A piecewise conservative method for unconstrained convex optimization

TL;DR

This paper introduces a novel conservative dynamical system-based optimization method with restart criteria, demonstrating faster convergence than existing accelerated gradient methods in convex and composite problems.

Contribution

It develops a new continuous-time conservative optimization framework with a practical restart scheme, extending it to composite problems and outperforming standard algorithms in experiments.

Findings

Faster convergence than restarted NAG-C on smooth convex problems.

Better performance than FISTA with adaptive restart on composite problems.

The proposed method guarantees convergence with the new restart criterion.

Abstract

We consider a continuous-time optimization method based on a dynamical system, where a massive particle starting at rest moves in the conservative force field generated by the objective function, without any kind of friction. We formulate a restart criterion based on the mean dissipation of the kinetic energy, and we prove a global convergence result for strongly-convex functions. Using the Symplectic Euler discretization scheme, we obtain an iterative optimization algorithm. We have considered a discrete mean dissipation restart scheme, but we have also introduced a new restart procedure based on ensuring at each iteration a decrease of the objective function greater than the one achieved by a step of the classical gradient method. For the discrete conservative algorithm, this last restart criterion is capable of guaranteeing a convergence result. We apply the same restart scheme to the Nesterov Accelerated Gradient (NAG-C), and we use this restarted NAG-C as benchmark in the numerical experiments. In the smooth convex problems considered, our method shows a faster convergence rate than the restarted NAG-C. We propose an extension of our discrete conservative algorithm to composite optimization: in the numerical tests involving non-strongly convex functions with $\ell^1$-regularization, it has better performances than the well known efficient Fast Iterative Shrinkage-Thresholding Algorithm, accelerated with an adaptive restart scheme.