Wasserstein Proximal Coordinate Gradient Algorithms
arxiv(2024)
摘要
Motivated by approximation Bayesian computation using mean-field variational
approximation and the computation of equilibrium in multi-species systems with
cross-interaction, this paper investigates the composite geodesically convex
optimization problem over multiple distributions. The objective functional
under consideration is composed of a convex potential energy on a product of
Wasserstein spaces and a sum of convex self-interaction and internal energies
associated with each distribution. To efficiently solve this problem, we
introduce the Wasserstein Proximal Coordinate Gradient (WPCG) algorithms with
parallel, sequential and random update schemes. Under a quadratic growth (QC)
condition that is weaker than the usual strong convexity requirement on the
objective functional, we show that WPCG converges exponentially fast to the
unique global optimum. In the absence of the QG condition, WPCG is still
demonstrated to converge to the global optimal solution, albeit at a slower
polynomial rate. Numerical results for both motivating examples are consistent
with our theoretical findings.
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