Dissipative Gradient Descent Ascent Method: A Control Theory Inspired
  Algorithm for Min-max Optimization
Tianqi Zheng Nicolas Loizou Pengcheng You Enrique Mallada
Gradient Descent Ascent (GDA) methods for min-max optimization problems
typically produce oscillatory behavior that can lead to instability, e.g., in
bilinear settings. To address this problem, we introduce a dissipation term
into the GDA updates to dampen these oscillations. The proposed Dissipative GDA
(DGDA) method can be seen as performing standard GDA on a state-augmented and
regularized saddle function that does not strictly introduce additional
convexity/concavity. We theoretically show the linear convergence of DGDA in
the bilinear and strongly convex-strongly concave settings and assess its
performance by comparing DGDA with other methods such as GDA, Extra-Gradient
(EG), and Optimistic GDA. Our findings demonstrate that DGDA surpasses these
methods, achieving superior convergence rates. We support our claims with two
numerical examples that showcase DGDA's effectiveness in solving saddle point
problems.