Explicit Stabilised Gradient Descent for Faster Strongly Convex Optimisation

TL;DR

This paper presents RKCD, a new explicit stabilised gradient descent method inspired by numerical integrators, which achieves near-optimal convergence rates for strongly convex functions and outperforms Nesterov's method in experiments.

Contribution

Introduction of RKCD, a novel explicit stabilised integrator-based gradient descent method with near-optimal convergence for strongly convex optimization.

Findings

RKCD nearly matches conjugate gradient convergence rates.

RKCD outperforms Nesterov's accelerated gradient descent in experiments.

Optimal convergence rate achieved for partitioned RKCD on perturbed quadratic functions.

Abstract

This paper introduces the Runge-Kutta Chebyshev descent method (RKCD) for strongly convex optimisation problems. This new algorithm is based on explicit stabilised integrators for stiff differential equations, a powerful class of numerical schemes that avoid the severe step size restriction faced by standard explicit integrators. For optimising quadratic and strongly convex functions, this paper proves that RKCD nearly achieves the optimal convergence rate of the conjugate gradient algorithm, and the suboptimality of RKCD diminishes as the condition number of the quadratic function worsens. It is established that this optimal rate is obtained also for a partitioned variant of RKCD applied to perturbations of quadratic functions. In addition, numerical experiments on general strongly convex problems show that RKCD outperforms Nesterov's accelerated gradient descent.