Direct Runge-Kutta Discretization Achieves Acceleration

TL;DR

This paper demonstrates that directly discretizing a second-order ODE related to Nesterov's method with Runge-Kutta integrators can achieve accelerated convergence rates in optimization, especially under certain flatness conditions.

Contribution

It introduces a novel approach of discretizing a second-order ODE with Runge-Kutta methods for acceleration and establishes convergence rates under various smoothness and flatness conditions.

Findings

Discretization with Runge-Kutta achieves accelerated convergence rates.

A new flatness condition allows faster rates with gradient-only information.

Numerical experiments confirm theoretical convergence rates.

Abstract

We study gradient-based optimization methods obtained by directly discretizing a second-order ordinary differential equation (ODE) related to the continuous limit of Nesterov's accelerated gradient method. When the function is smooth enough, we show that acceleration can be achieved by a stable discretization of this ODE using standard Runge-Kutta integrators. Specifically, we prove that under Lipschitz-gradient, convexity and order-$(s+2)$ differentiability assumptions, the sequence of iterates generated by discretizing the proposed second-order ODE converges to the optimal solution at a rate of $\mathcal{O}({N^{-2\frac{s}{s+1}}})$, where $s$ is the order of the Runge-Kutta numerical integrator. Furthermore, we introduce a new local flatness condition on the objective, under which rates even faster than $\mathcal{O}(N^{-2})$ can be achieved with low-order integrators and only gradient information. Notably, this flatness condition is satisfied by several standard loss functions used in machine learning. We provide numerical experiments that verify the theoretical rates predicted by our results.