Efficient Online Inverse Optimization with $O(d)$ Regret

Yang Cai; Anupam Gupta; Vineet Gupta; Guru Guruganesh; Yanchen Jiang; Christopher Liaw; Aranyak Mehta; Renato Paes Leme; Grigoris Velegkas; Di Wang

arXiv · 2609.13440

Efficient Online Inverse Optimization with $O(d)$ Regret

Yang Cai·Anupam Gupta·Vineet Gupta·Guru Guruganesh·Yanchen Jiang·Christopher Liaw·Aranyak Mehta·Renato Paes Leme·Grigoris Velegkas·Di Wang

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Abstract

We give a deterministic algorithm for online inverse linear optimization with regret $O(d)$, uniform in the horizon and $O(d^{2})$ time per round. A bound of this order was obtained recently by Dewasurendra, settling a question of Gollapudi et al.\ and of Oki and Sakaue, but by an improper rule that enumerates covers at every scale and costs $T^{Θ(d)}$ a round; ours is the first efficient such bound and the first proper one. We build on the variable-metric framework of Sakaue et al., adding a self-normalized rank-one update, and we replace the $\log\det$ potential by the trace power $\tr(H^{-1/2})$, which is bounded outright and removes the $\ln T$. The bound also holds against an expert that does not optimize, and we give corruption-robust and rank-adaptive variants, and an application to convex minimization.

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Yang Cai, Anupam Gupta, Vineet Gupta, Guru Guruganesh, Yanchen Jiang, Christopher Liaw, Aranyak Mehta, Renato Paes Leme, Grigoris Velegkas, Di Wang. 2026-09-11. Efficient Online Inverse Optimization with $O(d)$ Regret. https://arxiv.org/abs/2609.13440

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