Provable Benefit of Orthogonal Initialization in Optimizing Deep Linear Networks

TL;DR

This paper proves that orthogonal initialization in deep linear networks guarantees faster convergence and requires less width for deep networks compared to Gaussian initialization, providing a theoretical basis for empirical practices.

Contribution

It offers the first rigorous proof that orthogonal initialization improves convergence speed in deep linear networks, independent of depth, unlike Gaussian initialization.

Findings

Orthogonal initialization speeds up convergence.

Width for efficient convergence is independent of depth with orthogonal init.

Gaussian init requires width to scale linearly with depth.

Abstract

The selection of initial parameter values for gradient-based optimization of deep neural networks is one of the most impactful hyperparameter choices in deep learning systems, affecting both convergence times and model performance. Yet despite significant empirical and theoretical analysis, relatively little has been proved about the concrete effects of different initialization schemes. In this work, we analyze the effect of initialization in deep linear networks, and provide for the first time a rigorous proof that drawing the initial weights from the orthogonal group speeds up convergence relative to the standard Gaussian initialization with iid weights. We show that for deep networks, the width needed for efficient convergence to a global minimum with orthogonal initializations is independent of the depth, whereas the width needed for efficient convergence with Gaussian initializations scales linearly in the depth. Our results demonstrate how the benefits of a good initialization can persist throughout learning, suggesting an explanation for the recent empirical successes found by initializing very deep non-linear networks according to the principle of dynamical isometry.