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Clipped Stochastic Methods for Variational Inequalities with Heavy-Tailed Noise
Eduard Gorbunov · Marina Danilova · David Dobre · Pavel Dvurechenskii · Alexander Gasnikov · Gauthier Gidel

Wed Nov 30 02:00 PM -- 04:00 PM (PST) @ Hall J #617

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. While high-probability convergence bounds are known to more accurately reflect the actual behavior of stochastic methods, 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 assumptions [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.

Author Information

Eduard Gorbunov (Mohamed bin Zayed University of Artificial Intelligence)
Marina Danilova (ICS RAS)
David Dobre (Mila)
Pavel Dvurechenskii (Weierstrass Institute, Berlin)

Since 2015 Research assistant, Research Group 6 "Stochastic Algorithms and Nonparametric Statistics", Weierstrass Institute for Applied Analysis and Stochastics, Berlin 2014 - 2015 Research assistant, Institute for Information Transmission Problems, Moscow, Russia 2009 - 2015 Junior researcher, Moscow Institute of Physics and Technology, Moscow, Russia 2013 Ph.D., Moscow Institute of Physics and Technology, Moscow, Russia 2010 Master's Diploma, Moscow Institute of Physics and Technology, Moscow, Russia 2008 Bachelor's Diploma, Moscow Institute of Physics and Technology, Moscow, Russia

Alexander Gasnikov (Moscow Institute of Physics and Technology)
Gauthier Gidel (Mila)

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