This paper considers stochastic subgradient mirror-descent method for solving
constrained convex minimization problems. In particular, a stochastic
subgradient mirror-descent method with weighted iterate-averaging is
investigated and its per-iterate convergence rate is analyzed. The novel part
of the approach is in the choice of weights that are used to construct the
averages. Through the use of these weighted averages, we show that the known
optimal rates can be obtained with simpler algorithms than those currently
existing in the literature. Specifically, by suitably choosing the stepsize
values, one can obtain the rate of the order $1/k$ for strongly convex
functions, and the rate $1/\sqrt{k}$ for general convex functions (not
necessarily differentiable). Furthermore, for the latter case, it is shown that
a stochastic subgradient mirror-descent with iterate averaging converges (along
a subsequence) to an optimal solution, almost surely, even with the stepsize of
the form $1/\sqrt{1+k}$, which was not previously known. The stepsize choices
that achieve the best rates are those proposed by Paul Tseng for acceleration
of proximal gradient methods.

本文探讨随机次梯度镜像下降方法在解决受限凸规划问题中的应用及其收敛性分析，特别研究了带有加权迭代平均的随机次梯度镜像下降方法，并分析了其迭代收敛速度和优化效果。通过合适的步长取值，实现了具有强凸性和一般凸性的函数的最优平均速率，并且通过加权平均的方式，优于已有的算法。