  Optimization is at the heart of machine learning, statistics and many applied
scientific disciplines. It also has a long history in physics, ranging from the
minimal action principle to finding ground states of disordered systems such as
spin glasses. 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: Forward-backward splitting, Tseng
splitting, Douglas-Rachford, alternating direction method of multipliers, and
the more recent Davis-Yin. These methods sit on a higher level of abstraction
compared to gradient-based ones, with deep roots in nonlinear functional
analysis. We show that all of these methods are actually 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. We show that similar discretization schemes applied to
Newton's equation with an additional dissipative force, which we refer to as
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. Furthermore, we extend these methods to
stochastic 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. Our results therefore provide a
unified framework from which several important optimization methods are nothing
but simulations of classical dissipative systems.
