Enhancing finite-difference based derivative-free optimization methods with machine learning

TL;DR

This paper introduces a machine learning-enhanced approach to finite-difference derivative-free optimization, using surrogate models trained on collected data to improve convergence and performance of existing methods.

Contribution

It proposes a simple auxiliary procedure that leverages collected data to train surrogate models, enhancing finite-difference DFO methods with iterative improvements.

Findings

Significant performance improvements demonstrated in numerical experiments.

Using approximate gradients to train surrogates yields better optimization results.

The method effectively combines surrogate modeling with traditional derivative-free techniques.

Abstract

Derivative-Free Optimization (DFO) involves methods that rely solely on evaluations of the objective function. One of the earliest strategies for designing DFO methods is to adapt first-order methods by replacing gradients with finite-difference approximations. The execution of such methods generates a rich dataset about the objective function, including iterate points, function values, approximate gradients, and successful step sizes. In this work, we propose a simple auxiliary procedure to leverage this dataset and enhance the performance of finite-difference-based DFO methods. Specifically, our procedure trains a surrogate model using the available data and applies the gradient method with Armijo line search to the surrogate until it fails to ensure sufficient decrease in the true objective function, in which case we revert to the original algorithm and improve our surrogate based on the new available information. As a proof of concept, we integrate this procedure with the derivative-free method proposed in (Optim. Lett. 18: 195--213, 2024). Numerical results demonstrate significant performance improvements, particularly when the approximate gradients are also used to train the surrogates.