MASLA: Subdifferential Langevin Sampling

• MASLA is a subdifferential Langevin algorithm that generalizes MALA to locally Lipschitz and non-differentiable potentials.
• It replaces the gradient with an element from a conservative field and applies a Metropolis–Hastings correction to restore exact stationarity.
• The algorithm demonstrates competitive empirical performance in challenging non-smooth settings and offers extensions for high-dimensional sampling problems.

Metropolis-adjusted Subdifferential Langevin Algorithm (MASLA) is a Metropolis–Hastings generalization of the Metropolis-Adjusted Langevin Algorithm (MALA) for target densities of the form $\pi(x)\propto e^{-U(x)}$ when the potential $U:\mathbb{R}^d\to[0,\infty)$ is only locally Lipschitz and may be non-convex and non-differentiable. In this formulation, the gradient $\nabla U$ is replaced by an element of a set-valued conservative field $D_U(x)$ that generalizes the Clarke subgradient, and exact stationarity is restored by an MH correction (Ning, 9 Jul 2025). The acronym MASLA has also been used in related optimal-scaling work for a Moreau–Yosida regularized proximal MALA construction in which $U$ is replaced by its envelope $U_\lambda$ and the proposal is expressed through $\operatorname{prox}_{\lambda U}$ (Crucinio et al., 2023).

1. Problem setting and motivation

The target problem is to sample from a probability density $\pi$ on $\mathbb{R}^d$ of the form

$\pi(x)\propto e^{-U(x)},$

with $U:\mathbb{R}^d\to[0,\infty)$0 assumed only to be locally Lipschitz. By Rademacher’s theorem, $U:\mathbb{R}^d\to[0,\infty)$1 is differentiable almost everywhere, but global smoothness, and even convexity, are not assumed. This setting is outside the standard regularity regime of classical MALA, which requires $U:\mathbb{R}^d\to[0,\infty)$2 so that the Euler–Maruyama proposal

$U:\mathbb{R}^d\to[0,\infty)$3

is well defined (Ning, 9 Jul 2025).

The motivation for MASLA is tied to three limitations of unadjusted or smoothness-dependent Langevin discretizations in the non-smooth regime. When $U:\mathbb{R}^d\to[0,\infty)$4 is non-smooth, the unadjusted Langevin algorithm can fail to have the correct invariant law, can even be transient, and cannot exploit subgradient information. MASLA addresses these points by replacing $U:\mathbb{R}^d\to[0,\infty)$5 with an element of $U:\mathbb{R}^d\to[0,\infty)$6 and by applying a Metropolis–Hastings accept–reject step, thereby extending Langevin-type sampling to locally Lipschitz, generally non-differentiable, and non-convex log-densities (Ning, 9 Jul 2025).

2. Algorithmic construction

MASLA is motivated by the generalized Langevin differential inclusion

$U:\mathbb{R}^d\to[0,\infty)$7

where $U:\mathbb{R}^d\to[0,\infty)$8 is a conservative field for $U:\mathbb{R}^d\to[0,\infty)$9. The discrete proposal with step size $\nabla U$0 selects a subgradient $\nabla U$1 and performs the Euler–Maruyama step

$\nabla U$2

If $\nabla U$3 is differentiable at $\nabla U$4, the natural choice is $\nabla U$5. At a non-differentiable point, one may compute a Clarke subgradient via automatic differentiation or backpropagation, or pick any element of a conservative field, for example from the convex hull of nearby gradients (Ning, 9 Jul 2025).

Let $\nabla U$6 denote the density of the proposal in (1). The MH correction is then

$\nabla U$7

The next state is $\nabla U$8 with probability $\nabla U$9 and remains at $D_U(x)$0 otherwise. This structure preserves the Langevin-type local move while reinstating exact stationarity through detailed balance (Ning, 9 Jul 2025).

3. Stationarity, reversibility, and convergence status

The principal theoretical result states that if $D_U(x)$1 is $D_U(x)$2 on an open set whose complement has Lebesgue measure zero, and if $D_U(x)$3 has a density with respect to Lebesgue measure, then for almost every $D_U(x)$4 the MASLA chain is reversible with respect to $D_U(x)$5, and hence $D_U(x)$6 is its stationary distribution (Ning, 9 Jul 2025).

The proof strategy proceeds by showing that the proposal kernel $D_U(x)$7 admits a density $D_U(x)$8. This is done by verifying that the map

$D_U(x)$9

is a local diffeomorphism for almost every $U$0 and almost every $U$1, using the inverse-function theorem away from a null set of Hessian singularities. Detailed balance,

$U$2

then yields stationarity (Ning, 9 Jul 2025).

The convergence-rate picture is more limited. Explicit nonasymptotic Wasserstein or total-variation error bounds are not given. The stated position is that, under standard dissipativity and smoothness-a.e. conditions, one expects MASLA to inherit the geometric ergodicity and $U$3 mixing-time behavior proved for MALA and related nonsmooth extensions (Ning, 9 Jul 2025). This remains an expectation rather than a theorem in the cited work.

4. Implementation and computational profile

In practice, the step size $U$4 is tuned to achieve an average acceptance rate near $U$5, identified as the optimal value for MALA in high dimensions. Small $U$6 reduces discretization bias but slows mixing, whereas large $U$7 lowers acceptance. The paper presents this as a practical guideline rather than a derived optimality result for the full non-smooth setting (Ning, 9 Jul 2025).

The per-iteration cost consists of one gradient or subgradient evaluation together with $U$8 operations to sample $U$9 and compute Gaussian densities for $U_\lambda$0. The implementation notes also identify several high-dimensional accelerations: subsampled or stochastic subgradient estimates, preconditioning, and blockwise updates. These are presented as compatible extensions of the core scheme rather than as parts of the proved theory (Ning, 9 Jul 2025).

5. Empirical evaluation

The reported experiments include a composite TV–$U_\lambda$1 example in dimension $U_\lambda$2 with

$U_\lambda$3

where $U_\lambda$4, $U_\lambda$5, and $U_\lambda$6. MASLA is compared with Grad-sub and Prox-sub from Habring et al. (2024) in Wasserstein-2 distance, and with P-MALA from Pereyra (2016) and MYULA from Durmus and Pereyra (2022) in total-variation distance. The stated result is that MASLA matches or slightly improves on state-of-the-art convergence rates across $U_\lambda$7 and exhibits clear exponential ergodicity (Ning, 9 Jul 2025).

A second benchmark studies the non-proximal one-dimensional potential $U_\lambda$8. This function is locally Lipschitz and path-differentiable, but its proximal operator at $U_\lambda$9 is multivalued, $\operatorname{prox}_{\lambda U}$0, so prox-based samplers cannot apply. In this setting, Unadjusted Subdifferential Langevin (USLA) diverges from the true law, whereas MASLA with the MH correction closely tracks the exact target density. The reported quantitative errors are $\operatorname{prox}_{\lambda U}$1 and $\operatorname{prox}_{\lambda U}$2 for MASLA, while the corresponding USLA errors exceed $\operatorname{prox}_{\lambda U}$3 (Ning, 9 Jul 2025).

6. Relation to Moreau–Yosida and proximal formulations

A source of potential confusion is that the acronym MASLA also appears in the optimal-scaling literature for a Moreau–Yosida Metropolis-adjusted Langevin scheme. In that construction, $\operatorname{prox}_{\lambda U}$4 is assumed convex, proper, and lower-semicontinuous, and one introduces the Moreau–Yosida envelope

$\operatorname{prox}_{\lambda U}$5

The regularized potential is $\operatorname{prox}_{\lambda U}$6 with

$\operatorname{prox}_{\lambda U}$7

and $\operatorname{prox}_{\lambda U}$8 is Lipschitz continuous with constant $\operatorname{prox}_{\lambda U}$9. The proposal is then

$\pi$0

or, equivalently,

$\pi$1

followed by the standard MH correction (Crucinio et al., 2023).

Crucinio et al. analyze optimal scaling for product-form targets $\pi$2 under regimes where

$\pi$3

For smooth targets, the optimal acceptance is reported as $\pi$4 in one regime and $\pi$5 in others; for the non-smooth Laplace target $\pi$6, the only non-degenerate scaling has $\pi$7, with asymptotically optimal acceptance $\pi$8. The practical guidance given there is to choose $\pi$9 proportional to $\mathbb{R}^d$0 for non-smooth targets and to $\mathbb{R}^d$1 for smooth targets, and to set $\mathbb{R}^d$2 unless the target is very smooth (Crucinio et al., 2023).

These two usages of MASLA are related but not identical. The Moreau–Yosida construction smooths the target first and works through proximal maps; the 2025 “Metropolis-adjusted Subdifferential Langevin Algorithm” uses a conservative field directly and is explicitly designed for locally Lipschitz, generally non-differentiable, non-convex potentials. This suggests two distinct nonsmooth Metropolized Langevin paradigms within the same acronymic neighborhood: proximal regularization and direct subdifferential dynamics.

7. Applications, extensions, and open questions

The stated domain in which MASLA is especially useful consists of high-dimensional sampling problems where $\mathbb{R}^d$3 is non-smooth or non-convex, such as $\mathbb{R}^d$4-penalized posteriors, ReLU-net losses, and nonsmooth composite objectives, provided that a subgradient can be computed through automatic differentiation (Ning, 9 Jul 2025).

The same source identifies several extensions: nonasymptotic error bounds under non-convex dissipativity assumptions, stochastic-gradient MASLA using mini-batch subgradients, Riemann-manifold or preconditioned MASLA via variable-metric conservative fields, and subdifferential Hamiltonian Monte Carlo. The open questions are the dimension dependence of mixing times for general conservative fields, optimal acceptance rates in nonsmooth regimes, and adaptive tuning of $\mathbb{R}^d$5 in the presence of set-valued drift (Ning, 9 Jul 2025).

Within the broader MCMC landscape, MASLA therefore occupies a specific niche: it retains the exact-stationarity guarantee of a Metropolis correction while extending Langevin methodology beyond differentiable targets. The main unresolved issue is not whether such a scheme can be defined, but how far its theoretical performance can be characterized under the weak regularity and non-convexity assumptions that motivate it.