Active-Trace Complexity Bounds for Moreau--Yosida Unadjusted Langevin Sampling
We study the Moreau--Yosida unadjusted Langevin algorithm (MYULA) for the nonsmooth composite target \[ \pi(dx)\propto \exp\{-f(x)-g(x)\}\,dx, \qquad x\in\mathbb R^d, \] where \(f\) is \(m\)-strongly convex with \(L_f\)-Lipschitz gradient and \(g\) is convex and \(G\)-Lipschitz. Let \(g_\lambda\) be the Moreau envelope of \(g\), \(\pi_\lambda\) the corresponding smoothed target, and \(a_\lambda=\operatorname{tr}H_\lambda\), where \(H_\lambda\) is the a.e./weak Hessian of \(g_\lambda\). We show that the leading MYULA discretization error is controlled by the reference active trace \(B_{\mathrm{ref}}\), the average of \(a_\lambda\) along the heat substep of one MYULA update started from \(\pi_\lambda\), rather than by the global curvature bound \(d/\lambda\). If \(M_\lambda\) is an a.e. upper bound for \(a_\lambda\), then, up to logarithmic factors, \[ N \lesssim \frac{1}{m} \left[ L_f + \frac{ \tau_f+G^2+B_{\mathrm{ref}} }{ \varepsilon_{\mathrm{alg}}^2 } + \frac{M_\lambda}{\varepsilon_{\mathrm{alg}}} \right], \qquad \tau_f:= \sup_x\operatorname{tr}\nabla^2 f(x), \] iterations suffice to ensure \(\sqrt m\,W_2(\mu_N,\pi_\lambda)\leq\varepsilon_{\mathrm{alg}}\), where \(\mu_N\) is the law of the \(N\)-th iterate and \(W_2\) is the quadratic Wasserstein distance. We also prove the Moreau-bias bound \[ \sqrt m\,W_2(\pi_\lambda,\pi) \leq \frac{G^2\lambda}{4}. \] Thus, choosing \(\lambda\asymp\varepsilon/G^2\) gives an end-to-end guarantee for \(\pi\). The universal estimate \(B_{\mathrm{ref}}\leq d/\lambda\) 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 \(\lambda\), yielding \(\widetilde O(\varepsilon^{-2})\) for the same classical MYULA kernel.