Global Convergence of Gradient Descent for Deep Linear Residual Networks

Abstract

We analyze the global convergence of gradient descent for deep linear residual networks by proposing a new initialization: zero-asymmetric (ZAS) initialization. It is motivated by avoiding stable manifolds of saddle points. We prove that under the ZAS initialization, for an arbitrary target matrix, gradient descent converges to an $\varepsilon$-optimal point in $O\left( L^3 \log(1/\varepsilon) \right)$ iterations, which scales polynomially with the network depth $L$. Our result and the $\exp(\Omega(L))$ convergence time for the standard initialization (Xavier or near-identity) \cite{shamir2018exponential} together demonstrate the importance of the residual structure and the initialization in the optimization for deep linear neural networks, especially when $L$ is large.