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

arXiv: https://arxiv.org/abs/2608.13467 · HTML: https://mlsanity.com/paper/2608.13467/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 33 references that are in the map.
- First posted on arXiv 13 Aug 2026.

## Builds on (most cited first, 14 of 14)

- [Non-asymptotic convergence analysis for the Unadjusted Langevin Algorithm](https://mlsanity.com/paper/1507.05021/) (2015)
- [Analysis of Langevin Monte Carlo via convex optimization](https://mlsanity.com/paper/1802.09188/) (2018)
- [Efficient Bayesian computation by proximal Markov chain Monte Carlo: when Langevin meets Moreau](https://mlsanity.com/paper/1612.07471/) (2016)
- [Proximal Markov chain Monte Carlo algorithms](https://mlsanity.com/paper/1306.0187/) (2013)
- [Sampling from a log-concave distribution with compact support with proximal Langevin Monte Carlo](https://mlsanity.com/paper/1705.08964/) (2017)
- [Primal Dual Interpretation of the Proximal Stochastic Gradient Langevin Algorithm](https://mlsanity.com/paper/2006.09270/) (2020)
- [An Efficient Sampling Algorithm for Non-smooth Composite Potentials](https://mlsanity.com/paper/1910.00551/) (2019)
- [Stochastic Proximal Langevin Algorithm: Potential Splitting and Nonasymptotic Rates](https://mlsanity.com/paper/1905.11768/) (2019)
- [Subgradient Langevin Methods for Sampling from Non-smooth Potentials](https://mlsanity.com/paper/2308.01417/) (2023)
- [Bregman Proximal Langevin Monte Carlo via Bregman--Moreau Envelopes](https://mlsanity.com/paper/2207.04387/) (2022)
- [Diffusion at Absolute Zero: Langevin Sampling Using Successive Moreau Envelopes [conference paper]](https://mlsanity.com/paper/2502.01358/) (2025)
- [Accelerated Bayesian imaging by relaxed proximal-point Langevin sampling](https://mlsanity.com/paper/2308.09460/) (2023)
- [Improved Guarantees for Langevin Monte Carlo with Average Smoothness](https://mlsanity.com/paper/2605.31413/) (2026)
- [A proximal gradient algorithm for composite log-concave sampling](https://mlsanity.com/paper/2605.12461/) (2026)

## Built on by (most cited first, 2 of 2)

- [Poisson-Corrector Complexity Bounds for Moreau--Yosida Unadjusted Langevin Sampling](https://mlsanity.com/paper/2609.12594/) (2026)
- [Near-Linear Accuracy Bounds for Moreau--Yosida Unadjusted Langevin Sampling](https://mlsanity.com/paper/2609.40193/) (2026)

## Abstract (arXiv)

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.