  Attention to data-driven optimization approaches, including the well-known
stochastic gradient descent method, has grown significantly over recent
decades, but data-driven constraints have rarely been studied, because of the
computational challenges of projections onto the feasible set defined by these
hard constraints. In this paper, we focus on the non-smooth convex-concave
stochastic minimax regime and formulate the data-driven constraints as
expectation constraints. The minimax expectation constrained problem subsumes a
broad class of real-world applications, including two-player zero-sum game and
data-driven robust optimization. We propose a class of efficient primal-dual
algorithms to tackle the minimax expectation-constrained problem, and show that
our algorithms converge at the optimal rate of
$\mathcal{O}(\frac{1}{\sqrt{N}})$. We demonstrate the practical efficiency of
our algorithms by conducting numerical experiments on large-scale real-world
applications.
