Deep Learning has recently yielded important advances in single-agent learning challenges, much of that progress being fueled by the empirical success of gradient descent and its variants in computing local optima of non-convex optimization problems. In multi-agent learning applications, the role of single-objective optimization is played by equilibrium computation, yet our understanding of its complexity in settings that are relevant for Deep Learning remains sparse. In this talk we focus on min-max optimization of nonconvex-nonconcave objectives, which has found applications in GANs, and other adversarial learning problems. Here, not only are there no known gradient-descent based methods converging to even local and approximate min-max equilibria, but the computational complexity of identifying them remains poorly understood. We show that finding approximate local min-max equilibria of Lipschitz and smooth objectives requires a number of queries to the function and its gradient that is exponential in the relevant parameters, in sharp contrast to the polynomial number of queries required to find approximate local minima of non-convex objectives. Our oracle lower bound is a byproduct of a complexity-theoretic result showing that finding approximate local min-max equilibria is computationally equivalent to finding Brouwer fixed points, and Nash equilibria in non zero-sum games, and thus PPAD-complete. Minimal complexity theory knowledge will be assumed in the talk.
(This is joint work with Stratis Skoulakis and Manolis Zampetakis)
Constantinos (aka “Costis”) Daskalakis is a Professor of Electrical Engineering and Computer Science at MIT. He holds a Diploma in Electrical and Computer Engineering from the National Technical University of Athens, and a PhD in Electrical Engineering and Computer Science from UC Berkeley. He works on Computation Theory and its interface with Game Theory, Economics, Probability Theory, Machine Learning and Statistics. He has been honored with the ACM Doctoral Dissertation Award, the Kalai Prize from the Game Theory Society, the Sloan Fellowship in Computer Science, the SIAM Outstanding Paper Prize, the Microsoft Research Faculty Fellowship, the Simons Investigator Award, the Rolf Nevanlinna Prize from the International Mathematical Union, the ACM Grace Murray Hopper Award, and the Bodossaki Foundation Distinguished Young Scientists Award.