GradientLess Descent

Introduction

Motivation and Set-up

A general optimisation question can be formulated by asking to minimise an objective function [math]\displaystyle{ f : \mathbb{R}^n \to \mathbb{R} }[/math], which means finding: \begin{align*} x^* = \mathrm{argmin}_{x \in \mathbb{R}^n} f(x) \end{align*}

Depending on the nature of [math]\displaystyle{ f }[/math], different settings may be considered:

• Convex vs non-convex objective functions;
• Differentiable vs non-differentiable objective functions;
• Allowed function or gradient computations;
• Noisy/Stochastic oracle access.

For the purpose of this paper, we consider convex smooth objective noiseless functions, where we have access to function computations but not gradient computations. This class of functions is quite common in practice; for instance, they make special appearances in the reinforcement learning literature.