Rate of convergence for particle approximation of PDEs in Wasserstein space

Abstract

We prove a rate of convergence for the N-particle approximation of a second-order partial differential equation in the space of probability measures, such as the master equation or Bellman equation of the mean-field control problem under common noise. The rate is of order $1/N$ for the pathwise error on the solution v and of order $1/\sqrt{N}$ for the $L^2$ -error on its L-derivative $\partial_\mu v$ . The proof relies on backward stochastic differential equation techniques.

Approximately optimal distributed stochastic controls beyond the mean field setting

“…This then implies the optimal rate of |V − V MF | = O(1/n) for the convergence problem; see Theorem 5.6 for a precise statement. This appears to be a new result at this level of generality, though we stress that it relies crucially on some smoothness and most importantly the displacement convexity of F and G. It is expected, though not documented, that the same assumptions would lead to the existence of a smooth solution of the Hamilton-Jacobi equation on P 2 (R d ) which is (at least formally) satisfied by the value function of the mean field control problem; a smooth solution of this equation can be used to prove the same O(1/n) convergence rate using the method developed in [8] for mean field games and explained in [26] in the control setting. Interestingly, our method makes no use of this Hamilton-Jacobi equation.…”

“…The full-information convergence problem (i.e., the convergence of V to V MF ), has by now been resolved in a qualitative sense, in quite general settings for mean field games [29,18] and control [28,19]. The existing quantitative results for games [8,17] rely on the analysis of the so-called master equation, and the results for control [26,7,9] rely similarly on the Hamilton-Jacobi equation solved by the mean field value function, though we mention that alternative BSDE techniques of [38] have yielded quantitative convergence results for mean field games. In our notation, these works studied directly the convergence of V to V MF , without any intermediate use of V dist .…”

Linear-quadratic mean field games of controls with non-monotone data

“…, and a η , a η 1 , bη , bη 1 , qη , gη are defined by (27) where η1 (•) and η2 (•) are replaced by ηt 0 ,x 0 1 (•) and ηt 0 ,x 0 2 (•), respectively. In addition, FBSDE (35) is associated to the following PDE…”

“…There have been a lot of literature already on the convergence rate from N -particle systems to its mean field limit for both mean field control problems and mean field games. We refer to [7,11,24,27,28,38] for the convergence rate problems in either the mean field control setting under some convexity condition or in the mean field game setting under certain monotonicity condition. When the convexity/monotonicity condition fails to be satisfied, some qualitative results have been obtained in [1,12,20,22,23,31,37,45].…”

Importance Sampling for the Empirical Measure of Weakly Interacting Diffusions

“…Perhaps due to the computational intractability of both the Dawson-Gärtner [29] and Budhiraja-Dupuis-Fischer [16] rate functions, practical applications of the LDP for the many-particle limit of the empirical measure (2) associated with (1) have been few and far between (for some exceptions see, e.g., [30,49,50,70]). On the other hand, in recent years the HJB equation on Wasserstein space of [73] (along with the related equations in [21,67]) has received an immense amount of attention both in terms of numerical applications [20,22,47,54,55,60,66] and theoretical results [5,18,19,26,27,56,65,73,80]. The Dawson-Gärtner rate function has previously been related to the theory of mean field games and control through the observation that it can be viewed in terms of derivatives of the free energy associated to the limiting McKean-Vlasov equation viewed as a gradient flow on Wasserstein space in some settings [1,2,4,[41][42][43][44]52].…”