Finite and Corruption-Robust Regret Bounds in Online Inverse Linear Optimization under M-Convex Action Sets

TL;DR

This paper establishes finite, dimension-dependent regret bounds for online inverse linear optimization over M-convex sets, extending to adversarial corruption scenarios with adaptive detection.

Contribution

It proves that finite regret bounds of order O(d log d) are achievable for M-convex feasible sets, resolving an open question in the field.

Findings

Finite regret bound of O(d log d) for M-convex sets.

Extension to adversarially corrupted feedback with regret O((C+1)d log d).

Adaptive corruption detection without prior knowledge of C.

Abstract

We study online inverse linear optimization, also known as contextual recommendation, where a learner sequentially infers an agent's hidden objective vector from observed optimal actions over feasible sets that change over time. The learner aims to recommend actions that perform well under the agent's true objective, and the performance is measured by the regret, defined as the cumulative gap between the agent's optimal values and those achieved by the learner's recommended actions. Prior work has established a regret bound of $O(d\log T)$, as well as a finite but exponentially large bound of $\exp(O(d\log d))$, where $d$ is the dimension of the optimization problem and $T$ is the time horizon, while a regret lower bound of $\Omega(d)$ is known (Gollapudi et al. 2021; Sakaue et al. 2025). Whether a finite regret bound polynomial in $d$ is achievable or not has remained an open question. We partially resolve this by showing that when the feasible sets are M-convex -- a broad class that includes matroids -- a finite regret bound of $O(d\log d)$ is possible. We achieve this by combining a structural characterization of optimal solutions on M-convex sets with a geometric volume argument. Moreover, we extend our approach to adversarially corrupted feedback in up to $C$ rounds. We obtain a regret bound of $O((C+1)d\log d)$ without prior knowledge of $C$, by monitoring directed graphs induced by the observed feedback to detect corruptions adaptively.