We introduce a (de)-regularization of the Maximum Mean Discrepancy (DrMMD) and its Wasserstein gradient flow. Existing gradient flows that transport samples from source distribution to target distribution with only target samples, either lack tractable numerical implementation (f-divergence flows) or require strong assumptions and modifications, such as noise injection, to ensure convergence (Maximum Mean Discrepancy flows). In contrast, DrMMD flow can simultaneously (i) guarantee near-global convergence for a broad class of targets in both continuous and discrete time, and (ii) be implemented in closed form using only samples. The former is achieved by leveraging the connection between the DrMMD and the chi(2)-divergence, while the latter comes by treating DrMMD as MMD with a de-regularized kernel. Our numerical scheme employs an adaptive de-regularization schedule throughout the flow to optimally balance the trade-off between discretization errors and deviations from the chi(2) regime. The potential application of the DrMMD flow is demonstrated across several numerical experiments, including a large-scale setting of training student/teacher networks.
Under general assumptions on the target distribution p^⋆, we establish a sharp Lipschitz regularity theory for flow-matching vector fields and diffusion-model scores, with optimal dependence on time and dimension. As applications, we obtain Wasserstein discretization bounds for Euler-type samplers in dimension d: with N discretization steps, the error achieves the optimal rate √(d)/N up to logarithmic factors. Moreover, the constants do not deteriorate exponentially with the spatial extent of p^⋆. We also show that the one-sided Lipschitz control yields a globally Lipschitz transport map from the standard Gaussian to p^⋆, which implies Poincaré and log-Sobolev inequalities for a broad class of probability measures.
Many modern generative models can be viewed as minimizing divergences between probability distributions, yet they rely on different algorithmic and geometric principles. Wasserstein gradient flows provide a continuous-time formulation for optimizing over distributions, and can be approximated through their implicit discretization via the Jordan–Kinderlehrer–Otto (JKO) scheme. In this work, we present a unified theoretical framework for generative modeling based on Wasserstein gradient flows, which we refer to as Generative Wasserstein Flows. We show that a broad class of existing methods can be derived as instances of parametric JKO schemes for f-divergences objectives, and we establish equivalences between several recently proposed algorithms. We extend this framework beyond f-divergences to integral probability metrics, deriving new JKO-based generative algorithms for objectives such as Maximum Mean Discrepancy. We also clarify their connections with GANs. Finally, we analyze parametric Wasserstein flows, where the evolution is restricted to distributions generated by parameterized maps. We characterize the resulting dynamics as projected or preconditioned Wasserstein gradient flows, highlighting the role of the Wasserstein geometry in shaping the learning dynamics of generative models.
Path-dependent McKean–Vlasov (MKV) control models large interacting populations with history-dependent dynamics and costs. This paper develops a unified approximation-and-learning framework for continuous time path-dependent MKV problem under open-loop controls. First, an Euler discretization scheme with piecewise-constant controls is shown to achieve a non-asymptotic error of O(h^1/4). Second, we establish a discrete dynamic programming principle and prove value equivalence between open-loop and history-dependent feedback controls, enabling optimization on a reduced filtration. Third, an interacting particle system is introduced to approximate the continuous-time value, yielding an overall error bound of O(h^1/4) + O(M^-γ) for M particles and an explicitly given γ> 0. Finally, we propose a fully implementable neural-network policy-gradient method using pathwise features. Numerical experiments, including a path-dependent linear-quadratic benchmark, demonstrate the effectiveness of the algorithm.
This study proposes a formal, computationally efficient nonparametric omnibus test for treatment-effect heterogeneity that is compatible with a broad class of estimators, including modern machine-learning methods. The test is designed for settings in which identification can rely on high-dimensional controls while heterogeneity is assessed with respect to a low-dimensional subset of covariates. We derive the test statistic's asymptotic null distribution and develop a bootstrap procedure that is efficient because it avoids re-estimating nuisance parameters in each iteration. The testing approach applies to multiple empirical designs, including randomized experiments, selection-on-observables, difference-in-differences, and instrumental-variables settings. Monte Carlo simulations show that the test attains near-nominal size under the null and exhibits good power against heterogeneous alternatives. We further illustrate the procedure using two empirical applications on retirement savings and trade liberalization.
