Particle methods for optimization over the space of measures

Optimization problems over measures arise naturally in many areas of applied mathematics and machine learning, but solving them computationally requires dealing with the infinite-dimensional nature of the measure variables. This thesis focuses on particle methods, a class of algorithms that parametrize the variable by a finite sum of Dirac deltas and apply gradient-based updates to their positions. We analyze the convergence of these methods in several settings of practical relevance, leveraging ideas from optimal transport, and particularly the formal correspondence (known as Otto calculus) between Wasserstein geometry on the space of measures and Riemannian geometry. We first consider mean-field Langevin dynamics, which are the infinite-particle limits of noisy particle gradient flows. We obtain tight estimates for their local convergence rates in chi2-divergence, and we show a local convergence result for mean-field Langevin descent-ascent, a coupled system of dynamics whose global convergence is an open problem. We then study the problem of computing mixed Nash equilibria of two-player zero-sum games with continuous strategy sets, assuming a unique and finitely-supported equilibrium. We establish the local exponential convergence of a proximal point-type particle method that optimizes both particle positions and weights. We also analyze, in a general finite-dimensional context, the conditions for gradient methods in min-max optimization to be locally convergent. In a third setting, we examine how global convergence guarantees can be obtained for optimization over unnormalized measures, by applying noisy particle methods to liftings or bilevel formulations of the problem. Finally, we propose an adaptation of the Otto calculus framework enabling the formal computation of derivatives of arbitrary orders along probability flows. We apply this tool to make progress on a conjecture from information theory concerning the successive time-derivatives of the entropy along the heat flow.