Provable Convergence of Nesterov's Accelerated Gradient Method for Over-Parameterized Neural Networks

TL;DR

This paper proves that Nesterov's accelerated gradient method converges linearly to a global minimum in training over-parameterized neural networks, explaining its empirical success and showing it accelerates over standard gradient descent.

Contribution

The paper provides the first theoretical convergence guarantee for Nesterov's method in training over-parameterized neural networks, demonstrating its accelerated linear convergence rate.

Findings

NAG converges at a rate of (1 - Θ(1/√κ))^t

NAG's convergence rate is comparable to Heavy Ball method

Experimental results validate theoretical convergence guarantees

Abstract

Momentum methods, such as heavy ball method~(HB) and Nesterov's accelerated gradient method~(NAG), have been widely used in training neural networks by incorporating the history of gradients into the current updating process. In practice, they often provide improved performance over (stochastic) gradient descent~(GD) with faster convergence. Despite these empirical successes, theoretical understandings of their accelerated convergence rates are still lacking. Recently, some attempts have been made by analyzing the trajectories of gradient-based methods in an over-parameterized regime, where the number of the parameters is significantly larger than the number of the training instances. However, the majority of existing theoretical work is mainly concerned with GD and the established convergence result of NAG is inferior to HB and GD, which fails to explain the practical success of NAG. In this paper, we take a step towards closing this gap by analyzing NAG in training a randomly initialized over-parameterized two-layer fully connected neural network with ReLU activation. Despite the fact that the objective function is non-convex and non-smooth, we show that NAG converges to a global minimum at a non-asymptotic linear rate $(1-\Theta(1/\sqrt{\kappa}))^t$, where $\kappa > 1$ is the condition number of a gram matrix and $t$ is the number of the iterations. Compared to the convergence rate $(1-\Theta(1/{\kappa}))^t$ of GD, our result provides theoretical guarantees for the acceleration of NAG in neural network training. Furthermore, our findings suggest that NAG and HB have similar convergence rate. Finally, we conduct extensive experiments on six benchmark datasets to validate the correctness of our theoretical results.