Phase diagram of Stochastic Gradient Descent in high-dimensional two-layer neural networks

Rodrigo Veiga, Ludovic STEPHAN, Bruno Loureiro, Florent Krzakala, Lenka Zdeborova

Keywords: stochastic gradient descent, two-layer neural networks, overparametrization, gaussian inputs, statistical physics

TL;DR: We study the interaction between overparametrization and the choice of learning rate in two-layer neural networks, and the relation between the mean-field & Saad & Solla's theories

Abstract: Despite the non-convex optimization landscape, over-parametrized shallow networks are able to achieve global convergence under gradient descent. The picture can be radically different for narrow networks, which tend to get stuck in badly-generalizing local minima. Here we investigate the cross-over between these two regimes in the high-dimensional setting, and in particular investigate the connection between the so-called mean-field/hydrodynamic regime and the seminal approach of Saad \& Solla. Focusing on the case of Gaussian data, we study the interplay between the learning rate, the time scale, and the number of hidden units in the high-dimensional dynamics of stochastic gradient descent (SGD). Our work builds on a deterministic description of SGD in high-dimensions from statistical physics, which we extend and for which we provide rigorous convergence rates.

Community Implementations: [ 1 code implementation](https://www.catalyzex.com/paper/phase-diagram-of-stochastic-gradient-descent/code)