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In this work, we study the convergence in high probability of clipped gradient methods when the noise distribution has heavy tails, i.e., with bounded $$p$$th moments, for some $$1< p \leq 2$$. Prior works in this setting follow the same recipe of using concentration inequalities and an inductive argument with union bound to bound the iterates across all iterations. This method results in an increase in the failure probability by a factor of $$T$$, where $$T$$ is the number of iterations. We instead propose a new analysis approach based on bounding the moment generating function of a well chosen supermartingale sequence. We improve the dependency on $$T$$ in the convergence guarantee for a wide range of algorithms with clipped gradients, including stochastic (accelerated) mirror descent for convex objectives and stochastic gradient descent for nonconvex objectives. Our high probability bounds achieve the optimal convergence rates and match the best currently known in-expectation bounds. Our approach naturally allows the algorithms to use time-varying step sizes and clipping parameters when the time horizon is unknown, which appears difficult or even impossible using existing techniques from prior works. Furthermore, we show that in the case of clipped stochastic mirror descent, several problem constants, including the initial distance to the optimum, are not required when setting step sizes and clipping parameters.
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Gradient Descent in the Absence of Global Lipschitz Continuity of the Gradients
Gradient descent (GD) is a collection of continuous optimization methods that have achieved immeasurable success in practice. Owing to data science applications, GD with diminishing step sizes has become a prominent variant. While this variant of GD has been well studied in the literature for objectives with globally Lipschitz continuous gradients or by requiring bounded iterates, objectives from data science problems do not satisfy such assumptions. Thus, in this work, we provide a novel global convergence analysis of GD with diminishing step sizes for differentiable nonconvex functions whose gradients are only locally Lipschitz continuous. Through our analysis, we generalize what is known about gradient descent with diminishing step sizes, including interesting topological facts, and we elucidate the varied behaviors that can occur in the previously overlooked divergence regime. Thus, we provide a general global convergence analysis of GD with diminishing step sizes under realistic conditions for data science problems.
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- Award ID(s):
- 2023239
- PAR ID:
- 10529281
- Publisher / Repository:
- SIAM
- Date Published:
- Journal Name:
- SIAM Journal on Mathematics of Data Science
- Volume:
- 6
- Issue:
- 3
- ISSN:
- 2577-0187
- Page Range / eLocation ID:
- 602 to 626
- Subject(s) / Keyword(s):
- gradient descent diminishing step sizes convergence divergence
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
-
Accepted Manuscript
- Journal Article:
- https://doi.org/10.1137/22M1527210
