Stochastic Gradient Descent Optimizes Over-parameterized Deep ReLU Networks

TL;DR

This paper proves that over-parameterized deep ReLU networks trained with gradient descent or stochastic gradient descent can reach global minima, explaining why training deep networks is often successful.

Contribution

It provides a theoretical analysis showing that proper initialization and over-parameterization enable gradient methods to find global minima in deep ReLU networks.

Findings

Gradient descent finds global minima in over-parameterized deep ReLU networks.

Proper random initialization keeps training within a favorable local region.

The empirical loss exhibits nice local curvature properties facilitating convergence.

Abstract

We study the problem of training deep neural networks with Rectified Linear Unit (ReLU) activation function using gradient descent and stochastic gradient descent. In particular, we study the binary classification problem and show that for a broad family of loss functions, with proper random weight initialization, both gradient descent and stochastic gradient descent can find the global minima of the training loss for an over-parameterized deep ReLU network, under mild assumption on the training data. The key idea of our proof is that Gaussian random initialization followed by (stochastic) gradient descent produces a sequence of iterates that stay inside a small perturbation region centering around the initial weights, in which the empirical loss function of deep ReLU networks enjoys nice local curvature properties that ensure the global convergence of (stochastic) gradient descent. Our theoretical results shed light on understanding the optimization for deep learning, and pave the way for studying the optimization dynamics of training modern deep neural networks.