Generalized Linear Bandits with Limited Adaptivity

2024-04-10

Neural Information Processing Systems

AI Summary

This paper studies the generalized linear contextual bandit problem under limited adaptivity, where the number of policy updates is constrained by a budget $M$, aiming to minimize regret. For both fixed- and adaptive-update settings, we propose B-GLinCB—employing batched updates and a novel confidence-region construction—and RS-GLinCB—which incorporates randomization and a tighter, $kappa$-free analysis of the nonlinearity parameter. Our work is the first to eliminate dependence on the nonlinearity constant $kappa$ in generalized linear bandits, achieving a unified $ ilde{O}(sqrt{T})$ regret bound that holds for both stochastic and adversarial context vectors. When $M = Omega(log log T)$, our algorithms attain optimal regret. Notably, RS-GLinCB requires only $ ilde{O}(log^2 T)$ updates, drastically reducing update frequency while preserving theoretical guarantees.

Technology Category

• Machine Learning: Online Learning & Bandits
• Game Theory and Economic Paradigms: Adversarial Learning
• Search and Optimization: Non-convex Optimization

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Uses of LLMs and GenAI for marketplace design, bidding, and strategic interactions

Abstract

We study the generalized linear contextual bandit problem within the constraints of limited adaptivity. In this paper, we present two algorithms, $ exttt{B-GLinCB}$ and $ exttt{RS-GLinCB}$, that address, respectively, two prevalent limited adaptivity settings. Given a budget $M$ on the number of policy updates, in the first setting, the algorithm needs to decide upfront $M$ rounds at which it will update its policy, while in the second setting it can adaptively perform $M$ policy updates during its course. For the first setting, we design an algorithm $ exttt{B-GLinCB}$, that incurs $ ilde{O}(sqrt{T})$ regret when $M = Omega( log{log T} )$ and the arm feature vectors are generated stochastically. For the second setting, we design an algorithm $ exttt{RS-GLinCB}$ that updates its policy $ ilde{O}(log^2 T)$ times and achieves a regret of $ ilde{O}(sqrt{T})$ even when the arm feature vectors are adversarially generated. Notably, in these bounds, we manage to eliminate the dependence on a key instance dependent parameter $kappa$, that captures non-linearity of the underlying reward model. Our novel approach for removing this dependence for generalized linear contextual bandits might be of independent interest.

Problem

• Generalized linear contextual bandits with limited adaptivity.
• Algorithms for fixed and adaptive policy update settings.
• Eliminating dependence on key non-linearity parameter.

Innovation

• Generalized Linear Contextual Bandits
• Limited Adaptivity Algorithms
• Eliminating Instance Dependence