Tucker Hartland

Terrence Alsup, Tucker Hartland, Benjamin Peherstorfer et al.

Multilevel Stein variational gradient descent is a method for particle-based variational inference that leverages hierarchies of surrogate target distributions with varying costs and fidelity to computationally speed up inference. The contribution of this work is twofold. First, an extension of a previous cost complexity analysis is presented that applies even when the exponential convergence rate of single-level Stein variational gradient descent depends on iteration-varying parameters. Second, multilevel Stein variational gradient descent is applied to a large-scale Bayesian inverse problem of inferring discretized basal sliding coefficient fields of the Arolla glacier ice. The numerical experiments demonstrate that the multilevel version achieves orders of magnitude speedups compared to its single-level version.

Jingyi Wang, Haowei Wang, Nai-Yuan Chiang et al.

Efficient global optimization (EGO) is one of the most widely used noise-free Bayesian optimization algorithms.It comprises the Gaussian process (GP) surrogate model and expected improvement (EI) acquisition function. In practice, when EGO is applied, a scalar matrix of a small positive value (also called a nugget or jitter) is usually added to the covariance matrix of the deterministic GP to improve numerical stability. We refer to this EGO with a positive nugget as the practical EGO. Despite its wide adoption and empirical success, to date, cumulative regret bounds for practical EGO have yet to be established. In this paper, we present for the first time the cumulative regret upper bound of practical EGO. In particular, we show that practical EGO has sublinear cumulative regret bounds and thus is a no-regret algorithm for commonly used kernels including the squared exponential (SE) and Matérn kernels ($ν>\frac{1}{2}$). Moreover, we analyze the effect of the nugget on the regret bound and discuss the theoretical implication on its choice. Numerical experiments are conducted to support and validate our findings.