  This paper considers a general problem of convex stochastic optimization in a
relatively low-dimensional space (e.g., 100 variables). It is known that for
deterministic convex optimization problems of small dimensions, the fastest
convergence is achieved by the center of gravity type methods (e.g., Vaidya's
cutting plane method). For stochastic optimization problems, the question of
whether Vaidya's method can be used comes down to the question of how it
accumulates inaccuracy in the subgradient. The recent result of the authors
states that the errors do not accumulate on iterations of Vaidya's method,
which allows proposing its analog for stochastic optimization problems. The
primary technique is to replace the subgradient in Vaidya's method with its
probabilistic counterpart (the arithmetic mean of the stochastic subgradients).
The present paper implements the described plan, which ultimately leads to an
effective (if parallel computations for batching are possible) method for
solving convex stochastic optimization problems in relatively low-dimensional
spaces.
