Dimension-free regimes for unadjusted Langevin in ridge-separable models: A subspace robust Wasserstein analysis

The Unadjusted Langevin Algorithm (ULA) is widely used for high-dimensional Bayesian inference. Classical analyses in Wasserstein-2 distance predict an asymptotic bias and a mixing time that grow with dimension, but this metric is stronger than necessary for many statistical tasks. We instead analyze the bias in subspace robust Wasserstein distance, which controls the mean and arbitrary low-dimensional marginals and is therefore well suited to observables typically estimated in practice. For ridge-separable potentials formed with i.i.d. data, we prove high-probability bounds showing a bell-shaped mixing profile as the number of data points $N$ varies relative to the dimension $d$. In the overparameterized regime, the effective dimension is the rank $N$ of the data-dependent perturbation, leading to complexity depending on $N$ rather than $d$. In the large-sample regime, Gaussian-perturbation estimates yield dimension-free complexity once $N$ is sufficiently large. In contrast, near the proportional regime $\asymp d$, we prove that dimension-free guarantees are impossible in general.