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

We study the Moreau--Yosida unadjusted Langevin algorithm (MYULA) for the nonsmooth composite target

$$π(dx)\propto \exp\{-f(x)-g(x)\}\,dx, \qquad x\in\mathbb R^d,$$

where $f$ is $m$-strongly convex with $L_f$-Lipschitz gradient and $g$ is convex and $G$-Lipschitz. Let $g_λ$ be the Moreau envelope of $g$, $π_λ$ the corresponding smoothed target, and $a_λ=\operatorname{tr}H_λ$, where $H_λ$ is the a.e./weak Hessian of $g_λ$. We show that the leading MYULA discretization error is controlled by the reference active trace $B_{\mathrm{ref}}$, the average of $a_λ$ along the heat substep of one MYULA update started from $π_λ$, rather than by the global curvature bound $d/λ$. If $M_λ$ is an a.e. upper bound for $a_λ$, then, up to logarithmic factors,

$$N \lesssim \frac{1}{m} \left[ L_f + \frac{ τ_f+G^2+B_{\mathrm{ref}} }{ \varepsilon_{\mathrm{alg}}^2 } + \frac{M_λ}{\varepsilon_{\mathrm{alg}}} \right], \qquad τ_f:= \sup_x\operatorname{tr}\nabla^2 f(x),$$

iterations suffice to ensure $\sqrt m\,W_2(μ_N,π_λ)\leq\varepsilon_{\mathrm{alg}}$, where $μ_N$ is the law of the $N$-th iterate and $W_2$ is the quadratic Wasserstein distance. We also prove the Moreau-bias bound

$$\sqrt m\,W_2(π_λ,π) \leq \frac{G^2λ}{4}.$$

Thus, choosing $λ\asymp\varepsilon/G^2$ gives an end-to-end guarantee for $π$. The universal estimate $B_{\mathrm{ref}}\leq d/λ$ 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 $B_{\mathrm{ref}}$ independent of $λ$, yielding $\widetilde O(\varepsilon^{-2})$ for the same classical MYULA kernel.