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

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 $T^{\Theta(d)}$ 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.

Machine Learning (cs.LG); Data Structures and Algorithms (cs.DS)

arXiv:2609.13440 [cs.LG]

(or arXiv:2609.13440v1 [cs.LG] for this version)

https://doi.org/10.48550/arXiv.2609.13440 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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