Taming Nonconvex Stochastic Mirror Descent with General Bregman Divergence
It addresses a theoretical bottleneck in nonconvex optimization for machine learning researchers, enabling broader algorithm design with improved guarantees.
This paper tackles the convergence of Stochastic Mirror Descent (SMD) for nonconvex optimization by supporting general distance generating functions, overcoming prior limitations like excluding Shannon entropy, and establishes stronger convergence measures and extensions to high-probability and global convergence.
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-Łojasiewicz 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.