chosen treatment is maximized. Working under a continuous time limit, we characterize the optimal policies under the minimax regret criterion. We show that the same policies also remain optimal under both parametric and non-parametric outcome distributions in an asymptotic regime where sampling costs approach zero. The minimax optimal sampling rule is just the Neyman allocation: it is independent of sampling costs and does not adapt to observed outcomes. The decision-maker halts sampling when the product of the average treatment difference and the number of observations surpasses a specific threshold. The results derived also apply to the so-called best-arm identification problem, where the number of observations is exogenously specified.