  This paper presents a proximal-point-based catalyst scheme for simple
first-order methods applied to convex minimization and convex-concave minimax
problems. In particular, for smooth and (strongly)-convex minimization
problems, the proposed catalyst scheme, instantiated with a simple variant of
stochastic gradient method, attains the optimal rate of convergence in terms of
both deterministic and stochastic errors. For smooth and
strongly-convex-strongly-concave minimax problems, the catalyst scheme attains
the optimal rate of convergence for deterministic and stochastic errors up to a
logarithmic factor. To the best of our knowledge, this reported convergence
seems to be attained for the first time by stochastic first-order methods in
the literature. We obtain this result by designing and catalyzing a novel
variant of stochastic extragradient method for solving smooth and
strongly-monotone variational inequality, which may be of independent interest.
