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

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.

• The paper establishes an O(d log d) regret bound independent of the time horizon for uncorrupted online inverse linear optimization over M-convex action sets, improving prior exponential finite bounds.
• The proposed method combines M-convex exchange optimality with a center-of-gravity volume argument, while a simpler topological-sorting algorithm achieves O(d²) regret with lower computational demands.
• The paper extends the guarantee to adversarial corruption in C rounds, achieving O((C+1)d log d) regret through cycle detection and adaptive restarts, nearly matching the Ω(d) lower bound.

Problem setting and motivation

The paper studies online inverse linear optimization, also known as contextual recommendation, where a learner sequentially observes an agent's optimal actions $$x_t \in \arg\max_{x \in X_t} \langle w^*, x \rangle$$ over time-varying feasible sets $X_t$, and must recommend actions that perform well under the agent's hidden objective vector $w^*$. Regret is the cumulative gap $\sum_t \langle w^*, x_t - \hat x_t \rangle$ between the agent's optimal values and those achieved under the learner's estimates $\hat w_t$.

Prior work left a conspicuous gap: known upper bounds either depend on $T$ (e.g., $O(d \log T)$) or are finite but exponentially large ($\exp(O(d \log d))$), while a lower bound of $\Omega(d)$ holds (2602.01682). Whether a finite regret bound polynomial in $d$ is achievable was open. This paper resolves it for a broad and natural class of action sets: M-convex sets, which subsume matroid bases (uniform matroids for $X_t$0-sets, graphic matroids for spanning trees) and their integer-lattice extensions such as bounded-count allocations $X_t$1.

The motivation for restricting to M-convex sets is structural rather than arbitrary: when feasible sets are curved (e.g., ellipsoids), a single observation pins down the direction of $X_t$2 via its normal cone, making the problem trivial; polyhedral combinatorial sets are the genuinely hard case, and M-convexity supplies enough exchange structure to make progress while covering many practical problems.

Warm-up: an $X_t$3-regret algorithm

The key structural tool is Murota's characterization of optimality on M-convex sets: $X_t$4 maximizes $X_t$5 over an M-convex $X_t$6 if and only if $X_t$7 for every feasible unit exchange $X_t$8 (2602.01682). The algorithm maintains a set $X_t$9 of index pairs $w^*$0 such that $w^*$1 is implied by past observations, chooses $w^*$2 consistent with all pairs in $w^*$3, and augments $w^*$4 whenever the observed action reveals a new ordering constraint.

Two facts drive the analysis. First, the directed graph $w^*$5 is always acyclic—otherwise $w^*$6 would satisfy a strict cyclic chain of inequalities—so a consistent $w^*$7 always exists via topological sort. Second, by the optimality characterization, if no new pair is added at round $w^*$8 (i.e., $w^*$9), then $\sum_t \langle w^*, x_t - \hat x_t \rangle$0 is exactly the unique maximizer under $\sum_t \langle w^*, x_t - \hat x_t \rangle$1 (uniqueness follows from distinct components plus the M-convex exchange property). Hence nonzero regret occurs only when a new pair is learned, which can happen at most $\sum_t \langle w^*, x_t - \hat x_t \rangle$2 times, yielding $\sum_t \langle w^*, x_t - \hat x_t \rangle$3 (2602.01682).

This already improves on the exponential finite bound of prior work, but the choice of $\sum_t \langle w^*, x_t - \hat x_t \rangle$4 is still arbitrary; refining that choice yields the main result.

The $\sum_t \langle w^*, x_t - \hat x_t \rangle$5 bound via a volume argument

The refined algorithm takes $\sum_t \langle w^*, x_t - \hat x_t \rangle$6 as the center of gravity of the order-constrained polytope

$\sum_t \langle w^*, x_t - \hat x_t \rangle$7

with ties broken by arbitrarily small perturbation. Whenever a mistake occurs ($\sum_t \langle w^*, x_t - \hat x_t \rangle$8), some newly added pair $\sum_t \langle w^*, x_t - \hat x_t \rangle$9 satisfies $\hat w_t$0; since $\hat w_t$1 lies outside the halfspace $\hat w_t$2, Grünbaum's theorem implies $\hat w_t$3 (2602.01682).

To bound the total number of mistakes, the paper uses the Freudenthal triangulation: $\hat w_t$4 decomposes into $\hat w_t$5 equal-volume order simplices, and because all constraints in $\hat w_t$6 are consistent with the true ordering of $\hat w_t$7's components, $\hat w_t$8 contains at least one full order simplex. Thus at most $\hat w_t$9 mistakes occur, giving:

Main result: under M-convex action sets and uncorrupted feedback, regret $T$0, independent of $T$1 (2602.01682).

A practical caveat concerns computation: exact center-of-gravity computation is #P-hard even for order simplices, so the algorithm requires randomized approximation of the center of gravity in polynomial time; the volume analysis tolerates this via approximate versions of Grünbaum's theorem. The simpler $T$2 variant needs only topological sorting, costing $T$3 per round.

Corruption-robust extension

The paper then relaxes the assumption that observed actions are optimal, allowing adversarial corruption in up to $T$4 rounds (the standard corruption model from contextual search/pricing), without assuming the learner knows $T$5. The regret definition uses actual actions; the alternative definition based on unobserved optimal actions differs by only constant factors, since $T$6 and $T$7 in the worst case.

The mechanism rests on two lemmas about the directed graph induced by feedback over any interval $T$8 (2602.01682):

• Cycle implies corruption: if the graph built from arc sets $T$9 over the interval contains a directed cycle, at least one action in the interval is suboptimal.
• Acyclicity plus no new arcs implies zero regret: if the interval graph is acyclic and no new arcs appear at round $O(d \log T)$0, then $O(d \log T)$1 regardless of earlier corruptions.

The algorithm runs the uncorrupted procedure and restarts (resetting $O(d \log T)$2) whenever a cycle appears. Each inter-restart epoch accumulates at most $O(d \log T)$3 regret by the same volume argument—the volume-decrease lemma does not require optimality of observed actions—and each cycle certifies at least one corrupted round, so there are at most $O(d \log T)$4 restarts. The resulting guarantee is $O(d \log T)$5, recovering the clean bound at $O(d \log T)$6 and requiring no prior knowledge of $O(d \log T)$7 (2602.01682). Cycle detection adds only $O(d \log T)$8 per-round overhead via DFS or topological sort.

Compared with Gupta et al.'s parallel-copies framework for contextual pricing, which also incurs multiplicative $O(d \log T)$9 degradation, this approach instead exploits M-convex structure for detection and uses restarts rather than maintaining multiple algorithm copies. One limitation the authors note explicitly: Sakaue et al.'s more flexible corruption model, measured by cumulative suboptimality rather than round counts, is not covered; extending the finite-regret approach to that framework remains open.

Lower bound

The paper adapts the $\exp(O(d \log d))$0 lower-bound construction of Sakaue et al., which uses axis-aligned line-segment feasible sets, to the M-convex setting (2602.01682). The segment instances are not M-convex but are M$\exp(O(d \log d))$1-convex (box-type sets); since any M$\exp(O(d \log d))$2-convex set in $\exp(O(d \log d))$3 embeds as an M-convex set in $\exp(O(d \log d))$4, the lower bound carries over at dimension $\exp(O(d \log d))$5. Consequently, the $\exp(O(d \log d))$6 upper bound is tight up to a logarithmic factor, and the gap between the bounds is now only $\exp(O(d \log d))$7.

Limitations and open questions

Several caveats qualify the results. The $\exp(O(d \log d))$8 guarantee relies on randomized approximate center-of-gravity computation due to #P-hardness of the exact quantity. The corruption guarantee counts corrupted rounds rather than cumulative suboptimality, matching the weaker of the two models in the literature. The boundedness assumption (per-round regret $\exp(O(d \log d))$9) and the distinct-components assumption on $\Omega(d)$0 are standard but simplifying; the latter is handled via perturbation arguments justified by Lipschitz continuity of regret. Open questions identified by the authors include closing the $\Omega(d)$1 gap, determining the tight dependence on $\Omega(d)$2, achieving corruption-robust bounds under cumulative-suboptimality measures, and understanding what additional structure on $\Omega(d)$3 and the action sets permits learning faster than the $\Omega(d)$4 barrier—a question connected to preference-feedback settings, which two-action M-convex sets model as a special case.

Conclusion

This paper establishes that M-convexity of the action sets suffices for finite, dimension-polynomial regret in online inverse linear optimization: $\Omega(d)$5 without corruptions and $\Omega(d)$6 with adversarial corruptions detected adaptively, against an $\Omega(d)$7 lower bound. The technical contribution is an overview of Murota's local-exchange optimality characterization with a Grünbaum-style volume argument over order simplices, extended to corruptions through acyclicity monitoring of feedback-induced directed graphs. The results partially close a question open since Gollapudi et al. and Sakaue et al., and position discrete convexity as a productive structural assumption for online inverse optimization.