Bayesian Optimization under Heavy-tailed Payoffs
This addresses a practical problem for Bayesian optimization practitioners by extending robustness to heavy-tailed noise, though it is incremental as it builds on existing kernel methods.
The paper tackles black-box optimization with heavy-tailed noise, showing that an adapted GP-UCB algorithm achieves sublinear regret under bounded moments, but this is suboptimal compared to a fundamental lower bound; they resolve this by developing new algorithms that match the lower bound for squared exponential kernels, with numerical validation on synthetic and real-world data.
We consider black box optimization of an unknown function in the nonparametric Gaussian process setting when the noise in the observed function values can be heavy tailed. This is in contrast to existing literature that typically assumes sub-Gaussian noise distributions for queries. Under the assumption that the unknown function belongs to the Reproducing Kernel Hilbert Space (RKHS) induced by a kernel, we first show that an adaptation of the well-known GP-UCB algorithm with reward truncation enjoys sublinear $\tilde{O}(T^{\frac{2 + α}{2(1+α)}})$ regret even with only the $(1+α)$-th moments, $α\in (0,1]$, of the reward distribution being bounded ($\tilde{O}$ hides logarithmic factors). However, for the common squared exponential (SE) and Matérn kernels, this is seen to be significantly larger than a fundamental $Ω(T^{\frac{1}{1+α}})$ lower bound on regret. We resolve this gap by developing novel Bayesian optimization algorithms, based on kernel approximation techniques, with regret bounds matching the lower bound in order for the SE kernel. We numerically benchmark the algorithms on environments based on both synthetic models and real-world data sets.