We estimate convergence rates of value functions for particle approximations of a controlled nonlinear filtering problem. The state is a McKean--Vlasov diffusion on the flat torus, driven by hidden idiosyncratic noise and observed common noise. The filter---the conditional law of the state given the observations---serves as the state variable of the control problem, and the associated value function solves a second-order Hamilton--Jacobi--Bellman equation on the Wasserstein space. We approximate this problem by a centralized \(N\)-particle control problem with independent idiosyncratic noises and a common observation noise. The framework accommodates nonseparable rewards and controlled drifts. Since a single control is applied to the entire population, the Hamiltonian is defined by an optimization performed after integration over the population. Under smoothness of the data, uniform ellipticity, and regularity of this Hamiltonian, we establish uniform value-function error bounds of order \(N^{-1/6}\) for \(d=1\), \(N^{-1/6}(\log N)^{1/3}\) for \(d=2\), and \(N^{-1/(3d)}\) for \(d>2\). The proof combines a translation lift in the common-noise direction, Fourier--Wasserstein inf- and sup-convolutions, viscosity comparison, and particle derivative estimates uniform in \(N\).
Erhan Bayraktar, Ibrahim Ekren, Xihao He et al.· 0 citations
We study the computation of static mean-field equilibria on a compact state space by formulating the equilibrium condition as a variational inequality over probability measures. We propose an entropic variant of Korpelevich's extragradient algorithm---the Kullback--Leibler Mirror-Prox method---in which Euclidean projections are replaced by relative-entropy proximal steps. Each half-step is therefore an explicit exponential reweighting of the current measure, implemented on a finite state-space discretization. Under Lasry--Lions monotonicity and continuity assumptions, we prove convergence of mesh-refined ergodic averages and obtain finite-iteration Minty-residual and approximate-equilibrium bounds that jointly quantify iteration and discretization errors. Under strong monotonicity, we derive metric convergence rates for the last, best, and averaged iterates. We also develop a KL-type Tikhonov regularization that selects the equilibrium minimizing relative entropy with respect to a reference measure. The framework applies to potential and nonpotential cost operators and does not require differentiability or convexity of the cost in the individual state.
Erhan Bayraktar, Ibrahim Ekren, L. Vy et al.· 0 citations
