Towards Noise-adaptive, Problem-adaptive Stochastic Gradient Descent

• URL: http://arxiv.org/abs/2110.11442v1
• Date: Thu, 21 Oct 2021 19:22:14 GMT
• Title: Towards Noise-adaptive, Problem-adaptive Stochastic Gradient Descent
• Authors: Sharan Vaswani, Benjamin Dubois-Taine, Reza Babanezhad
• Abstract summary: We design step-size schemes that make gradient descent (SGD) adaptive to (i) the noise. We prove that $T$ iterations of SGD with Nesterov iterations can be near optimal. Compared to other step-size schemes, we demonstrate the effectiveness of a novel novel exponential step-size scheme.
• Score: 7.176107039687231
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We design step-size schemes that make stochastic gradient descent (SGD) adaptive to (i) the noise $\sigma^2$ in the stochastic gradients and (ii) problem-dependent constants. When minimizing smooth, strongly-convex functions with condition number $\kappa$, we first prove that $T$ iterations of SGD with Nesterov acceleration and exponentially decreasing step-sizes can achieve a near-optimal $\tilde{O}(\exp(-T/\sqrt{\kappa}) + \sigma^2/T)$ convergence rate. Under a relaxed assumption on the noise, with the same step-size scheme and knowledge of the smoothness, we prove that SGD can achieve an $\tilde{O}(\exp(-T/\kappa) + \sigma^2/T)$ rate. In order to be adaptive to the smoothness, we use a stochastic line-search (SLS) and show (via upper and lower-bounds) that SGD converges at the desired rate, but only to a neighbourhood of the solution. Next, we use SGD with an offline estimate of the smoothness and prove convergence to the minimizer. However, its convergence is slowed down proportional to the estimation error and we prove a lower-bound justifying this slowdown. Compared to other step-size schemes, we empirically demonstrate the effectiveness of exponential step-sizes coupled with a novel variant of SLS.

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