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arxiv: 2605.00723 · v1 · submitted 2026-05-01 · 📊 stat.ML · cs.LG· math.PR

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Decentralized Proximal Stochastic Gradient Langevin Dynamics

Mohammad Rafiqul Islam , Lingjiong Zhu
Authors on Pith no claims yet

Pith reviewed 2026-05-09 18:18 UTC · model grok-4.3

classification 📊 stat.ML cs.LGmath.PR
keywords decentralized MCMCproximal Langevin dynamicsconstrained samplingWasserstein convergencelog-concave distributionsdistributed Bayesian inferencestochastic gradient methods
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The pith

DE-PSGLD converges in 2-Wasserstein distance to a regularized Gibbs distribution for decentralized constrained sampling.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces DE-PSGLD, a decentralized Markov chain Monte Carlo algorithm designed to sample from log-concave distributions restricted to convex domains. It enforces the constraints through a shared proximal regularization based on the Moreau-Yosida envelope, which supports fully unconstrained updates while remaining consistent with the target posterior. The analysis supplies non-asymptotic convergence bounds in the 2-Wasserstein metric that apply both to the iterates of individual agents and to the network-wide average. The limit distribution is shown to be a regularized Gibbs measure, and the bias introduced by the proximal step is quantified explicitly. This construction matters for distributed Bayesian inference tasks where computation or data must remain partitioned across nodes without a central coordinator.

Core claim

DE-PSGLD performs stochastic gradient Langevin dynamics in a decentralized setting with Moreau-Yosida proximal regularization to handle convex constraints. The method is proved to converge non-asymptotically in 2-Wasserstein distance to the regularized Gibbs distribution for both single-agent paths and network averages, with an explicit bound on the proximal bias. Experiments on synthetic and real datasets confirm rapid posterior concentration together with high predictive accuracy.

What carries the argument

Moreau-Yosida proximal regularization that approximates the constrained posterior and permits unconstrained decentralized Langevin updates while preserving sampling consistency.

Load-bearing premise

The target distribution is log-concave on a convex domain and the Moreau-Yosida proximal map consistently approximates the constrained posterior without invalidating the convergence guarantees.

What would settle it

A concrete log-concave test distribution on a convex polytope where the empirical 2-Wasserstein distance of DE-PSGLD samples to the regularized target exceeds the derived non-asymptotic bound for sufficiently large iteration counts.

Figures

Figures reproduced from arXiv: 2605.00723 by Lingjiong Zhu, Mohammad Rafiqul Islam.

Figure 1
Figure 1. Figure 1: Different types of network structures Synthetic 1-Dimensional Sampling. We start our experiment by sampling from a non￾Gaussian target on a compact convex set K = [−1, 1] in one dimension, d = 1, with density π(x) ∝ e −f(x) where f(x) = x 2 2 + x 4 8 − x, and x ∈ K. The target is smooth and log-concave near the origin. To evaluate sampling accuracy, we compute the 2-Wasserstein distance in 1 dimension [Pan… view at source ↗
Figure 2
Figure 2. Figure 2: Evolution of the 2-Wasserstein distance between the target distribution and the samples view at source ↗
Figure 3
Figure 3. Figure 3: Comparison of the sampling from the target density view at source ↗
Figure 4
Figure 4. Figure 4: The contour plot prior distri￾bution constrained in the 2-norm ball around the origin view at source ↗
Figure 5
Figure 5. Figure 5: The contour plot of the poste￾rior distribution of the model parameter sampled using the PSGLD algorithm. 10 view at source ↗
Figure 6
Figure 6. Figure 6: The top row shows the contour plot of samples of a randomly chosen agent out of 20 view at source ↗
Figure 7
Figure 7. Figure 7: The plots show the accuracy versus iterations using the DE-PSGLD and PSGLD al view at source ↗
read the original abstract

We propose Decentralized Proximal Stochastic Gradient Langevin Dynamics (DE-PSGLD), a decentralized Markov chain Monte Carlo (MCMC) algorithm for sampling from a log-concave probability distribution constrained to a convex domain. Constraints are enforced through a shared proximal regularization based on the Moreau-Yosida envelope, enabling unconstrained updates while preserving consistency with the target constrained posterior. We establish non-asymptotic convergence guarantees in the 2-Wasserstein distance for both individual agent iterates and their network averages. Our analysis shows that DE-PSGLD converges to a regularized Gibbs distribution and quantifies the bias introduced by the proximal approximation. We evaluate DE-PSGLD for different sampling problems on synthetic and real datasets. As the first decentralized approach for constrained domains, our algorithm exhibits fast posterior concentration and high predictive accuracy.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper proposes Decentralized Proximal Stochastic Gradient Langevin Dynamics (DE-PSGLD), a decentralized MCMC method for sampling log-concave distributions constrained to convex domains. Constraints are handled via shared Moreau-Yosida proximal regularization, allowing unconstrained updates. The main results are non-asymptotic 2-Wasserstein convergence bounds for both individual agent iterates and network averages to a regularized Gibbs measure, together with a quantification of the proximal bias; the method is evaluated on synthetic and real datasets for posterior concentration and predictive accuracy.

Significance. If the stated non-asymptotic bounds and bias quantification hold, the contribution is significant: it supplies the first decentralized algorithm for constrained-domain sampling together with explicit rates that incorporate decentralization, stochastic gradients, and proximal regularization. The explicit bias term and the empirical demonstration of fast concentration provide a concrete foundation for distributed Bayesian inference under constraints.

major comments (2)
  1. [Convergence analysis / main theorem] The non-asymptotic W2 bound (main convergence theorem) is derived under a fixed Moreau-Yosida parameter λ. Recovering the original constrained posterior requires λ → 0, yet no joint schedule relating λ, step-size h, and iteration count T is supplied that simultaneously drives bias (O(λ)), discretization error, and consensus error to zero while preserving the claimed rate. This gap directly affects the consistency claim with the target posterior.
  2. [Bias analysis section] The bias quantification between the regularized Gibbs measure and the original constrained posterior is stated, but the dependence of the overall error bound on λ (including how proximal error interacts with the decentralized stochastic-gradient terms) is not made explicit enough to verify that the total error remains controlled under any vanishing-λ regime.
minor comments (2)
  1. [Experiments] The experimental section would benefit from explicit reporting of the λ values used, sensitivity plots with respect to λ, and a clear statement of how λ is chosen relative to the step-size in the reported runs.
  2. [Preliminaries] Notation for the communication graph, mixing matrix, and proximal operator should be introduced once and used uniformly; a short table of symbols would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address the major comments point by point below and will incorporate revisions to improve clarity on the analysis.

read point-by-point responses

  1. Referee: The non-asymptotic W2 bound (main convergence theorem) is derived under a fixed Moreau-Yosida parameter λ. Recovering the original constrained posterior requires λ → 0, yet no joint schedule relating λ, step-size h, and iteration count T is supplied that simultaneously drives bias (O(λ)), discretization error, and consensus error to zero while preserving the claimed rate. This gap directly affects the consistency claim with the target posterior.

    Authors: Our main theorem provides explicit non-asymptotic 2-Wasserstein bounds to the regularized Gibbs measure for any fixed λ > 0, incorporating decentralization, stochastic gradients, and proximal regularization. The proximal bias to the original constrained posterior is quantified separately as O(λ). We acknowledge that an explicit joint schedule for λ, h, and T is not derived in the current analysis, as this would require a refined time-varying regularization argument beyond the fixed-λ focus of the work. The existing bounds nevertheless permit balancing errors by selecting sufficiently small fixed λ. We will add a discussion remark providing guidance on parameter regimes (e.g., λ = o(1) relative to discretization and consensus terms) to support consistency in the limit λ → 0 after mixing, and will clarify the consistency claim accordingly. revision: partial

  2. Referee: The bias quantification between the regularized Gibbs measure and the original constrained posterior is stated, but the dependence of the overall error bound on λ (including how proximal error interacts with the decentralized stochastic-gradient terms) is not made explicit enough to verify that the total error remains controlled under any vanishing-λ regime.

    Authors: We agree that the interaction of the proximal bias with the decentralized stochastic-gradient and consensus terms could be stated more explicitly. In the revised manuscript we will expand the bias analysis section to combine the O(λ) bias term directly with the main convergence bound, yielding an overall error expression that makes the dependence on λ transparent and confirms control under vanishing-λ regimes when λ is chosen appropriately relative to h and T. revision: yes

Circularity Check

0 steps flagged

No circularity: standard analysis applied to proximal decentralized setting

full rationale

The paper derives non-asymptotic 2-Wasserstein convergence of DE-PSGLD iterates and network averages to the regularized Gibbs measure induced by the Moreau-Yosida envelope of the constrained log-concave potential. This target is exactly the stationary distribution of the algorithm by construction, and the bias to the original constrained posterior is quantified separately as an approximation error. No step reduces a claimed prediction or uniqueness result to a fitted parameter, self-citation chain, or definitional tautology. The analysis combines standard SGLD discretization bounds, proximal regularization properties, and decentralized consensus arguments without importing load-bearing results from the authors' prior work or smuggling ansatzes. The derivation remains self-contained against external benchmarks for log-concave sampling.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Based solely on the abstract; the method rests on standard domain assumptions for log-concave sampling and convex optimization. No free parameters, invented entities, or ad-hoc axioms are described.

axioms (2)
  • domain assumption Target distribution is log-concave
    Required for convergence of Langevin-type dynamics and stated as the setting for the algorithm.
  • domain assumption Constraint set is convex
    Enables use of proximal operators and Moreau-Yosida regularization for the constrained domain.

pith-pipeline@v0.9.0 · 5431 in / 1421 out tokens · 52028 ms · 2026-05-09T18:18:51.937139+00:00 · methodology

discussion (0)

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