  We consider strongly-convex-strongly-concave saddle point problems assuming
we have access to unbiased stochastic estimates of the gradients. We propose a
stochastic accelerated primal-dual (SAPD) algorithm and show that SAPD
sequence, generated using constant primal-dual step sizes, linearly converges
to a neighborhood of the unique saddle point. Interpreting the size of the
neighborhood as a measure of robustness to gradient noise, we obtain explicit
characterizations of robustness in terms of SAPD parameters and problem
constants. Based on these characterizations, we develop computationally
tractable techniques for optimizing the SAPD parameters, i.e., the primal and
dual step sizes, and the momentum parameter, to achieve a desired trade-off
between the convergence rate and robustness on the Pareto curve. This allows
SAPD to enjoy fast convergence properties while being robust to noise as an
accelerated method. SAPD admits convergence guarantees for the distance metric
with a variance term optimal up to a logarithmic factor, which can be removed
by employing a restarting strategy. We also discuss how convergence and
robustness results extend to the convex-concave setting. Finally, we illustrate
our framework on distributionally robust logistic regression.
