Understanding the Acceleration Phenomenon via High-Resolution Differential Equations

TL;DR

This paper introduces high-resolution differential equations to better understand and distinguish between accelerated gradient methods, revealing new insights into their convergence behaviors and leading to the development of novel optimization algorithms.

Contribution

The paper develops high-resolution ODEs that accurately differentiate between Nesterov's accelerated method and the heavy-ball method, and uncovers new convergence properties and algorithmic variants.

Findings

High-resolution ODEs distinguish between NAG-SC and heavy-ball methods.

NAG-C minimizes squared gradient norm at an inverse cubic rate.

New optimization methods maintain NAG-C's accelerated convergence.

Abstract

Gradient-based optimization algorithms can be studied from the perspective of limiting ordinary differential equations (ODEs). Motivated by the fact that existing ODEs do not distinguish between two fundamentally different algorithms---Nesterov's accelerated gradient method for strongly convex functions (NAG-SC) and Polyak's heavy-ball method---we study an alternative limiting process that yields high-resolution ODEs. We show that these ODEs permit a general Lyapunov function framework for the analysis of convergence in both continuous and discrete time. We also show that these ODEs are more accurate surrogates for the underlying algorithms; in particular, they not only distinguish between NAG-SC and Polyak's heavy-ball method, but they allow the identification of a term that we refer to as "gradient correction" that is present in NAG-SC but not in the heavy-ball method and is responsible for the qualitative difference in convergence of the two methods. We also use the high-resolution ODE framework to study Nesterov's accelerated gradient method for (non-strongly) convex functions, uncovering a hitherto unknown result---that NAG-C minimizes the squared gradient norm at an inverse cubic rate. Finally, by modifying the high-resolution ODE of NAG-C, we obtain a family of new optimization methods that are shown to maintain the accelerated convergence rates of NAG-C for smooth convex functions.