PMBO: Enhancing Black-Box Optimization through Multivariate Polynomial Surrogates

We introduce a surrogate-based black-box optimization method, termed Polynomial-model-based optimization (PMBO). The algorithm alternates polynomial approximation with Bayesian optimization steps, using Gaussian processes to model the error between the objective and its polynomial fit. We describe the algorithmic design of PMBO and compare the results of the performance of PMBO with several optimization methods for a set of analytic test functions. The results show that PMBO outperforms the classic Bayesian optimization and is robust with respect to the choice of its correlation function family and its hyper-parameter setting, which, on the contrary, need to be carefully tuned in classic Bayesian optimization. Remarkably, PMBO performs comparably with state-of-the-art evolutionary algorithms such as the Covariance Matrix Adaptation -- Evolution Strategy (CMA-ES). This finding suggests that PMBO emerges as the pivotal choice among surrogate-based optimization methods when addressing low-dimensional optimization problems. Hereby, the simple nature of polynomials opens the opportunity for interpretation and analysis of the inferred surrogate model, providing a macroscopic perspective on the landscape of the objective function.

Polynomial-Model-Based Optimization for Blackbox Objectives

For a wide range of applications the structure of systems like Neural Networks or complex simulations, is unknown and approximation is costly or even impossible. Black-box optimization seeks to find optimal (hyper-) parameters for these systems such that a pre-defined objective function is minimized. Polynomial-Model-Based Optimization (PMBO) is a novel blackbox optimizer that finds the minimum by fitting a polynomial surrogate to the objective function. Motivated by Bayesian optimization the model is iteratively updated according to the acquisition function Expected Improvement, thus balancing the exploitation and exploration rate and providing an uncertainty estimate of the model. PMBO is benchmarked against other state-of-the-art algorithms for a given set of artificial, analytical functions. PMBO competes successfully with those algorithms and even outperforms all of them in some cases. As the results suggest, we believe PMBO is the pivotal choice for solving blackbox optimization tasks occurring in a wide range of disciplines.