High-Probability Convergence for Composite and Distributed Stochastic Minimization and Variational Inequalities with Heavy-Tailed Noise

Eduard Gorbunov, Abdurakhmon Sadiev, Marina Danilova, Samuel Horváth, Gauthier Gidel, Pavel Dvurechensky, Alexander Gasnikov, Peter Richtárik

Primary Area: optimization

Code Of Ethics: I acknowledge that I and all co-authors of this work have read and commit to adhering to the ICLR Code of Ethics.

Keywords: heavy-tailed noise, variational inequalities, extragradient method, gradient descent-ascent, high-probability bounds, clipping, composite minimization, distributed minimization

Submission Guidelines: I certify that this submission complies with the submission instructions as described on https://iclr.cc/Conferences/2024/AuthorGuide.

Abstract: High-probability analysis of stochastic first-order optimization methods under mild assumptions on the noise has been gaining a lot of attention in recent years. Typically, gradient clipping is one of the key algorithmic ingredients to derive good high-probability guarantees when the noise is heavy-tailed. However, if implemented naively, clipping can spoil the convergence of the popular methods for composite and distributed optimization (Prox-SGD/Parallel SGD) even in the absence of any noise. Due to this reason, many works on high-probability analysis consider only unconstrained non-distributed problems, and the existing results for composite/distributed problems do not include some important special cases (like strongly convex problems) and are not optimal. To address this issue, we propose new stochastic methods for composite and distributed optimization based on the clipping of stochastic gradient differences and prove tight high-probability convergence results (including nearly optimal ones) for the new methods. Using similar ideas, we also develop new methods for composite and distributed variational inequalities and analyze the high-probability convergence of these methods.

Anonymous Url: I certify that there is no URL (e.g., github page) that could be used to find authors' identity.

No Acknowledgement Section: I certify that there is no acknowledgement section in this submission for double blind review.

Submission Number: 5223