Tractable Instances of Bilinear Maximization: Implementing LinUCB on Ellipsoids

• URL: http://arxiv.org/abs/2511.07504v1
• Date: Wed, 12 Nov 2025 01:01:55 GMT
• Title: Tractable Instances of Bilinear Maximization: Implementing LinUCB on Ellipsoids
• Authors: Raymond Zhang, Hédi Hadiji, Richard Combes,
• Abstract summary: We show that for some sets $mathcalX$ e.g. $ell_p$ balls with $p>2$, no efficient algorithms exist unless $mathcalP = mathcalNP$.<n>We provide two novel algorithms solving this problem efficiently when $mathcalX$ is a centered ellipsoid.
• Score: 12.091821597708337
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We consider the maximization of $x^\top θ$ over $(x,θ) \in \mathcal{X} \times Θ$, with $\mathcal{X} \subset \mathbb{R}^d$ convex and $Θ\subset \mathbb{R}^d$ an ellipsoid. This problem is fundamental in linear bandits, as the learner must solve it at every time step using optimistic algorithms. We first show that for some sets $\mathcal{X}$ e.g. $\ell_p$ balls with $p>2$, no efficient algorithms exist unless $\mathcal{P} = \mathcal{NP}$. We then provide two novel algorithms solving this problem efficiently when $\mathcal{X}$ is a centered ellipsoid. Our findings provide the first known method to implement optimistic algorithms for linear bandits in high dimensions.

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