  Recent focus on robustness to adversarial attacks for deep neural networks
produced a large variety of algorithms for training robust models. Most of the
effective algorithms involve solving the min-max optimization problem for
training robust models (min step) under worst-case attacks (max step). However,
they often suffer from high computational cost from running several inner
maximization iterations (to find an optimal attack) inside every outer
minimization iteration. Therefore, it becomes difficult to readily apply such
algorithms for moderate to large size real world data sets. To alleviate this,
we explore the effectiveness of iterative descent-ascent algorithms where the
maximization and minimization steps are executed in an alternate fashion to
simultaneously obtain the worst-case attack and the corresponding robust model.
Specifically, we propose a novel discrete-time dynamical system-based algorithm
that aims to find the saddle point of a min-max optimization problem in the
presence of uncertainties. Under the assumptions that the cost function is
convex and uncertainties enter concavely in the robust learning problem, we
analytically show that our algorithm converges asymptotically to the robust
optimal solution under a general adversarial budget constraints as induced by
$\ell_p$ norm, for $1\leq p\leq \infty$. Based on our proposed analysis, we
devise a fast robust training algorithm for deep neural networks. Although such
training involves highly non-convex robust optimization problems, empirical
results show that the algorithm can achieve significant robustness compared to
other state-of-the-art robust models on benchmark data sets.
