Online Inverse Integer Linear Optimization via Small-Gradient Skipping: Constant Regret and Finite Mistakes
Abstract
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 Ω(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.
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BibTeX
@misc{kitaoka2026online,
title = {{Online Inverse Integer Linear Optimization via Small-Gradient Skipping: Constant Regret and Finite Mistakes}},
author = {Akira Kitaoka},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.09809},
eprint = {2609.09809},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.09809},
}
RIS
TY - UNPB
TI - Online Inverse Integer Linear Optimization via Small-Gradient Skipping: Constant Regret and Finite Mistakes
AU - Akira Kitaoka
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2609.09809
UR - https://arxiv.org/abs/2609.09809
AB - 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 Ω(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(d^2), 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.
ER -