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

• URL: http://arxiv.org/abs/2202.00293v4
• Date: Wed, 14 Jun 2023 14:15:08 GMT
• Title: Phase diagram of Stochastic Gradient Descent in high-dimensional two-layer neural networks
• Authors: Rodrigo Veiga, Ludovic Stephan, Bruno Loureiro, Florent Krzakala, Lenka Zdeborov\'a
• Abstract summary: We investigate the connection between the mean-fieldhydrodynamic regime and the seminal approach of Saad & Solla. Our work builds on a deterministic description of rates in high-dimensionals from statistical physics.
• Score: 22.823904789355495
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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