Contextual Combinatorial Bandits with Changing Action Sets via Gaussian Processes
This addresses the challenge of efficient decision-making in dynamic environments like online advertising or recommendation systems, though it is incremental as it builds on existing bandit and GP frameworks.
The paper tackles the problem of contextual combinatorial bandits with time-varying action sets by proposing an algorithm called O'CLOK-UCB that uses Gaussian Processes, achieving a regret bound of Õ(√(λ*(K)KTγ_KT(∪_{t≤T}ℱ_t))) and showing experimental improvements over prior UCB-based methods.
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 $\tilde{O}(\sqrt{λ^*(K)KTγ_{KT}(\cup_{t\leq T}\mathcal{X}_t)} )$ regret with high probability, where $γ_{KT}(\cup_{t\leq T}\mathcal{X}_t)$ 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 $λ^*(K)$ 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.