Stochastic Bias-Reduced Gradient Methods

• URL: http://arxiv.org/abs/2106.09481v1
• Date: Thu, 17 Jun 2021 13:33:05 GMT
• Title: Stochastic Bias-Reduced Gradient Methods
• Authors: Hilal Asi, Yair Carmon, Arun Jambulapati, Yujia Jin and Aaron Sidford
• Abstract summary: We develop a new primitive for optimization: a low-bias, low-cost smoothing of ther $x_star$ of any bound of the Moreau-Yoshida function.
• Score: 44.35885731095432
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We develop a new primitive for stochastic optimization: a low-bias, low-cost estimator of the minimizer $x_\star$ of any Lipschitz strongly-convex function. In particular, we use a multilevel Monte-Carlo approach due to Blanchet and Glynn to turn any optimal stochastic gradient method into an estimator of $x_\star$ with bias $\delta$, variance $O(\log(1/\delta))$, and an expected sampling cost of $O(\log(1/\delta))$ stochastic gradient evaluations. As an immediate consequence, we obtain cheap and nearly unbiased gradient estimators for the Moreau-Yoshida envelope of any Lipschitz convex function, allowing us to perform dimension-free randomized smoothing. We demonstrate the potential of our estimator through four applications. First, we develop a method for minimizing the maximum of $N$ functions, improving on recent results and matching a lower bound up logarithmic factors. Second and third, we recover state-of-the-art rates for projection-efficient and gradient-efficient optimization using simple algorithms with a transparent analysis. Finally, we show that an improved version of our estimator would yield a nearly linear-time, optimal-utility, differentially-private non-smooth stochastic optimization method.

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