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We consider the optimization problem of minimizing a functional defined overa family of probability distributions, where the objective functional isassumed to possess a variational form. Such a distributional optimizationproblem arises widely in machine learning and statistics, with Monte-Carlosampling, variational inference, policy optimization, and generativeadversarial network as examples. For this problem, we propose a novelparticle-based algorithm, dubbed as variational transport, which approximatelyperforms Wasserstein gradient descent over the manifold of probabilitydistributions via iteratively pushing a set of particles. Specifically, weprove that moving along the geodesic in the direction of functional gradientwith respect to the second-order Wasserstein distance is equivalent to applyinga pushforward mapping to a probability distribution, which can be approximatedaccurately by pushing a set of particles. Specifically, in each iteration ofvariational transport, we first solve the variational problem associated withthe objective functional using the particles, whose solution yields theWasserstein gradient direction. Then we update the current distribution bypushing each particle along the direction specified by such a solution. Bycharacterizing both the statistical error incurred in estimating theWasserstein gradient and the progress of the optimization algorithm, we provethat when the objective function satisfies a functional version of thePolyak-\L{}ojasiewicz (PL) (Polyak, 1963) and smoothness conditions,variational transport converges linearly to the global minimum of the objectivefunctional up to a certain statistical error, which decays to zero sublinearlyas the number of particles goes to infinity.

Variational Transport: A Convergent Particle-BasedAlgorithm for Distributional Optimization