Regret Bounds and Experimental Design for Estimate-then-Optimize

TL;DR

This paper derives a new regret bound for the estimate-then-optimize approach in decision-making, and proposes an experimental design method to minimize regret, demonstrated through examples and a pandemic control case.

Contribution

It introduces a novel regret bound for smooth optimization problems and a general experimental design procedure to reduce regret in estimate-then-optimize methods.

Findings

Derived a bound on regret for estimate-then-optimize.

Proposed an experimental design method to minimize regret.

Validated approach with examples and pandemic control application.

Abstract

In practical applications, data is used to make decisions in two steps: estimation and optimization. First, a machine learning model estimates parameters for a structural model relating decisions to outcomes. Second, a decision is chosen to optimize the structural model's predicted outcome as if its parameters were correctly estimated. Due to its flexibility and simple implementation, this ``estimate-then-optimize'' approach is often used for data-driven decision-making. Errors in the estimation step can lead estimate-then-optimize to sub-optimal decisions that result in regret, i.e., a difference in value between the decision made and the best decision available with knowledge of the structural model's parameters. We provide a novel bound on this regret for smooth and unconstrained optimization problems. Using this bound, in settings where estimated parameters are linear transformations of sub-Gaussian random vectors, we provide a general procedure for experimental design to minimize the regret resulting from estimate-then-optimize. We demonstrate our approach on simple examples and a pandemic control application.