Recent work has shown that tools from dynamical systems can be used to analyze accelerated optimization algorithms. For example, it has been shown that the continuous limit of Nesterov’s accelerated gradient (NAG) gives an ODE whose convergence rate matches that of NAG for convex, unconstrained, and smooth problems. Conversely, it has been shown that NAG can be obtained as the discretization of an ODE, however since different discretizations lead to different algorithms, the choice of the discretization becomes important. The first part of this talk will extend this type of analysis to convex, constrained and non-smooth problems by using Lyapunov stability theory to analyze continuous limits of the Alternating Direction Method of Multipliers (ADMM). The second part of this talk will show that many existing and new optimization algorithms can be obtained by suitably discretizing a dissipative Hamiltonian. As an example, we will present a new method called Relativistic Gradient Descent (RGD), which empirically outperforms momentum, RMSprop, Adam and AdaGrad on several non-convex problems.

This is joint work with Guilherme França, Daniel Robinson and Jeremias Sulam.