Convergence and Implicit Bias of Gradient Flow on Overparametrized Linear Networks

TL;DR

This paper analyzes how overparametrization and initialization influence the convergence and implicit bias of gradient flow in linear neural networks, revealing conditions that lead to min-norm solutions.

Contribution

It provides a novel analysis connecting initialization, overparametrization, and convergence, establishing bounds and invariant sets for linear networks trained with gradient flow.

Findings

Exponential convergence rate depends on initialization imbalance and margin.

Proper initialization constrains parameters to an invariant set leading to min-norm solutions.

Large width and scaled random initialization ensure proximity to the invariant set during training.

Abstract

Neural networks trained via gradient descent with random initialization and without any regularization enjoy good generalization performance in practice despite being highly overparametrized. A promising direction to explain this phenomenon is to study how initialization and overparametrization affect convergence and implicit bias of training algorithms. In this paper, we present a novel analysis of single-hidden-layer linear networks trained under gradient flow, which connects initialization, optimization, and overparametrization. Firstly, we show that the squared loss converges exponentially to its optimum at a rate that depends on the level of imbalance and the margin of the initialization. Secondly, we show that proper initialization constrains the dynamics of the network parameters to lie within an invariant set. In turn, minimizing the loss over this set leads to the min-norm solution. Finally, we show that large hidden layer width, together with (properly scaled) random initialization, ensures proximity to such an invariant set during training, allowing us to derive a novel non-asymptotic upper-bound on the distance between the trained network and the min-norm solution.