An Algebraically Converging Stochastic Gradient Descent Algorithm for Global Optimization

TL;DR

This paper introduces a novel stochastic gradient descent algorithm with adaptive, state-dependent randomness that guarantees algebraic convergence to global optima in nonconvex problems, supported by theoretical proofs and numerical tests.

Contribution

It presents a new gradient descent method with adaptive stochasticity and proves its global convergence with algebraic rate, improving upon classical approaches.

Findings

Proven algebraic convergence rate in probability and parameter space.

Algorithm demonstrates robustness and efficiency in numerical experiments.

Adaptive, state-dependent noise improves global optimization performance.

Abstract

We propose a new gradient descent algorithm with added stochastic terms for finding the global optimizers of nonconvex optimization problems. A key component in the algorithm is the adaptive tuning of the randomness based on the value of the objective function. In the language of simulated annealing, the temperature is state-dependent. With this, we prove the global convergence of the algorithm with an algebraic rate both in probability and in the parameter space. This is a significant improvement over the classical rate from using a more straightforward control of the noise term. The convergence proof is based on the actual discrete setup of the algorithm, not just its continuous limit as often done in the literature. We also present several numerical examples to demonstrate the efficiency and robustness of the algorithm for reasonably complex objective functions.