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Nguegnang, Gabin Maxime ORCID logoORCID: https://orcid.org/0000-0002-6310-075X; Rauhut, Holger ORCID logoORCID: https://orcid.org/0000-0003-4750-5092 und Terstiege, Ulrich ORCID logoORCID: https://orcid.org/0000-0003-4750-5092 (18. Juli 2024): Convergence of gradient descent for learning linear neural networks. In: Advances in Continuous and Discrete Models, Bd. 2024, 23 [PDF, 2MB]

Abstract

We study the convergence properties of gradient descent for training deep linear neural networks, i.e., deep matrix factorizations, by extending a previous analysis for the related gradient flow. We show that under suitable conditions on the stepsizes gradient descent converges to a critical point of the loss function, i.e., the square loss in this article. Furthermore, we demonstrate that for almost all initializations gradient descent converges to a global minimum in the case of two layers. In the case of three or more layers, we show that gradient descent converges to a global minimum on the manifold matrices of some fixed rank, where the rank cannot be determined a priori.

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