stat.MLOct 7, 2026

Reflected Anchored Langevin Algorithms

Authors: Changwei Tu, Xiaoyu Wang, Yingli Wang, Xicheng Zhang, Lingjiong Zhu

Organizations: Hong Kong University of Science and Technology (Guangzhou), Guangzhou, Guangdong Province, People’s Republic of China · School of Mathematical Sciences, Fudan University, Shanghai, People’s Republic of China · School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, People’s Republic of China · Department of Mathematics, Florida State University, Tallahassee, Florida, United States of America

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Abstract

First order Langevin algorithms for constrained sampling in machine learning, such as projected Langevin Monte Carlo which are based on discretizations of reflected Langevin dynamics, require differentiable log densities that limits their applicability. This paper introduces reflected anchored Langevin dynamics (RALD), a reflected diffusion that converges to non-differentiable targets on constrained domains. The method uses a smooth anchored reference potential and multiplies the drift and noise covariance of its reflected Langevin dynamics by the same state dependent scaling factor. Its Euler-Maruyama discretization with projection gives reflected anchored Langevin Monte Carlo (RALMC) algorithm. We prove explicit convergence bounds and iteration complexity for RALMC in the 2-Wasserstein distance to the target distribution. Numerical experiments are provided to illustrate the theoretical predictions and the empirical performance of the method.

Figures & tables

Figure 1 : Conditional one step crossing at a density bottleneck. The layer BB separates the two high density regions D−D_{-} and D+D_{+} . Starting from a fixed x∈Bx\in B , both algorithms are evaluated on the same event of landing in D+D_{+} . The point y∗y_{*} represents a closest point of D+‾\overline{D_{+}} to xx (it need not belong to D+D_{+} when D+D_{+} is open), so the distance is rx=dist⁡(x,D+)r_{x}=\operatorname{dist}(x,D_{+}) . Because D+⊂int⁡(C)D_{+}\subset\operatorname{int}(\mathcal{C}) , projection does not change this landing event.
Figure 2 : Visualized final sampled density plots for the first 2 dimensions in ball constraint
Figure 3 : Convergence of RALMC and PLMC in W2\mathcal{W}_{2} distance for a truncated Laplace distribution within a ball constraint
Figure 4 : Cross sections of the target truncated Gaussian mixture model along the first two dimensions
Figure 5 : Performance of RALMC with a constant potential and PLMC, measured by the W2\mathcal{W}_{2} distance to the target Gaussian mixture model, for different stepsizes η\eta over 10001000 epochs.
StepsizeMethod# of Entered D+D^{+}Enter epoch
1×10−41\times 10^{-4}RALMC-Const293.00±12.36293.00\pm 12.36712.15±14.13712.15\pm 14.13
 PLMC204.79±11.06204.79\pm 11.06834.47±10.29834.47\pm 10.29
5×10−45\times 10^{-4}RALMC-Const519.67±8.70519.67\pm 8.70361.23±14.62361.23\pm 14.62
 PLMC427.95±11.95427.95\pm 11.95503.66±15.44503.66\pm 15.44
1×10−31\times 10^{-3}RALMC-Const581.70±4.21581.70\pm 4.21219.32±10.62219.32\pm 10.62
 PLMC510.79±8.77510.79\pm 8.77362.68±15.49362.68\pm 15.49
Table 1 : First enter performance from B={x∈C:−0.25≤x1≤0.25}B=\{x\in\mathcal{C}:-0.25\leq x_{1}\leq 0.25\} to D+={x∈C:x1<−0.5}D^{+}=\{x\in\mathcal{C}:x_{1}<-0.5\} . Results are mean ±\pm standard deviation over independent runs. Paths that do not enter D+D^{+} are censored at the terminal epoch.
Figure 6 : Performance on accuracy of constrained Bayesian logistic regression under different prior distributions through RALMC with Gaussian smoothing, RALMC with constant potential and PLMC for the test set of the MAGIC Gamma Telescope dataset using the stepsize η=1×10−6\eta=1\times 10^{-6} in 200200 epochs
Figure 7 : Performance on accuracy of constrained Bayesian logistic regression under different prior distributions through RALMC with Gaussian smoothing, RALMC with constant potential and PLMC for the test set of the MAGIC Gamma Telescope dataset using the stepsize η=5×10−4\eta=5\times 10^{-4} in 100100 epochs
Figure 8 : Performance on accuracy of constrained Bayesian logistic regression under different prior distributions through RALMC with Gaussian smoothing, RALMC with constant potential and PLMC for the test set of the Breast Cancer Wisconsin (Diagnostic) dataset using the stepsize η=1×10−4\eta=1\times 10^{-4} in 200200 epochs
Figure 9 : Performance on accuracy of constrained Bayesian logistic regression under different prior distributions through RALMC with Gaussian smoothing, RALMC with constant potential and PLMC for the test set of the Breast Cancer Wisconsin (Diagnostic) dataset using the stepsize η=2×10−2\eta=2\times 10^{-2} in 100100 epochs

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