Contextually Adaptive Algorithms for Gaussian Process Bandit Optimization Under Heavy-Tailed Noise

Hyeon-jun Park, Kyungjae Lee

Abstract: We consider a Gaussian process (GP) bandit optimization problem when the objective function lives in a reproducing kernel Hilbert space (RKHS), assuming that the payoffs follow a heavy-tailed distribution with a bounded (1+ϵ)-th moment for some ϵ∈(0,1]. Existing algorithms for this setting face practical challenges due to their significant computational demands and inconsistent theoretical guarantee to translation of noise distribution. To address these issues, we introduce two robust algorithms. The first algorithm utilizes a truncation estimator, achieving the same regret bound as that of the existing algorithm up to logarithmic terms with reduced time complexity. The second algorithm employs a median-of-means estimator and achieves more stable regret bound to alteration of noise distribution with lower time and space complexities compared to existing methods. Finally, we empirically validate the performance of our proposed algorithms against previous methods in both synthetic and real-world datasets.