Reflected Anchored Langevin Algorithm: Cracking the Constrained Non-Smooth Sampling Problem

The Old Problem of Constrained Sampling: When the Log-Density Is No Longer Differentiable

In probabilistic modeling and Bayesian inference for machine learning, sampling from constrained target distributions is a fundamental yet challenging task. Classical first-order Langevin algorithms — such as Projected Langevin Monte Carlo, which is based on a discretization of reflected Langevin dynamics — perform well on constrained sampling problems, but come with one hard prerequisite: the log-density of the target distribution must be differentiable.

This requirement may seem technical, but it severely limits the applicability of these methods. Many important distributions encountered in practice have log-densities that are non-differentiable at certain points — for example, models involving absolute values, piecewise functions, or certain sparse priors (such as Laplace-type priors). As soon as such non-smooth targets appear, gradient-based Langevin methods cannot be directly applied. This is precisely the pain point that a new arXiv paper (arXiv:2610.09522v1) sets out to address.

Langevin Dynamics is one of the core tools in Bayesian sampling. Its basic idea originates from the Brownian motion equations in physics: by adding random noise on top of gradient descent, one constructs a stochastic differential equation (SDE) whose stationary distribution equals the target posterior. Intuitively, the gradient term "climbs toward high-probability regions," while the noise term ensures exploration does not prematurely converge to a local optimum. When the target domain is constrained (e.g., samples must lie within a convex set or a simplex), reflected Langevin dynamics introduces a "reflection" mechanism at the boundary — analogous to light reflecting off a mirror — ensuring that trajectories stay within the feasible domain. Projected Langevin Monte Carlo is its discretized implementation: each step performs a gradient update with noise perturbation, then applies a projection operator to pull any out-of-bounds point back into the domain. Convergence analysis for these methods is well-established, but it assumes the log-density of the target distribution (i.e., the negative energy function) has a gradient everywhere — a condition that many real-world models (such as Lasso sparse priors and quantile regression losses) fail to satisfy.

RALD: Bypassing Non-Differentiability with a Smooth Anchored Reference Potential

The core method proposed in the paper is the Reflected Anchored Langevin Dynamics (RALD). As the name suggests, it is a reflected diffusion process that converges to non-differentiable target distributions on constrained domains.

RALD's elegance lies in two key design choices:

First, it introduces a smooth anchored reference potential. Rather than directly handling the non-differentiable original target, this smooth reference potential provides a numerically friendly "anchor point" that allows the dynamics to proceed stably without being disrupted by the target's non-smoothness.

Second, it simultaneously multiplies both the drift term and the noise covariance of the reflected Langevin dynamics by the same state-dependent scaling factor. This unified scaling of drift and noise is the key structural arrangement that ensures the process still converges to the correct target distribution — it preserves the detailed balance property of the diffusion process, so that scaling does not alter the final stationary distribution.

The core idea behind Anchored Langevin Dynamics is to decompose the target distribution into two parts: an easy-to-handle "reference distribution" (corresponding to the smooth anchored potential) and a "residual" that captures the complexity of the original target. The algorithm only computes gradients for the smooth reference potential, while absorbing the non-smooth part into the scaling factor in a gradient-free manner (e.g., via subgradients or zeroth-order estimates), thereby circumventing the numerical difficulties posed by non-differentiable points. The state-dependent scaling factor is also known in the literature as preconditioning, analogous to AdaGrad or RMSProp in optimization: larger step sizes are used in directions where gradient information is rich, while step sizes automatically shrink near non-smooth points where information is scarce. The critical structural constraint is that the drift term and noise covariance must use the same scaling matrix — this ensures that the stationary solution of the Fokker-Planck equation is not altered by the scaling. If the two were scaled differently, the stationary distribution would be distorted and would no longer correspond to the target distribution.

From Continuous Dynamics to Executable Algorithm: RALMC

The theoretical continuous diffusion process cannot be run directly on a computer and must be discretized. The authors apply the classical Euler-Maruyama discretization combined with a projection operation to handle domain constraints, yielding a practically executable algorithm — the Reflected Anchored Langevin Monte Carlo (RALMC).

The projection step ensures that after each iteration, samples are pulled back into the constraint domain, consistent with the boundary-handling spirit of reflected dynamics. The entire procedure transforms an abstract stochastic differential equation into an iterable sampling routine that balances theoretical rigor with engineering practicality.

Provable Convergence Guarantees

The most substantial contribution of this paper lies at the theoretical level. The authors do not stop at the empirical observation that "the method works" — they provide explicit convergence bounds and iteration complexity analysis for RALMC.

Critically, these convergence results measure the gap between the sampling distribution and the target distribution in terms of the 2-Wasserstein distance. The Wasserstein distance has a solid foundation in optimal transport theory; compared to metrics such as KL divergence, it is more sensitive to the geometric structure of distributions and better suited for characterizing convergence behavior in constrained sampling problems. Being able to provide explicit Wasserstein convergence rates in a non-differentiable, constrained setting means that the method is not only runnable, but that "how fast it converges and how many iterations are needed" are both theoretically grounded.

The 2-Wasserstein distance (W₂) comes from optimal transport theory and measures the minimum cost of "transporting mass" between two probability distributions, where the cost function is the squared Euclidean distance. Compared to KL divergence, W₂ is more sensitive to the geometry of the distribution's support: even when the supports of two distributions do not overlap, W₂ still yields a meaningful finite distance — a property that is especially important for analyzing sampling on constrained domains. The challenge of establishing W₂ convergence bounds in non-differentiable settings is that standard proof pathways (such as the Girsanov transform or logarithmic Sobolev inequalities) typically rely on smoothness assumptions on the target distribution. The authors must compensate for the roughness of the target through the smoothness of the anchored potential — without a global gradient Lipschitz condition — and leverage the geometric properties of the reflecting boundary condition to control the accumulation of discretization error, ultimately obtaining explicit polynomial convergence rates in terms of the number of iterations $N$, step size $h$, and problem dimension $d$.

Numerical Validation

Beyond theoretical proofs, the authors also provide numerical experiments to demonstrate two aspects: verifying whether the convergence behavior predicted by theory matches practice, and showcasing the empirical performance of the method. This dual structure of "theoretical bounds + experimental validation" is standard practice for papers on sampling algorithms and lends RALMC a relatively complete foundation of credibility.

Significance and Positioning

From a research trajectory perspective, RALD/RALMC continues the ongoing push to improve constrained sampling algorithms. Prior work has largely been blocked by the "differentiability" barrier, and this paper offers a new theoretically grounded path for constrained sampling of non-smooth targets through the combination of a smooth anchored potential and a unified scaling factor.

For researchers who need to handle sparse priors, non-smooth regularization, and similar scenarios on constrained domains, this type of method holds potential practical value. Of course, the paper remains at the stage of theory and preliminary experiments; its performance on large-scale real-world problems, as well as the sensitivity of the scaling factor design to different target distributions, remain open questions for future research.