Enhancing Gaussian Process Surrogates for Optimization and Posterior Approximation via Random Exploration
Hwanwoo Kim, Daniel Sanz-Alonso
TL;DR
This work addresses the efficiency gap in noise-free Bayesian optimization by introducing two random-exploration-enhanced algorithms, GP-UCB+ and EXPLOIT+, which improve surrogate accuracy and accelerate convergence while maintaining implementation simplicity. It also extends to Bayesian inference by using optimization iterates to build GP surrogates for unnormalized log-posterior densities, and derives rigorous bounds on the Hellinger distance between true and surrogate posteriors. Theoretical results establish near-optimal simple-regret rates for Matérn kernels and exponential-type bounds for squared-exponential kernels, alongside practical performance gains on high-dimensional benchmarks, hyperparameter tuning, and differential-equation inference. The proposed approach provides a versatile experimental-design framework that yields accurate local posterior behavior around modes without sacrificing global fidelity, with concrete benefits for problems with expensive or intractable likelihoods. Overall, the methods offer a principled, scalable path to faster optimization and reliable posterior approximation in complex scientific and engineering domains.
Abstract
This paper proposes novel noise-free Bayesian optimization strategies that rely on a random exploration step to enhance the accuracy of Gaussian process surrogate models. The new algorithms retain the ease of implementation of the classical GP-UCB algorithm, but the additional random exploration step accelerates their convergence, nearly achieving the optimal convergence rate. Furthermore, to facilitate Bayesian inference with an intractable likelihood, we propose to utilize optimization iterates for maximum a posteriori estimation to build a Gaussian process surrogate model for the unnormalized log-posterior density. We provide bounds for the Hellinger distance between the true and the approximate posterior distributions in terms of the number of design points. We demonstrate the effectiveness of our Bayesian optimization algorithms in non-convex benchmark objective functions, in a machine learning hyperparameter tuning problem, and in a black-box engineering design problem. The effectiveness of our posterior approximation approach is demonstrated in two Bayesian inference problems for parameters of dynamical systems.