We establish near-linear accuracy bounds for the classical Moreau--Yosida unadjusted Langevin algorithm (MYULA). The target is
π ∝ e − f − g π\propto e^{-f-g} π ∝ e − f − g , where
f ∈ C 2 ( R d ) f\in C^2(\mathbb{R}^d) f ∈ C 2 ( R d ) is
m m m -strongly convex with Lipschitz gradient and
g g g is convex and globally Lipschitz. Under an explicit parameter-dependent step-size condition, we bound the invariant-measure bias relative to the Moreau-smoothed target by
O ~ ( h ) \widetilde O(h) O ( h ) , with only logarithmic dependence on the inverse smoothing parameter in the error coefficient. Combining this estimate with the Moreau approximation bias and Wasserstein contraction gives
O ~ ( ε − 1 ) \widetilde O(\varepsilon^{-1}) O ( ε − 1 ) iterations to make the
N N N th-iterate law
μ N μ_N μ N satisfy
m W 2 ( μ N , π ) ≤ ε \sqrt m\,W_2(μ_N,π)\le\varepsilon m W 2 ( μ N , π ) ≤ ε , for fixed model parameters and initialization. We bound the stationary error directly, without assuming third derivatives or a Lipschitz Hessian. Each iteration uses one gradient evaluation and one exact proximal evaluation. The key idea in our analysis is to convert a second-order stationary residual into a Wasserstein bound using a Poisson-based estimate.