Stochasticity helps to navigate rough landscapes: comparing gradient-descent-based algorithms in the phase retrieval problem

• URL: http://arxiv.org/abs/2103.04902v1
• Date: Mon, 8 Mar 2021 17:06:18 GMT
• Title: Stochasticity helps to navigate rough landscapes: comparing gradient-descent-based algorithms in the phase retrieval problem
• Authors: Francesca Mignacco, Pierfrancesco Urbani, Lenka Zdeborov\'a
• Abstract summary: We investigate how analytically-based algorithms such as dynamical descent, its persistent gradient, and Langevin landscape descent can be studied. We apply statistical-field theory from statistical trajectories to these algorithms in their full-time limit, with a start, and for large system sizes.
• Score: 8.164433158925593
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: In this paper we investigate how gradient-based algorithms such as gradient descent, (multi-pass) stochastic gradient descent, its persistent variant, and the Langevin algorithm navigate non-convex losslandscapes and which of them is able to reach the best generalization error at limited sample complexity. We consider the loss landscape of the high-dimensional phase retrieval problem as a prototypical highly non-convex example. We observe that for phase retrieval the stochastic variants of gradient descent are able to reach perfect generalization for regions of control parameters where the gradient descent algorithm is not. We apply dynamical mean-field theory from statistical physics to characterize analytically the full trajectories of these algorithms in their continuous-time limit, with a warm start, and for large system sizes. We further unveil several intriguing properties of the landscape and the algorithms such as that the gradient descent can obtain better generalization properties from less informed initializations.

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