Tractable Instances of Bilinear Maximization: Implementing LinUCB on Ellipsoids

2025-11-10

AI Summary

This paper addresses a core subproblem in linear bandits: maximizing the bilinear function (x^ op heta) over a convex set (X) and an ellipsoid (Theta). First, the authors prove that the problem is NP-hard when (X) is a (p)-norm ball with (p > 2), unless P = NP. Then, for the structured setting where both (X) and (Theta) are centered ellipsoids, they propose two efficient algorithms grounded in convex optimization and duality theory—exploiting ellipsoidal geometry to achieve (O(d^2)) time complexity. Their approach overcomes the computational bottleneck of optimistic algorithms like LinUCB in high dimensions. Crucially, it yields the first scalable, theoretically sound implementation framework for high-dimensional linear bandits: maintaining the optimal (O(sqrt{T})) regret bound while significantly improving practical runtime efficiency.

Technology Category

• Machine Learning: Online Learning & Bandits
• Search and Optimization: Non-convex Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

We consider the maximization of $x^ op heta$ over $(x, heta) in mathcal{X} imes Theta$, with $mathcal{X} subset mathbb{R}^d$ convex and $Theta subset mathbb{R}^d$ 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.

Problem

• Efficiently solving bilinear maximization over convex sets
• Addressing computational hardness in linear bandit optimization
• Implementing optimistic algorithms for high-dimensional ellipsoids

Innovation

• Efficient algorithms for bilinear maximization on ellipsoids
• Novel methods implementing LinUCB in high dimensions
• Optimistic algorithm solutions for centered ellipsoid sets