Active-Trace Complexity Bounds for Moreau--Yosida Unadjusted Langevin Sampling

We study the Moreau--Yosida unadjusted Langevin algorithm (MYULA) for the nonsmooth composite target [ π(dx)propto exp-f(x)-g(x),dx, qquad x∈mathbb Rd, ] where (f) is (m)-strongly convex with (Lf)-Lipschitz gradient and (g) is convex and (G)-Lipschitz. Let (gλ) be the Moreau envelope of (g), (πλ) the corresponding smoothed target, and (aλ=operatornametrHλ), where (Hλ) is the a.e./weak Hessian of (gλ). We show that the leading MYULA discretization error is controlled by the reference active trace (Bref), the average of (aλ) along the heat substep of one MYULA update started from (πλ), rather than by the global curvature bound (d/λ). If (Mλ) is an a.e. upper bound for (aλ), then, up to logarithmic factors, [ N lesssim frac1m left[ Lf + frac τf+G2+Bref varepsilonalg2 + fracMλvarepsilonalg right], qquad τf:= supxoperatornametrnabla2 f(x), ] iterations suffice to ensure (√ m,W2(μN,πλ)≤varepsilonalg), where (μN) is the law of the (N)-th iterate and (W2) is the quadratic Wasserstein distance. We also prove the Moreau-bias bound [ √ m,W2(πλ,π) ≤ fracG2λ4. ] Thus, choosing (λasympvarepsilon/G2) gives an end-to-end guarantee for (π). The universal estimate (Bref≤ d/λ) yields (widetilde O(varepsilon-3)) accuracy dependence. For the structured piecewise-linear, lasso-type, group, and total-variation penalties considered here, curvature--tube estimates make (Bref) independent of (λ), yielding (widetilde O(varepsilon-2)) for the same classical MYULA kernel.