Taming Nonconvex Stochastic Mirror Descent with General Bregman Divergence

• URL: http://arxiv.org/abs/2402.17722v1
• Date: Tue, 27 Feb 2024 17:56:49 GMT
• Title: Taming Nonconvex Stochastic Mirror Descent with General Bregman Divergence
• Authors: Ilyas Fatkhullin, Niao He
• Abstract summary: This paper revisits the convergence of gradient Forward Mirror (SMD) in the contemporary non optimization setting. For the training, we develop provably convergent algorithms for the problem of linear networks.
• Score: 25.717501580080846
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: This paper revisits the convergence of Stochastic Mirror Descent (SMD) in the contemporary nonconvex optimization setting. Existing results for batch-free nonconvex SMD restrict the choice of the distance generating function (DGF) to be differentiable with Lipschitz continuous gradients, thereby excluding important setups such as Shannon entropy. In this work, we present a new convergence analysis of nonconvex SMD supporting general DGF, that overcomes the above limitations and relies solely on the standard assumptions. Moreover, our convergence is established with respect to the Bregman Forward-Backward envelope, which is a stronger measure than the commonly used squared norm of gradient mapping. We further extend our results to guarantee high probability convergence under sub-Gaussian noise and global convergence under the generalized Bregman Proximal Polyak-{\L}ojasiewicz condition. Additionally, we illustrate the advantages of our improved SMD theory in various nonconvex machine learning tasks by harnessing nonsmooth DGFs. Notably, in the context of nonconvex differentially private (DP) learning, our theory yields a simple algorithm with a (nearly) dimension-independent utility bound. For the problem of training linear neural networks, we develop provably convergent stochastic algorithms.

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