We consider a class of stochastic smooth convex optimization problems under
rather general assumptions on the noise in the stochastic gradient observation.
As opposed to the classical problem setting in which the variance of noise is
assumed to be uniformly bounded, herein we assume that the variance of
stochastic gradients is related to the "sub-optimality" of the approximate
solutions delivered by the algorithm. Such problems naturally arise in a
variety of applications, in particular, in the well-known generalized linear
regression problem in statistics. However, to the best of our knowledge, none
of the existing stochastic approximation algorithms for solving this class of
problems attain optimality in terms of the dependence on accuracy, problem
parameters, and mini-batch size.
We discuss two non-Euclidean accelerated stochastic approximation
routines--stochastic accelerated gradient descent (SAGD) and stochastic
gradient extrapolation (SGE)--which carry a particular duality relationship. We
show that both SAGD and SGE, under appropriate conditions, achieve the optimal
convergence rate, attaining the optimal iteration and sample complexities
simultaneously. However, corresponding assumptions for the SGE algorithm are
more general; they allow, for instance, for efficient application of the SGE to
statistical estimation problems under heavy tail noises and discontinuous score
functions. We also discuss the application of the SGE to problems satisfying
quadratic growth conditions, and show how it can be used to recover sparse
solutions. Finally, we report on some simulation experiments to illustrate
numerical performance of our proposed algorithms in high-dimensional settings.

本文讨论了一类随机光滑凸优化问题，其噪声的方差与算法产生的近似解的次优性有关，提出了两个非欧几里德加速随机逼近算法，即随机加速梯度下降（SAGD）和随机梯度外推（SGE），并证明了在适当的条件下，这两个算法可以同时达到最优的迭代和样本复杂度。同时本文还提出了应用 SGE 进行恢复稀疏解的方法。