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

A Fast Saddle-Point Dynamical System Approach to Robust Deep Learning