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Motivated by the computation of the non-parametric maximum likelihoodestimator (NPMLE) and the Bayesian posterior in statistics, this paper exploresthe problem of convex optimization over the space of all probabilitydistributions. We introduce an implicit scheme, called the implicit KL proximaldescent (IKLPD) algorithm, for discretizing a continuous-time gradient flowrelative to the Kullback-Leibler divergence for minimizing a convex targetfunctional. We show that IKLPD converges to a global optimum at a polynomialrate from any initialization; moreover, if the objective functional is stronglyconvex relative to the KL divergence, for example, when the target functionalitself is a KL divergence as in the context of Bayesian posterior computation,IKLPD exhibits globally exponential convergence. Computationally, we propose anumerical method based on normalizing flow to realize IKLPD. Conversely, ournumerical method can also be viewed as a new approach that sequentially trainsa normalizing flow for minimizing a convex functional with a strong theoreticalguarantee.

Minimizing Convex Functionals over Space of Probability Measures via KL Divergence Gradient Flow