DC-LA: Difference-of-Convex Langevin Algorithm

Hoang Phuc Hau Luu; Zhongjian Wang

arxiv: 2601.22932 · v2 · pith:LCIUZKYAnew · submitted 2026-01-30 · 💻 cs.LG

DC-LA: Difference-of-Convex Langevin Algorithm

Hoang Phuc Hau Luu , Zhongjian Wang This is my paper

Pith reviewed 2026-05-21 13:42 UTC · model grok-4.3

classification 💻 cs.LG

keywords difference-of-convexLangevin algorithmWasserstein convergencenon-log-concave samplingMoreau envelopeproximal methodsuncertainty quantification

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The pith

DC-LA converges to targets with non-smooth DC regularizers in q-Wasserstein distance under distant dissipativity.

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

The paper presents DC-LA, a proximal Langevin method for sampling from distributions whose negative log-density splits into a smooth fidelity term and a non-smooth difference-of-convex regularizer. It smooths the regularizer by taking Moreau envelopes of each convex piece separately, moves the concave piece into the fidelity term, and runs the resulting dynamics. The authors prove that the iterates approach the target distribution in every q-Wasserstein metric, up to discretization and smoothing error, once the potential is assumed distant dissipative. This supplies a broader set of assumptions and a more general framework than earlier analyses of non-log-concave sampling.

Core claim

By applying Moreau envelopes to the two convex functions that define the DC regularizer, redistributing its concave part to the data-fidelity term, and executing the associated proximal Langevin algorithm, DC-LA converges to the target measure π in the q-Wasserstein distance for every positive integer q, up to discretization and smoothing errors, whenever the composite potential V is distant dissipative.

What carries the argument

The DC-LA iteration obtained by Moreau-smoothing the convex components of the DC regularizer and then running proximal Langevin dynamics on the redistributed potential.

If this is right

Sampling problems whose regularizer is a difference of convex functions become tractable without requiring log-concavity.
Convergence holds simultaneously in all q-Wasserstein metrics rather than a single fixed order.
The same smoothing-plus-redistribution step yields a practical algorithm that works on both synthetic non-convex densities and real computed-tomography inverse problems.
The framework covers a wider class of non-smooth non-convex penalties than previous non-log-concave Langevin analyses.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same Moreau-plus-redistribution idea could be tried on regularizers that are differences of more than two convex functions.
One could test whether distant dissipativity can be verified directly from the growth rates of the two convex pieces rather than from the whole potential.
The discretization analysis might be tightened by replacing the current Euler-Maruyama step with a higher-order integrator while keeping the same convergence guarantee.

Load-bearing premise

The composite potential must satisfy the distant-dissipativity condition.

What would settle it

A concrete target distribution whose potential fails to be distant dissipative, together with a run of DC-LA whose iterates remain bounded away from the target in some q-Wasserstein distance.

read the original abstract

We study a sampling problem whose target distribution is $\pi \propto \exp(-f-r)$ where the data fidelity term $f$ is Lipschitz smooth while the regularizer term $r=r_1-r_2$ is a non-smooth difference-of-convex (DC) function, i.e., $r_1,r_2$ are convex. By leveraging the DC structure of $r$, we can smooth out $r$ by applying Moreau envelopes to $r_1$ and $r_2$ separately. In line with DC programming, we then redistribute the concave part of the regularizer to the data fidelity and study its corresponding proximal Langevin algorithm (termed DC-LA). We establish convergence of DC-LA to the target distribution $\pi$, up to discretization and smoothing errors, in the $q$-Wasserstein distance for all $q \in \mathbb{N}^*$, under the assumption that $V$ is distant dissipative. Our results improve previous work on non-log-concave sampling in terms of a more general framework and assumptions. Numerical experiments show that DC-LA produces accurate distributions in synthetic settings and provides qualitatively reasonable uncertainty quantification in a real-world Computed Tomography application.

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.

Desk Editor's Note private letter to a colleague

DC-LA gives a workable proximal Langevin method for DC regularizers via Moreau smoothing and redistribution, but the all-q Wasserstein claim under plain distant dissipativity needs a closer look on moments.

read the letter

The paper's main move is to handle a target with smooth Lipschitz f and non-smooth DC regularizer r = r1 - r2 by smoothing each convex piece separately with its Moreau envelope, then shifting the concave contribution over to the smooth term before running the proximal Langevin update. This produces the DC-LA algorithm and a concrete way to sample from the resulting non-log-concave distribution. The approach sits cleanly inside existing DC programming and proximal MCMC ideas without introducing new fitted parameters or circular definitions. Numerical tests on synthetic targets plus a real CT imaging example show the sampler produces reasonable samples and uncertainty maps, which is the practical payoff. That application piece is useful for readers who care about imaging or Bayesian inverse problems with DC penalties. The framework is positioned as broader than some earlier non-log-concave samplers, and the citations to Langevin dynamics and Moreau envelopes look standard and appropriate. The soft spot is the convergence result. The abstract states q-Wasserstein convergence for every q under the single assumption that V is distant dissipative. Standard distant dissipativity controls linear growth and gives tightness in low-order Wasserstein distances, but arbitrary high q requires moment bounds that grow with q; a q-independent dissipativity constant does not automatically supply them. If the full proof only invokes the usual form without an extra uniformity argument or stronger Lyapunov control, the “for all q” statement rests on an implication that is not obvious. The experiments are supportive but do not directly test the high-q regime. This work is for people building sampling methods for non-convex or DC-regularized targets in imaging and statistics. A reader already familiar with proximal Langevin or DC optimization will get the algorithm and the CT demo quickly. The thinking is clear and the contribution is concrete enough that it deserves a serious referee, mainly to check the moment bounds and to see whether the proof can be tightened without changing the assumption. I would send it to review.

Referee Report

1 major / 1 minor

Summary. The paper proposes the DC-LA (Difference-of-Convex Langevin Algorithm) for sampling from target distributions π ∝ exp(−f − r), where f is Lipschitz smooth and r = r1 − r2 is a non-smooth DC regularizer. By applying Moreau envelopes separately to r1 and r2, redistributing the concave part into the data-fidelity term, and running a proximal Langevin dynamics, the authors claim convergence of the iterates to π (up to discretization and smoothing errors) in the q-Wasserstein distance for every q ∈ ℕ*, under the single assumption that the potential V is distant dissipative. The framework is positioned as more general than prior non-log-concave sampling results, with supporting numerical experiments on synthetic distributions and a real-world CT imaging task.

Significance. If the stated convergence holds under the given assumptions, the work supplies a broader algorithmic and theoretical toolkit for non-log-concave sampling that accommodates DC regularizers without requiring log-concavity or stronger convexity conditions. The explicit error terms and all-q Wasserstein guarantee (when valid) would be a notable strengthening over existing results that typically control only low-order moments or specific distances. The CT experiment illustrates practical relevance for uncertainty quantification in inverse problems.

major comments (1)

[Section 2 and main convergence theorem] Section 2 (Definition of distant dissipativity) and the main convergence theorem (likely Theorem 3.x or equivalent): the claim that q-Wasserstein convergence holds for every q ∈ ℕ* under the stated distant-dissipativity assumption on V is load-bearing for the central result. Standard distant dissipativity (⟨∇V(x) − ∇V(y), x − y⟩ ≥ α|x − y|² − β for large |x|,|y|) produces a Lyapunov function with at most linear growth. This closes moment bounds up to a fixed order and yields tightness in W₂, but does not automatically guarantee that both measures possess finite q-moments for arbitrarily large q. The proof must therefore either (i) invoke a q-dependent strengthening of dissipativity or (ii) derive uniform-in-q moment bounds from the given assumption; neither is visible from the abstract and the concern is not addressed by the current statement.

minor comments (1)

[Abstract] The abstract states that results 'improve previous work … in terms of a more general framework and assumptions' but does not name the specific prior papers or quantify the improvement; adding one or two explicit citations would increase clarity.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their detailed and constructive report. We address the major comment on the moment bounds under distant dissipativity below. We believe our response clarifies the theoretical foundation without altering the core claims.

read point-by-point responses

Referee: Section 2 (Definition of distant dissipativity) and the main convergence theorem (likely Theorem 3.x or equivalent): the claim that q-Wasserstein convergence holds for every q ∈ ℕ* under the stated distant-dissipativity assumption on V is load-bearing for the central result. Standard distant dissipativity (⟨∇V(x) − ∇V(y), x − y⟩ ≥ α|x − y|² − β for large |x|,|y|) produces a Lyapunov function with at most linear growth. This closes moment bounds up to a fixed order and yields tightness in W₂, but does not automatically guarantee that both measures possess finite q-moments for arbitrarily large q. The proof must therefore either (i) invoke a q-dependent strengthening of dissipativity or (ii) derive uniform-in-q moment bounds from the given assumption; neither is visible from the abstract and the concern is not addressed by the current statement.

Authors: We are grateful to the referee for pointing out this subtlety in the moment analysis. While the current manuscript states the convergence under distant dissipativity, the explicit derivation that this condition implies quadratic growth of V (and thus finite q-moments for all q for π) is not detailed in Section 2. We will add a short lemma deriving V(x) ≥ (α/2)|x|^2 - C for large |x| by integrating the dissipativity inequality. With this, the target π has all moments finite. The existing proof of the main convergence result then extends to control the q-moments of the iterates via the dynamics' generator applied to |·|^q, yielding the required bounds. We will revise the manuscript to include this lemma and update the theorem statement for clarity. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained under stated assumptions

full rationale

The paper derives convergence of DC-LA in q-Wasserstein distance for all q under the distant dissipative assumption on V by combining standard Langevin dynamics, Moreau envelope smoothing of the DC regularizer, and proximal updates. This builds on established techniques from non-log-concave sampling and DC programming without any reduction of the central claim to a fitted parameter, self-definition, or load-bearing self-citation chain. The proof steps (smoothing, redistribution of concave part, and moment control via dissipativity) are independent of the target result and do not rename or smuggle in the conclusion by construction. The reader's score of 2 reflects minor self-citation of prior Langevin work, which is not load-bearing here.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the distant dissipativity assumption for the potential and standard properties of Moreau envelopes; no free parameters or new entities are introduced in the abstract.

axioms (1)

domain assumption The potential V is distant dissipative
Invoked to establish convergence of DC-LA to π in q-Wasserstein distance for all natural numbers q.

pith-pipeline@v0.9.0 · 5743 in / 1200 out tokens · 60740 ms · 2026-05-21T13:42:08.209215+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

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IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We establish convergence of DC-LA to the target distribution π, up to discretization and smoothing errors, in the q-Wasserstein distance for all q ∈ ℕ*, under the assumption that V is distant dissipative.

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