How Much Over-parameterization Is Sufficient to Learn Deep ReLU Networks?

TL;DR

This paper demonstrates that deep ReLU networks can be effectively learned with a polylogarithmic over-parameterization, significantly reducing the network width requirements for optimization and generalization guarantees.

Contribution

It establishes sharper guarantees for deep ReLU networks trained by gradient descent, showing polylogarithmic width suffices under certain assumptions, advancing practical understanding.

Findings

Polylogarithmic width suffices for deep ReLU network learning.

Optimization and generalization guarantees hold with reduced over-parameterization.

Results extend previous work to more practical deep learning settings.

Abstract

A recent line of research on deep learning focuses on the extremely over-parameterized setting, and shows that when the network width is larger than a high degree polynomial of the training sample size $n$ and the inverse of the target error $\epsilon^{-1}$, deep neural networks learned by (stochastic) gradient descent enjoy nice optimization and generalization guarantees. Very recently, it is shown that under certain margin assumptions on the training data, a polylogarithmic width condition suffices for two-layer ReLU networks to converge and generalize (Ji and Telgarsky, 2019). However, whether deep neural networks can be learned with such a mild over-parameterization is still an open question. In this work, we answer this question affirmatively and establish sharper learning guarantees for deep ReLU networks trained by (stochastic) gradient descent. In specific, under certain assumptions made in previous work, our optimization and generalization guarantees hold with network width polylogarithmic in $n$ and $\epsilon^{-1}$. Our results push the study of over-parameterized deep neural networks towards more practical settings.