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

TL;DR

This work studies online inverse linear optimization with M-convex action sets, establishing a finite regret bound that scales polynomially in the dimension. By combining a structural characterization of M-convex optima with a geometric volume argument, the authors show an O(d log d) regret bound under uncorrupted feedback and extend it to corruption-robust settings with O((C+1)d log d) regret without knowing C in advance, via a restart mechanism that monitors acyclicity in a feedback-derived graph. A matching Omega(d) lower bound is adapted to the M-convex case, indicating the main bound is near-tight. The results apply to broad combinatorial action sets, including matroids and lattice-extensions, offering practical regret guarantees for contextual recommendation in online settings.

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 $Ω(d)$ 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.

Figures (2)

• Figure 1: The decomposition of $[0,1]^3$ into six order simplices. The order simplex for $w(1) > w(2) > w(3)$ is highlighted in red.
• Figure 2: Examples of directed graphs $([d], A_{t+1})$ in uncorrupted and corrupted sequences; note that the graph for $A_1 = \emptyset$ is omitted. Let $d = 3$ and $w^* \in \mathbb{R}^3$ satisfy $w^*(1) > w^*(2) > w^*(3)$. Feasible sets $X_t$ are given in the top row. In the uncorrupted case, $A_{t+1}$ remains acyclic for all $t$. In the corrupted case, $x_2$ (shown in red) is corrupted, adding a wrong arc $3 \to 2$ to $A_3$. Still, $A_3$ remains acyclic and does not yet indicate the corruption. This does not affect the choice from $X_3 = \{\mathbf{e}_1, \mathbf{e}_2\}$ in the next round, and we correctly select $\hat{x}_3 = x_3 = \mathbf{e}_1$. Later, when $x_4$ is observed, the correct arc $2 \to 3$ is added, creating a cycle $2 \to 3 \to 2$ in $A_5$, revealing the corruption.

Theorems & Definitions (17)

• Remark 2.1: On the regret metric
• Definition 2.3: Murota2003-bq
• Proposition 2.4: Murota2003-bq
• Theorem 3.1
• proof
• Lemma 4.1
• proof
• Theorem 4.2
• proof
• Lemma 5.1