Contextual Combinatorial Bandits with Changing Action Sets via Gaussian Processes

TL;DR

This paper introduces a Gaussian Process-based algorithm for combinatorial contextual bandits with changing action sets, achieving low regret and outperforming existing methods through theoretical guarantees and experiments.

Contribution

The paper proposes the O'CLOK-UCB algorithm and its sparse GP variant for combinatorial bandits with dynamic action sets, providing regret bounds and empirical improvements.

Findings

O'CLOK-UCB achieves near-optimal regret bounds.

Sparse GP variant significantly speeds up computation.

Algorithms outperform previous state-of-the-art in experiments.

Abstract

We consider a contextual bandit problem with a combinatorial action set and time-varying base arm availability. At the beginning of each round, the agent observes the set of available base arms and their contexts and then selects an action that is a feasible subset of the set of available base arms to maximize its cumulative reward in the long run. We assume that the mean outcomes of base arms are samples from a Gaussian Process (GP) indexed by the context set ${\cal X}$, and the expected reward is Lipschitz continuous in expected base arm outcomes. For this setup, we propose an algorithm called Optimistic Combinatorial Learning and Optimization with Kernel Upper Confidence Bounds (O'CLOK-UCB) and prove that it incurs $\tilde{O}(\sqrt{\lambda^*(K)KT\gamma_{KT}(\cup_{t\leq T}\mathcal{X}_t)} )$ regret with high probability, where $\gamma_{KT}(\cup_{t\leq T}\mathcal{X}_t)$ is the maximum information gain associated with the sets of base arm contexts $\mathcal{X}_t$ that appeared in the first $T$ rounds, $K$ is the maximum cardinality of any feasible action over all rounds, and $\lambda^*(K)$ is the maximum eigenvalue of all covariance matrices of selected actions up to time $T$, which is a function of $K$. To dramatically speed up the algorithm, we also propose a variant of O'CLOK-UCB that uses sparse GPs. Finally, we experimentally show that both algorithms exploit inter-base arm outcome correlation and vastly outperform the previous state-of-the-art UCB-based algorithms in realistic setups.