# A Theoretical and Empirical Comparison of Gradient Approximations in Derivative-Free Optimization

## TL;DR

This paper compares various gradient approximation methods for noisy functions in derivative-free optimization, analyzing their theoretical bounds and empirical performance to guide effective method selection.

## Contribution

It provides a comprehensive theoretical analysis and empirical evaluation of multiple gradient approximation techniques, highlighting their convergence guarantees and practical effectiveness.

## Key findings

- Finite difference and Gaussian smoothing methods offer reliable gradient estimates.
- Sampling strategies significantly impact convergence speed and accuracy.
- Numerical experiments demonstrate the practical performance differences among methods.

## Abstract

In this paper, we analyze several methods for approximating gradients of noisy functions using only function values. These methods include finite differences, linear interpolation, Gaussian smoothing and smoothing on a sphere. The methods differ in the number of functions sampled, the choice of the sample points, and the way in which the gradient approximations are derived. For each method, we derive bounds on the number of samples and the sampling radius which guarantee favorable convergence properties for a line search or fixed step size descent method. To this end, we use the results in [Berahas et al., 2019] and show how each method can satisfy the sufficient conditions, possibly only with some sufficiently large probability at each iteration, as happens to be the case with Gaussian smoothing and smoothing on a sphere. Finally, we present numerical results evaluating the quality of the gradient approximations as well as their performance in conjunction with a line search derivative-free optimization algorithm.