Tight Generalization Bounds for Noiseless Inverse Optimization

2026-05-09

AI Summary

This study addresses the noiseless inverse optimization problem, aiming to accurately infer the parameters of a decision-maker’s objective function from observed context–action pairs and to establish theoretical guarantees on its generalization performance. By integrating statistical learning theory with an analogy to best-arm identification, the work derives, for the first time, a tight high-probability generalization upper bound of $O(d/T)$ and proves that this rate is a fundamental lower bound for all consistent estimators. These findings reveal that, under this setting, stochastic inverse optimization inherently exhibits adversarial characteristics. The paper further proposes a parameter-free, computationally efficient algorithm and validates both the theoretical bounds and the predicted convergence rate through empirical experiments.

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.

Problem

• inverse optimization
• generalization bounds
• noiseless setting
• regret analysis
• parameter estimation

Innovation

• inverse optimization
• generalization bound
• tight rate
• regret analysis
• parameter-free algorithm