Optimization as Estimation with Gaussian Processes in Bandit Settings (Supplement)

In this supplement, we provide proofs for all theorems and lemmas in the main paper, more exhaustive experimental results and details on the experiments. 1 Proofs 1.1 Proofs from Section 2 Lemma 2.1. In any round t, the point selected by EST is the same as the point selected by a variant of GP-UCB with λ t = min x∈Xˆmt−µt−1(x) σt−1(x). Conversely, the candidate selected by GP-UCB is the same as the candidate selected by a variant of EST withˆm t = max x∈X µ t−1 (x) + λ t σ t−1 (x). Proof. We omit the subscripts t for simplicity. Let a be the point selected by GP-UCB, and b selected by EST. Without loss of generality, we assume a and b are unique. With λ = min x∈Xˆm−µ(x) σ(x) , GP-UCB chooses to evaluate a = arg max x∈X µ(x)+λσ(x) = arg min x∈Xˆm − µ(x) σ(x). This is becausê m = max x∈X µ(x) + λσ(x) = µ(a) + λσ(a).