Momentum methods are popularly used in accelerating stochastic iterative methods. Although a fair amount of literature is dedicated to momentum in stochastic optimisation, there are limited results that quantify the benefits of using heavy ball momentum in the specific case of stochastic approximation algorithms. We first show that the convergence rate with optimal step size does not improve when momentum is used (under some assumptions). Secondly, to quantify the behaviour in the initial phase we analyse the sample complexity of iterates with and without momentum. We show that the sample complexity bound for SA without momentum is $\tilde{\mathcal{O}}(\frac{1}{\alpha\lambda_{min}(A)})$ while for SA with momentum is $\tilde{\mathcal{O}}(\frac{1}{\sqrt{\alpha\lambda_{min}(A)}})$, where $\alpha$ is the step size and $\lambda_{min}(A)$ is the smallest eigenvalue of the driving matrix $A$. Although the sample complexity bound for SA with momentum is better for small enough $\alpha$, it turns out that for optimal choice of $\alpha$ in the two cases, the sample complexity bounds are of the same order.

翻译：在加速随机迭代方法方面,流行的调子方法是流行的。虽然大量文献致力于在随机优化方面形成势头,但数量化结果有限,无法量化在随机近似算法的具体情况下使用重球动力的好处。我们首先显示,在使用动力时(根据一些假设),最佳步骤尺寸的趋同率没有改善。第二,为了量化初始阶段的行为,我们用和没有动力的方式分析迭代的样本复杂性。我们显示,没有动力的SA的样本复杂性是美元(frac{O})(1. ALpha{1\ lambda}(A)),而没有动力的SA则使用重球动力的好处是$(tilde_mathcal}}(O) (\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\