Tight Generalization Bounds for Noiseless Inverse Optimization

TL;DR

This paper establishes tight generalization bounds for noiseless inverse optimization, demonstrating that the error rate scales as d/T, and introduces a parameter-free algorithm with validated theoretical guarantees.

Contribution

The paper provides the first tight generalization bounds for noiseless inverse optimization and extends these results to regret analysis, with a new efficient algorithm.

Findings

The generalization error rate is O(d/T), which is tight.

The bounds match adversarial setting lower bounds, indicating the problem's difficulty.

A parameter-free algorithm with lower complexity is proposed and validated.

Abstract

Inverse optimization (IO) seeks to infer the parameters of a decision-maker's objective from observed context--action data. We study noiseless IO, where demonstrations are generated by a ground-truth objective. We provide a high-probability ${O}(\frac{d}{T})$ generalization bound for the induced action set, where $d$ is the number of unknown parameters and $T$ is the size of the training dataset. We strengthen these guarantees under additional conditions that ensure uniqueness of the chosen action, bringing our IO guarantees in line with best-arm identification results in the bandit literature. We further show that the ${O}(\frac{d}{T})$ rate is tight over all consistent estimators considered here, and extend the result to both instantaneous and cumulative regret. Notably, the resulting regret lower bound matches the corresponding upper bounds in the adversarial setting, indicating that the stochastic IO setting is effectively adversarial for the class of estimators studied here. Finally, we propose a parameter-free algorithm with lower per-iteration complexity than generic solvers. Experiments validate the predicted rates and illustrate the tightness of our bounds.