GradientLess Descent

From statwiki
Revision as of 17:23, 31 October 2020 by Jlavilez (talk | contribs)
Jump to navigation Jump to search

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.

Zeroth-Order Optimisation

GradientLess Descent Algorithm

Proof of correctness