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

Rodrigo Veiga; Ludovic Stephan; Bruno Loureiro; Florent Krzakala; Lenka Zdeborová

doi:10.52202/068431-1689

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arXiv · 2202.00293

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

Rodrigo Veiga·Ludovic Stephan·Bruno Loureiro·Florent Krzakala·Lenka Zdeborová

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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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stat.ML cond-mat.dis-nn cs.LG

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Rodrigo Veiga, Ludovic Stephan, Bruno Loureiro, Florent Krzakala, Lenka Zdeborová. 2022-02-01. Phase diagram of Stochastic Gradient Descent in high-dimensional two-layer neural networks. https://doi.org/10.52202/068431-1689

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