Noise-Adaptive (Accelerated) Stochastic Heavy-Ball Momentum

Abstract

We analyze the convergence of stochastic heavy ball (SHB) momentum in the smooth, strongly-convex setting. Kidambi et al. showed that SHB (with small mini-batches) cannot attain an accelerated rate of convergence even for quadratics, and conjecture that the practical gains of SHB are a by-product of mini-batching. We substantiate this claim by showing that SHB can obtain an accelerated rate when the mini-batch size is larger than some threshold. In particular, for strongly-convex quadratics with condition number $\kappa$, we prove that SHB with the standard step-size and momentum parameters results in an $O\left(\exp(-\frac{T}{\sqrt{\kappa}}) + \sigma\right)$ convergence rate, where $T$ is the number of iterations and $\sigma^2$ is the variance in the stochastic gradients. To ensure convergence to the minimizer, we propose a multi-stage approach that results in a noise-adaptive $O\left(\exp\left(-\frac{T}{\sqrt{\kappa}}\right) + \frac{\sigma}{T}\right)$ rate. For general strongly-convex functions, we use the averaging interpretation of SHB along with exponential step-sizes to prove an $O\left(\exp\left(-\frac{T}{\kappa}\right) + \frac{\sigma^2}{T}\right)$ convergence to the minimizer. Finally, we empirically demonstrate the effectiveness of the proposed algorithms.