Tight Regret Bound for Online Inverse Linear Optimization via Multiscale Matrix Weights
AuthorsShinsaku Sakaue
AffiliationsSep 2026
Resources
A new multiscale matrix-weights algorithm learns hidden linear preferences online with provably optimal dimension-dependent regret.
Key results
Explicit constant multiplying √d, valid for every horizon without horizon knowledge
Matches the Ω(√d) lower bound for horizons T≥d
Fixed matrix-weight update rate η=2^-9
Per-round linear-optimization oracle bound for the rational implementation
What the paper found
This paper studies online inverse linear optimization when a learner recommends an action, observes only an action that maximizes an unknown fixed linear utility, and receives no utility values. For arbitrary adaptively chosen compact action sets in the d-dimensional Euclidean unit ball, it introduces an improper randomized learner with expected regret at most 2097152√d for every horizon, without knowing the horizon. This matches the Ω(√d) lower bound, improving the previous O(d) dimension dependence and remaining valid with ties among optimal feedback actions. The method combines matrix multiplicative weights across geometrically spaced accuracy scales: at scale k it builds antisymmetric comparison matrices on polynomial feature spaces of degree 2·4^k+2, chooses recommendation probabilities by a linear program enforcing skew-symmetric balance, and updates score matrices using a fixed learning rate of 0.001953125. The scale schedule K_t=⌈2 log2(t+1)⌉ makes truncation errors summable, while Golden–Thompson potential analysis yields the dimension-optimal rate. A rational-oracle implementation terminates each round and preserves the guarantee, but can require (dT)^O(d) linear-optimization oracle calls, so polynomial-time implementation remains open; robustness to suboptimal feedback is also unresolved. The paper reports using GPT-6 Astra in ChatGPT, OpenAI Codex, and Claude Fable 5.1 for presentation assistance, not for the core algorithmic guarantee.
Original abstract
We study online inverse linear optimization with a fixed unknown linear utility: in each round, an environment presents a compact action set, the learner recommends an action from it, and the environment returns an action that maximizes the utility over the same set. When the utility vector and the actions lie in the $d$-dimensional Euclidean unit ball, we give a randomized algorithm whose regret---the cumulative utility shortfall relative to optimal actions---is $O(\sqrt d)$ in expectation for every time horizon, without knowledge of the horizon. The dependence on $d$ is optimal up to a constant factor by the known $Ω(\sqrt d)$ lower bound for horizons $T\ge d$. Our algorithm maintains matrix multiplicative weights on polynomial feature spaces at geometrically spaced scales. It selects a recommendation distribution by solving a linear program and updates its score matrices by comparing the available actions with the feedback action. With rational oracle outputs and feedback actions, an implementation computable relative to a linear-optimization oracle preserves the $O(\sqrt d)$ regret bound. Whether the same rate is attainable with running time polynomial in the dimension, horizon, and input length remains open.
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