Stochastic gradient descent with momentum, also known as Stochastic Heavy Ball method (SHB), is one of the most popular algorithms for solving large-scale stochastic optimization problems in various machine learning tasks. In practical scenarios, tuning the step-size and momentum parameters of the method is a prohibitively expensive and time-consuming process. In this work, inspired by the recent advantages of stochastic Polyak step-size in the performance of stochastic gradient descent (SGD), we propose and explore new Polyak-type variants suitable for the update rule of the SHB method. In particular, using the Iterate Moving Average (IMA) viewpoint of SHB, we propose and analyze three novel step-size selections: MomSPS$_{\max}$, MomDecSPS, and MomAdaSPS. For MomSPS$_{\max}$, we provide convergence guarantees for SHB to a neighborhood of the solution for convex and smooth problems (without assuming interpolation). If interpolation is also satisfied, then using MomSPS$_{\max}$, SHB converges to the true solution at a fast rate matching the deterministic HB. The other two variants, MomDecSPS and MomAdaSPS, are the first adaptive step-sizes for SHB that guarantee convergence to the exact minimizer without prior knowledge of the problem parameters and without assuming interpolation. The convergence analysis of SHB is tight and obtains the convergence guarantees of SGD with stochastic Polyak step-sizes as a special case. We supplement our analysis with experiments that validate the theory and demonstrate the effectiveness and robustness of the new algorithms.

在本文中，我们提出了一种基于随机梯度下降算法的新型多步骤选择方法来解决大规模随机优化问题，该方法不需要预先了解问题参数并且具有收敛性保证。

随机Polyak步长和动量：收敛保证和实际性能