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We study the problem of global optimization, where we analyze the performanceof the Piyavskii--Shubert algorithm and its variants. For any given timeduration $T$, instead of the extensively studied simple regret (which is thedifference of the losses between the best estimate up to $T$ and the globalminimum), we study the cumulative regret up to time $T$. For $L$-Lipschitzcontinuous functions, we show that the cumulative regret is $O(L\log T)$. For$H$-Lipschitz smooth functions, we show that the cumulative regret is $O(H)$.We analytically extend our results for functions with Holder continuousderivatives, which cover both the Lipschitz continuous and the Lipschitz smoothfunctions, individually. We further show that a simpler variant of thePiyavskii-Shubert algorithm performs just as well as the traditional variantsfor the Lipschitz continuous or the Lipschitz smooth functions. We furtherextend our results to broader classes of functions, and show that, ouralgorithm efficiently determines its queries; and achieves nearly minimaxoptimal (up to log factors) cumulative regret, for general convex or evenconcave regularity conditions on the extrema of the objective (whichencompasses many preceding regularities). We consider further extensions byinvestigating the performance of the Piyavskii-Shubert variants in thescenarios with unknown regularity, noisy evaluation and multivariate domain.

Cumulative Regret Analysis of the Piyavskii--Shubert Algorithm and Its Variants for Global Optimization