Clipped Stochastic Methods for Variational Inequalities with Heavy-Tailed Noise

• URL: http://arxiv.org/abs/2206.01095v1
• Date: Thu, 2 Jun 2022 15:21:55 GMT
• Title: Clipped Stochastic Methods for Variational Inequalities with Heavy-Tailed Noise
• Authors: Eduard Gorbunov, Marina Danilova, David Dobre, Pavel Dvurechensky, Alexander Gasnikov, Gauthier Gidel
• Abstract summary: We prove the first high-probability results with logarithmic dependence on the confidence level for methods for solving monotone and structured non-monotone VIPs. Our results match the best-known ones in the light-tails case and are novel for structured non-monotone problems. In addition, we numerically validate that the gradient noise of many practical formulations is heavy-tailed and show that clipping improves the performance of SEG/SGDA.
• Score: 64.85879194013407
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Stochastic first-order methods such as Stochastic Extragradient (SEG) or Stochastic Gradient Descent-Ascent (SGDA) for solving smooth minimax problems and, more generally, variational inequality problems (VIP) have been gaining a lot of attention in recent years due to the growing popularity of adversarial formulations in machine learning. However, while high-probability convergence bounds are known to reflect the actual behavior of stochastic methods more accurately, most convergence results are provided in expectation. Moreover, the only known high-probability complexity results have been derived under restrictive sub-Gaussian (light-tailed) noise and bounded domain Assump. [Juditsky et al., 2011]. In this work, we prove the first high-probability complexity results with logarithmic dependence on the confidence level for stochastic methods for solving monotone and structured non-monotone VIPs with non-sub-Gaussian (heavy-tailed) noise and unbounded domains. In the monotone case, our results match the best-known ones in the light-tails case [Juditsky et al., 2011], and are novel for structured non-monotone problems such as negative comonotone, quasi-strongly monotone, and/or star-cocoercive ones. We achieve these results by studying SEG and SGDA with clipping. In addition, we numerically validate that the gradient noise of many practical GAN formulations is heavy-tailed and show that clipping improves the performance of SEG/SGDA.

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