The Wasserstein Proximal Gradient Algorithm
This work addresses the gap in understanding discrete-time behavior and minimization power of discretized gradient flows in optimal transport, which is incremental for researchers in machine learning and optimization.
The paper tackles the problem of discretizing Wasserstein gradient flows for minimizing objective functions over probability measures, proposing a Forward Backward scheme that handles smooth and nonsmooth geodesically convex terms, and shows it achieves convergence guarantees similar to proximal gradient methods in Euclidean spaces.
Wasserstein gradient flows are continuous time dynamics that define curves of steepest descent to minimize an objective function over the space of probability measures (i.e., the Wasserstein space). This objective is typically a divergence w.r.t. a fixed target distribution. In recent years, these continuous time dynamics have been used to study the convergence of machine learning algorithms aiming at approximating a probability distribution. However, the discrete-time behavior of these algorithms might differ from the continuous time dynamics. Besides, although discretized gradient flows have been proposed in the literature, little is known about their minimization power. In this work, we propose a Forward Backward (FB) discretization scheme that can tackle the case where the objective function is the sum of a smooth and a nonsmooth geodesically convex terms. Using techniques from convex optimization and optimal transport, we analyze the FB scheme as a minimization algorithm on the Wasserstein space. More precisely, we show under mild assumptions that the FB scheme has convergence guarantees similar to the proximal gradient algorithm in Euclidean spaces.