Confidence bounds are an essential tool for rigorously quantifying the uncertainty of predictions. In this capacity, they can inform the exploration-exploitation trade-off and form a core component in many sequential learning and decision-making algorithms. Tighter confidence bounds give rise to algorithms with better empirical performance and better performance guarantees. In this work, we use martingale tail bounds and finite-dimensional reformulations of infinite-dimensional convex programs to establish new confidence bounds for sequential kernel regression. We prove that our new confidence bounds are always tighter than existing ones in this setting. We apply our confidence bounds to the kernel bandit problem, where future actions depend on the previous history. When our confidence bounds replace existing ones, the KernelUCB (GP-UCB) algorithm has better empirical performance, a matching worst-case performance guarantee and comparable computational cost. Our new confidence bounds can be used as a generic tool to design improved algorithms for other kernelised learning and decision-making problems.

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