Stochastic Polyak Step-size for SGD: An Adaptive Learning Rate for Fast Convergence

TL;DR

This paper introduces a stochastic Polyak step-size for SGD that adapts the learning rate using known optimal values, providing fast convergence guarantees across various settings and excelling in over-parameterized models.

Contribution

The paper proposes a novel stochastic Polyak step-size for SGD, with theoretical convergence guarantees and superior performance in over-parameterized models.

Findings

SPS enables SGD to converge faster in strongly convex, convex, and non-convex settings.

The method achieves fast convergence without needing problem-specific constants.

Experimental results show SPS outperforms state-of-the-art optimization methods.

Abstract

We propose a stochastic variant of the classical Polyak step-size (Polyak, 1987) commonly used in the subgradient method. Although computing the Polyak step-size requires knowledge of the optimal function values, this information is readily available for typical modern machine learning applications. Consequently, the proposed stochastic Polyak step-size (SPS) is an attractive choice for setting the learning rate for stochastic gradient descent (SGD). We provide theoretical convergence guarantees for SGD equipped with SPS in different settings, including strongly convex, convex and non-convex functions. Furthermore, our analysis results in novel convergence guarantees for SGD with a constant step-size. We show that SPS is particularly effective when training over-parameterized models capable of interpolating the training data. In this setting, we prove that SPS enables SGD to converge to the true solution at a fast rate without requiring the knowledge of any problem-dependent constants or additional computational overhead. We experimentally validate our theoretical results via extensive experiments on synthetic and real datasets. We demonstrate the strong performance of SGD with SPS compared to state-of-the-art optimization methods when training over-parameterized models.