Gradient flows and proximal splitting methods: a unified view on accelerated and stochastic optimization

Optimization is at the heart of machine learning, statistics, and several applied scientific disciplines. Proximal algorithms form a class of methods that are broadly applicable and are particularly well-suited to nonsmooth, constrained, large-scale, and distributed optimization problems. There are essentially five proximal algorithms currently known, each proposed in seminal work: forward-backward splitting, Tseng splitting, Douglas-Rachford, alternating direction method of multipliers, and the more recent Davis-Yin. Such methods sit on a higher level of abstraction compared to gradient-based methods, having deep roots in nonlinear functional analysis. In this paper, we show that all of these algorithms can be derived as different discretizations of a single differential equation, namely the simple gradient flow which dates back to Cauchy (1847). An important aspect behind many of the success stories in machine learning relies on "accelerating" the convergence of first order methods. However, accelerated methods are notoriously difficult to analyze, counterintuitive, and without an underlying guiding principle. We show that by employing similar discretization schemes to Newton's classical equation of motion with an additional dissipative force, which we refer to as the accelerated gradient flow, allow us to obtain accelerated variants of all these proximal algorithms; the majority of which are new although some recover known cases in the literature. Moreover, we extend these algorithms to stochastic optimization settings, allowing us to make connections with Langevin and Fokker-Planck equations. Similar ideas apply to gradient descent, heavy ball, and Nesterov's method which are simpler. These results thus provide a unified framework from which several optimization methods can be derived from basic physical systems.