Trust-Region Optimization for Smooth Potential-Interaction Energies in Wasserstein Space

We study a class of stochastic nonsmooth optimization problems in which an outer variable minimizes the expectation of a pointwise maximum. This minimization--expectation--maximization (minEmax) problem arises in Wasserstein distributionally robust optimization and adversarially robust training, and it cannot in general be reformulated as a finite-dimensional minimax problem when the underlying distribution is not empirical. We propose a stochastic smoothing proximal gradient method based on log-mean-exp smoothing of the value function. Under compactness and Lipschitz-type assumptions, we present nonasymptotic analysis in terms of Goldstein stationarity and show that every almost-sure cluster point generated by our method is a Clarke stationary point; by Clarke regularity, such a point is also directional stationary for the original problem. Numerical experiments on newsvendor, robust regression, and adversarially robust learning problems show that the proposed method is competitive with existing baselines.

We study the long-time behavior of Wasserstein gradient flows for interaction energies $$\mathsf E[μ] = \frac12\iint_{M\times M}K(x,y)\,\mathrm dμ(x)\,\mathrm dμ(y)$$ on a closed manifold $M$. For kernels diagonal in a Laplace eigenbasis with nonnegative spectral coefficients, we prove a differential inequality relating the relative entropy to the energy gap. Consequently, for any nonnegative initial density $u_0\in L^p(M)$, $p>1$, the energy gap is integrable in time and satisfies $$\mathsf E[μ_t]-\mathsf E_{\min}=o(t^{-1}).$$ If all spectral coefficients are positive, the flow converges weakly to the constant measure. These interaction energies need not be geodesically convex in Wasserstein space, and the associated flows contain no diffusion; their global convergence therefore does not follow from standard Wasserstein gradient flow theory. The kernels covered by our results include zonal kernels on spheres, kernels arising in transformer models, regularized Riesz kernels, and inverse fractional Laplacian kernels. We also investigate the sharpness of the $o(t^{-1})$ rate. For any smooth kernel in this class with infinitely many positive spectral coefficients and any $δ>0$, we construct a solution of the linearized flow whose energy is comparable to $t^{-1-δ}$ along a sequence of times tending to infinity. Moreover, for any $δ>0$, by choosing a suitable inverse fractional Laplacian kernel on the flat torus, we construct an exact solution of the nonlinear Wasserstein gradient flow whose energy is comparable to $t^{-1-δ}$. The nonlinear construction is based on uniform-in-time estimates for the evolution of the dyadic Fourier coefficient blocks and a blockwise energy-persistence argument. These estimates also yield a uniform-in-time quantitative comparison between the nonlinear Wasserstein gradient flow and its linearization.

Optimizing functionals over the space of probability measures is now ubiquitous in machine learning. A widely used approach is to perform the optimization directly over the Wasserstein space, but many objective functionals of practical interest are non-convex along Wasserstein geodesics, making the analysis of standard first-order methods challenging. In this work, we study a class of objectives over the Wasserstein space that admit a difference-of-convex (DC) decomposition and we lift the classical convex-concave procedure (CCCP) to this setting. Under smoothness and strong convexity assumptions on the convex components of the decomposition, we prove almost stationarity along the iterates of the resulting algorithm. Our main focus is on the Maximum Mean Discrepancy (MMD) and the Energy Distance (ED) functionals, for which we develop explicit Wasserstein DC decompositions, and establish local convergence of the scheme under mild assumptions. Empirically, we show that well-chosen DC decompositions yield faster and more stable convergence than Wasserstein gradient descent on these MMD objectives.