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We consider problems of minimizing functionals $\mathcal{F}$ of probabilitymeasures on the Euclidean space. To propose an accelerated gradient descentalgorithm for such problems, we consider gradient flow of transport maps thatgive push-forward measures of an initial measure. Then we propose adeterministic accelerated algorithm by extending Nesterov's accelerationtechnique with momentum. This algorithm do not based on the Wassersteingeometry. Furthermore, to estimate the convergence rate of the acceleratedalgorithm, we introduce new convexity and smoothness for $\mathcal{F}$ based ontransport maps. As a result, we can show that the accelerated algorithmconverges faster than a normal gradient descent algorithm. Numericalexperiments support this theoretical result.

Accelerated gradient descent method for functionals of probability measures by new convexity and smoothness based on transport maps