Function Optimization with Posterior Gaussian Derivative Process

TL;DR

This paper introduces a Bayesian optimization algorithm leveraging Gaussian derivative processes to efficiently locate stationary points of functions, with proven convergence and applicability to high-dimensional problems.

Contribution

The paper develops a novel Bayesian optimization method using Gaussian derivative processes, providing theoretical convergence guarantees and demonstrating effectiveness on complex, high-dimensional functions.

Findings

Algorithm converges almost surely to true optima.

Effective in high-dimensional settings up to 100 dimensions.

Provides Bayesian characterization of the number of optima.

Abstract

In this article, we propose and develop a novel Bayesian algorithm for optimization of functions whose first and second partial derivatives are known. The basic premise is the Gaussian process representation of the function which induces a first derivative process that is also Gaussian. The Bayesian posterior solutions of the derivative process set equal to zero, given data consisting of suitable choices of input points in the function domain and their function values, emulate the stationary points of the function, which can be fine-tuned by setting restrictions on the prior in terms of the first and second derivatives of the objective function. These observations motivate us to propose a general and effective algorithm for function optimization that attempts to get closer to the true optima adaptively with in-built iterative stages. We provide theoretical foundation to this algorithm, proving almost sure convergence to the true optima as the number of iterative stages tends to infinity. The theoretical foundation hinges upon our proofs of almost sure uniform convergence of the posteriors associated with Gaussian and Gaussian derivative processes to the underlying function and its derivatives in appropriate fixed-domain infill asymptotics setups; rates of convergence are also available. We also provide Bayesian characterization of the number of optima using information inherent in our optimization algorithm. We illustrate our Bayesian optimization algorithm with five different examples involving maxima, minima, saddle points and even inconclusiveness. Our examples range from simple, one-dimensional problems to challenging 50 and 100-dimensional problems.