A Variance-Reduced Stochastic Accelerated Primal Dual Algorithm

• URL: http://arxiv.org/abs/2202.09688v1
• Date: Sat, 19 Feb 2022 22:12:30 GMT
• Title: A Variance-Reduced Stochastic Accelerated Primal Dual Algorithm
• Authors: Bugra Can, Mert Gurbuzbalaban, Necdet Serhat Aybat
• Abstract summary: Such problems arise frequently in machine learning in the context of robust empirical risk minimization. We consider the accelerated primal dual (SAPD) algorithm as a robust method against gradient noise. We show that our method improves upon SAPD both in practice and in theory.
• Score: 3.2958527541557525
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: In this work, we consider strongly convex strongly concave (SCSC) saddle point (SP) problems $\min_{x\in\mathbb{R}^{d_x}}\max_{y\in\mathbb{R}^{d_y}}f(x,y)$ where $f$ is $L$-smooth, $f(.,y)$ is $\mu$-strongly convex for every $y$, and $f(x,.)$ is $\mu$-strongly concave for every $x$. Such problems arise frequently in machine learning in the context of robust empirical risk minimization (ERM), e.g. $\textit{distributionally robust}$ ERM, where partial gradients are estimated using mini-batches of data points. Assuming we have access to an unbiased stochastic first-order oracle we consider the stochastic accelerated primal dual (SAPD) algorithm recently introduced in Zhang et al. [2021] for SCSC SP problems as a robust method against gradient noise. In particular, SAPD recovers the well-known stochastic gradient descent ascent (SGDA) as a special case when the momentum parameter is set to zero and can achieve an accelerated rate when the momentum parameter is properly tuned, i.e., improving the $\kappa \triangleq L/\mu$ dependence from $\kappa^2$ for SGDA to $\kappa$. We propose efficient variance-reduction strategies for SAPD based on Richardson-Romberg extrapolation and show that our method improves upon SAPD both in practice and in theory.

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