# Online Inverse Integer Linear Optimization via Small-Gradient Skipping: Constant Regret and Finite Mistakes

arXiv: https://arxiv.org/abs/2609.09809 · HTML: https://mlsanity.com/paper/2609.09809/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 34 references that are in the map.
- First posted on arXiv 9 Sep 2026.

## Builds on (most cited first, 11 of 11)

- [Introduction to Online Convex Optimization](https://mlsanity.com/paper/1909.05207/) (2019)
- [Online Learning: A Modern Introduction Using Convex Optimization](https://mlsanity.com/paper/1912.13213/) (2019)
- [An Online-Learning Approach to Inverse Optimization](https://mlsanity.com/paper/1810.12997/) (2018)
- [Online Inverse Linear Optimization: Efficient Logarithmic-Regret Algorithm, Robustness to Suboptimality, and Lower Bound](https://mlsanity.com/paper/2501.14349/) (2025)
- [Revisiting Online Learning Approach to Inverse Linear Optimization: A Fenchel-Young Loss Perspective and Gap-Dependent Regret Analysis](https://mlsanity.com/paper/2501.13648/) (2025)
- [Finite and Corruption-Robust Regret Bounds in Online Inverse Linear Optimization under M-Convex Action Sets](https://mlsanity.com/paper/2602.01682/) (2026)
- [Simple Projection-Free Algorithm for Contextual Recommendation with Logarithmic Regret and Robustness](https://mlsanity.com/paper/2603.20826/) (2026)
- [Tight Generalization Bounds for Noiseless Inverse Optimization](https://mlsanity.com/paper/2605.08866/) (2026)
- [Multiscale Reward Hedging from Correct Demonstrations](https://mlsanity.com/paper/2608.06825/) (2026)
- [Exact Solution to Data-Driven Inverse Optimization of MILPs in Finite Time via Gradient-Based Methods](https://mlsanity.com/paper/2405.14273/) (2024)
- [Explicit Iteration Complexity of Exact Data-Driven Inverse Optimization for Integer Linear Programs](https://mlsanity.com/paper/2607.22263/) (2026)

## Built on by (most cited first, 2 of 2)

- [Tight Regret Bound for Online Inverse Linear Optimization via Multiscale Matrix Weights](https://mlsanity.com/paper/2609.26978/) (2026)
- [Optimal and Efficient Online Inverse Optimization](https://mlsanity.com/paper/2610.08735/) (2026)

## Abstract (arXiv)

In online inverse linear optimization, the learner predicts a weight at each round, observes the optimal action of the agent, and updates its prediction. In the general setting, the gap of log T between the regret upper bound O(d log T) and the lower bound Omega(d) is unresolved (here T is the total number of rounds and d is the dimension). When the action set is M-convex, the regret is known to be bounded by O(d log d), but the method attaining it computes a center of gravity at every round. This paper therefore proposes Small-Gradient Skipping (SGS), a mechanism that skips the update at rounds without a mistake, and applies it to online gradient descent, the online Newton step, and MetaGrad. When the forward problem is an integer linear program with a unique optimal solution, the number of mistakes is bounded, for all three, by a quantity independent of T; and for the online Newton step and for MetaGrad with SGS, the dimension dependence of the regret becomes O(d2), that is, the factor log T is removed. Moreover, when the action set is M-convex, the regret is bounded efficiently without computing a center of gravity.