  This paper focus on the convergence of stochastic approximation with Nesterov
momentum. Nesterov acceleration has proven effective in machine learning for
its ability to reduce computational complexity. The issue of delayed
information in the acceleration term remains a challenge to achieving the
almost sure convergence. Based on the delayed supermatingale convergence
lemmas, we give a series of framework for almost sure convergence. Our
framework applies to several widely-used random iterative methods, such as
stochastic subgradient methods, the proximal Robbins-Monro method for general
stochastic optimization, and the proximal stochastic subgradient method for
composite optimization. Through the applications of our framework, these
methods with Nesterov acceleration achieve almost sure convergence. And three
groups of numerical experiments is to check out theoretical results.
