A Continuized View on Nesterov Acceleration for Stochastic Gradient Descent and Randomized Gossip

TL;DR

This paper introduces a continuized version of Nesterov acceleration using continuous-time variables and differential equations, enabling new analysis and acceleration of stochastic gradient methods and asynchronous gossip algorithms.

Contribution

It proposes a novel continuized Nesterov acceleration framework that unifies continuous and discrete analyses, and applies it to accelerate gossip algorithms.

Findings

Continuized Nesterov acceleration allows precise parameter analysis.

Discretization retains Nesterov's structure with random parameters.

First rigorous acceleration of asynchronous gossip algorithms achieved.

Abstract

We introduce the continuized Nesterov acceleration, a close variant of Nesterov acceleration whose variables are indexed by a continuous time parameter. The two variables continuously mix following a linear ordinary differential equation and take gradient steps at random times. This continuized variant benefits from the best of the continuous and the discrete frameworks: as a continuous process, one can use differential calculus to analyze convergence and obtain analytical expressions for the parameters; and a discretization of the continuized process can be computed exactly with convergence rates similar to those of Nesterov original acceleration. We show that the discretization has the same structure as Nesterov acceleration, but with random parameters. We provide continuized Nesterov acceleration under deterministic as well as stochastic gradients, with either additive or multiplicative noise. Finally, using our continuized framework and expressing the gossip averaging problem as the stochastic minimization of a certain energy function, we provide the first rigorous acceleration of asynchronous gossip algorithms.