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 th