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Analysis of Langevin Monte Carlo via convex optimization
Alain Durmus, Szymon Majewski, Błażej Miasojedow
TL;DR
The paper addresses how to quantify approximation error when Langevin discretizations are used to sample target distributions, including non-smooth cases. It formulates ULA as first-order optimization in Wasserstein space, applies convex-optimization proof techniques, and develops two SGLD-related non-smooth methods. The resulting analysis provides non-asymptotic KL bounds for ULA and new computable bounds for non-smooth sampling schemes.
Problem
The paper studies how to justify approximate Langevin sampling when discretization destroys the target distribution's invariance, including sampling from non-smooth log-concave targets.
Method
The paper interprets ULA as a first-order optimization algorithm for a functional on Wasserstein space and analyzes it using convex-optimization techniques, extending the approach to two SGLD-related methods.
Results
The paper derives non-asymptotic KL bounds for ULA, including tighter strongly convex bounds and computable convex-case bounds, and analyzes two new non-smooth sampling algorithms.
Takeaways & Limitations
The optimization interpretation supplies a common analysis framework for ULA and extensions using stochastic gradients, subgradients, or proximal operators.
Takeaways & Limitations
The stochastic subgradient analysis requires controlling the ergodic average variance of stochastic gradient estimates, for example through a uniform bound under a stated condition.
Abstract
from arXiv · showhide
In this paper, we provide new insights on the Unadjusted Langevin Algorithm. We show that this method can be formulated as a first order optimization algorithm of an objective functional defined on the Wasserstein space of order $2$. Using this interpretation and techniques borrowed from convex optimization, we give a non-asymptotic analysis of this method to sample from logconcave smooth target distribution on $\mathbb{R}^d$. Based on this interpretation, we propose two new methods for sampling from a non-smooth target distribution, which we analyze as well. Besides, these new algorithms are natural extensions of the Stochastic Gradient Langevin Dynamics (SGLD) algorithm, which is a popular extension of the Unadjusted Langevin Algorithm. Similar to SGLD, they only rely on approximations of the gradient of the target log density and can be used for large-scale Bayesian inference.
1 Introduction
The paper studies sampling from target distributions using Langevin methods, addressing discretization error and difficult proposal selection in broader applications. It interprets ULA through Wasserstein-space optimization and introduces non-smooth extensions related to SGLD.
- Motivation: Sampling from π arises in Bayesian inference, machine learning, inverse problems, and computational physics, where Markov-chain methods construct kernels with π as invariant distribution.Choosing an appropriate Hastings–Metropolis proposal is described as difficult.
- Motivation: The Langevin diffusion converges to π under suitable conditions, but its paths are generally impractical to simulate directly.Euler–Maruyama discretization is therefore used, although the resulting Markov kernels no longer preserve π exactly.
- ULA: ULA applies Euler–Maruyama discretization with step sizes and Gaussian noise to approximately sample from π.The paper situates ULA within prior analyses of weak errors and non-asymptotic behavior.
- Contributions: Two new algorithms target non-smooth log-concave distributions with computable non-asymptotic bounds, extending SGLD.One handles Lipschitz convex potentials using unbiased stochastic subgradient estimates; the other uses stochastic gradients and a proximal operator.
- Contributions: The paper gives a new ULA interpretation and derives Kullback–Leibler bounds, recovering strongly convex dimension dependence while tightening bounds and implying Wasserstein and total-variation bounds.For convex U, it also gives computable bounds that improve several earlier results.
- Approach and evaluation: The paper analyzes ULA and its variants using gradient-flow ideas in Wasserstein space and supports the new methodologies with logistic-regression experiments.The paper also presents complexity tables for ULA and SGLD under strongly convex or convex settings.
2 Interpretation of ULA as an optimization algorithm
The paper interprets ULA as a first-order optimization method for an objective functional on Wasserstein space, linking its discretization to Langevin diffusion and gradient-flow analysis.
- Assumptions: ULA is analyzed under m-convexity of U, including ordinary convexity when m = 0.The paper additionally considers continuously differentiable potentials with L-Lipschitz gradients.
- Langevin diffusion: The Langevin diffusion has a unique strong solution under the stated convexity and gradient-Lipschitz assumptions, with π as its unique invariant measure.Its semigroup is reversible with respect to π and converges in total variation.
- Wasserstein formulation: The diffusion densities arise through a Wasserstein gradient-flow formulation involving the Boltzmann H-functional and potential energy functional.The associated minimization scheme is connected to proximal-type algorithms on Wasserstein space.
- Interpretation of ULA: The paper interprets Euler–Maruyama ULA as a first-order optimization algorithm for the objective functional F.This perspective differs from analyses based primarily on comparing the discretized chain with the continuous diffusion.
- Interpretation of ULA: Directly minimizing F is not practical because explicit recursions for its minimization scheme are as difficult to obtain as minimizing F itself.This motivates analyzing the computable Euler–Maruyama scheme instead.
3 Main results for the Unadjusted Langevin algorithm
The paper interprets ULA as first-order optimization on Wasserstein space, deriving non-asymptotic convergence bounds under convexity and smoothness assumptions. The analysis yields computable KL, Wasserstein, and total-variation guarantees, including improved condition-number dependence and complexity estimates.
- Optimization interpretation: ULA is analyzed as an optimization method for the functional F on P2(Rd), replacing Euclidean distance with W2 distance.The Euler–Maruyama kernel is decomposed into deterministic gradient descent for the energy and stochastic movement along the entropy gradient flow.
- Proof strategy: The proof transfers convex-optimization inequalities to Wasserstein space by bounding energy and entropy changes separately before combining them.The decomposition F(µRγ) − F(π) separates energy and entropy differences, and the supporting lemmas control each term.
- General convergence bounds: Under A1(m) and A2, Theorem 6 provides weighted convergence bounds for averaged ULA measures with non-increasing step sizes and weights.The theorem requires γ1 ≤ L−1 and λk+1(1 − mγk+1)/γk+1 ≤ λk/γk.
- Convex targets: For convex U, the paper gives computable KL bounds and recovers total-variation guarantees through Pinsker’s inequality.The stated complexity for KL is dO(ε−2), while the corresponding total-variation complexity is dO(ε−4).
- Strongly convex targets: For strongly convex U, the bounds improve dependence on the condition number L/m and establish Wasserstein convergence rates for decreasing step sizes.With γk = γ1/k^α, the paper obtains W2(µ0Qn_γ, π) ≤ Cn^−α/2 and uses burn-in to improve the convex-case result.
- Discretization bias: With constant step size, ULA has an invariant measure πγ generally different from π, but the paper gives computable bounds on their KL and Wasserstein distances.The bounds imply 2 KL(πγ|π) + mW2^2(πγ, π) ≤ 2Ldγ.
4 Extensions of ULA
The paper develops two stochastic extensions of ULA for non-smooth convex potentials: SSGLD uses stochastic sub-gradients, while SPGLD combines stochastic gradients with proximal operators. Both receive non-asymptotic analyses, with complexity bounds controlled by gradient variance and problem parameters.
- Extensions of ULA: SSGLD and SPGLD extend ULA to non-continuously differentiable convex potentials using only i.i.d. unbiased (sub-)gradient estimates.SSGLD relies on sub-gradients; SPGLD uses a proximal operator for the non-smooth component.
- Stochastic Sub-Gradient Langevin Dynamics: SSGLD applies to Lipschitz convex potentials with unbiased sub-gradient estimates and can have dimension dependence only through the initial condition and stochastic sub-gradient variance.The paper presents this as a property of the first proposed method.
- Stochastic Sub-Gradient Langevin Dynamics: SSGLD admits computable non-asymptotic KL and total-variation complexity bounds under bounded stochastic-gradient variance.With a warm start, the stated orders are (M^2 + D2)O(ε^-2) for KL and (M^2 + D2)O(ε^-4) for total variation.
- Refined bounds: Under cocoercivity assumptions, the paper reports improved variance dependence for both SSGLD and SPGLD bounds, while some dimension dependence can deteriorate.For SSGLD, the variance term scales as υΘ(δx⋆)^1/2 rather than supx υΘ(δx).
- Stochastic Proximal Gradient Langevin Dynamics: SPGLD decomposes U as U1 + U2, using stochastic gradients for U1 and the proximal operator associated with convex U2.This construction generalizes SGLD while accommodating a non-smooth component.
- Stochastic Proximal Gradient Langevin Dynamics: SPGLD has computable non-asymptotic bounds, with warm-start KL complexity of order (d + M^2 + D2)O(ε^-2) under bounded stochastic-gradient variance.The corresponding total-variation complexity is stated as (d + M^2 + D2)O(ε^-4).
5 Numerical experiments
The experiments evaluate SPGLD and SSGLD on Bayesian logistic regression using three UCI datasets and estimate two posterior-mean functionals. Errors are computed across repeated long runs against prox-MALA reference values, with figures reporting convergence, effective passes, and run-to-run variation.
- Experimental setup: The experiments study Bayesian logistic regression on Heart disease, Australian Credit Approval, and Musk datasets with dimensions 14, 34, and 166.The datasets contain 270, 690, and 476 observations, respectively.
- Experimental setup: SPGLD and SSGLD approximate p1, while only SPGLD is applied to p1,2 because the associated potential for p1 is Lipschitz.The paper uses the two methods according to the regularity of the target potential.
- Metrics and reference: The evaluation estimates posterior means I1 and I2 for β 7→β1 and β 7→(1/d)Σi βi, then computes absolute errors against prox-MALA reference values.The reference uses 10^7 iterations and a step size corresponding to an approximately 0.5 acceptance ratio.
- Evaluation protocol: Each parameter setting uses 100 replications, with runs of length 10^6 and constant step sizes τ(L + m)^-1 for τ = 0.01, 0.1, and 1.Stochastic (sub-)gradient minibatch sizes are selected from N, floor(N/10), and floor(N/100).
- Reported results: Figures 1 and 2 report mean absolute errors for I2 and I1 on Australian Credit Approval, respectively, using convergence curves, effective-pass comparisons, and boxplots.The layouts compare SPGLD alone for p1,2 with SPGLD and SSGLD for p1.
6 Discussion
The discussion presents a Wasserstein-space optimization interpretation of ULA and uses convex-optimization proof techniques to derive non-asymptotic sampling bounds. This perspective also supports new analyses for stochastic and non-smooth regimes and motivates optimization-inspired sampling methods.
- Discussion: The paper interprets ULA as a first-order optimization algorithm and derives non-asymptotic bounds using techniques from convex optimization.The approach gives simpler proofs for some existing ULA results and establishes previously unknown bounds.
- Discussion: The analysis covers stochastic-gradient ULA in the non-strongly convex case and non-smooth non-strongly convex sampling.The authors describe these as first non-asymptotic results to their knowledge.
- Discussion: The same perspective extends to stochastic non-smooth sampling, including the new SSGLD and SPGLD methodologies.Both algorithms are described as generalizations of SGLD for non-smooth potentials.
- Future directions: The authors identify translating efficient optimization algorithms into sampling algorithms with non-asymptotic guarantees as a direction for future research.This proposed direction follows from the connection between Langevin Monte Carlo and optimization.
7.1 Proof of of Lemma 1
The proof section establishes integrability and finite second-moment and entropy properties for the target distribution under the stated assumptions. It also handles measures without Lebesgue densities by assigning infinite values to both sides of the relevant identity.
- Proof of Lemma 1: Under A1(m) and integrability of e^-U, a lower bound on U together with A2 implies that π belongs to P2(Rd).The proof uses U(x) ≥ C1||x|| − C2.
- Proof of Lemma 1: The same growth argument establishes finiteness of the energy E(π) and entropy H(π).The proof additionally uses boundedness of x 7→ U(x)e^-U(x)/2 on the relevant domain.
- Proof of Lemma 1: If μ has no Lebesgue density, both sides of the identity in (10) are infinite; otherwise, the proof proceeds using equation (7).This separates the singular and absolutely continuous cases.
7.2 Proof of Corollary 8
The proof invokes Theorem 6, bounds the relevant sequences using an integral test, and combines these inequalities to conclude the result.
- Theorem 6 provides the initial bound used in the proof.
- An integral test gives Γ0,n ≥ C1n^(1−α) for some constant C1 ≥ 0.
- For α = 1/2, the sequence P_n^k is bounded by C2(1 + log(n)) for some constant C2 ≥ 0.
7.3 Proofs of Section 4.1
This section develops Wasserstein-functional bounds for the paper's stochastic operators under stated assumptions, then combines them with earlier results to complete convergence proofs and corollaries.
- For positive γ and γ~, the operator R̄γ,γ~ decomposes as S̄γTγ~, enabling separate analysis of its components.
- Lemma 26 bounds the functional decrease using the squared Wasserstein-distance reduction and a γ²-weighted term involving M² and υΘ(µ).
- Figures 5 and 6 report mean absolute errors for estimators of I2 and I1 on the Musk dataset, comparing SPGLD and SSGLD configurations.
- The proof expands the stochastic update norm and separates the mean subgradient contribution from gradient-estimation fluctuations.
- The resulting inequalities are combined with Theorem 6, convexity of Kullback–Leibler divergence, and monotonicity of λk/γk+1 to finish the proofs.
7.4 Proof of Section 4.2
The section proves bounds for the nonsmooth proximal construction by decomposing the objective into smooth, nonsmooth, and entropy terms, then combining recursive inequalities under strong convexity assumptions.
- The objective gap is decomposed into smooth potential, nonsmooth potential, and entropy differences.
- The smooth component uses Lipschitz-gradient and strong-convexity properties of U1 to control the stochastic forward step.
- The nonsmooth component is handled through a proximal-step inequality, with M2-Lipschitzness controlling the change in U2.
- Lemma 30 combines these component bounds into a contraction-type inequality for F after applying the composite operator.
- The proofs use KL convexity, step-size relations, induction, and error-budget bounds to establish the stated corollaries and theorem consequences.
A Definitions and useful results from theory of gradient flows
This section introduces absolute continuity, Wasserstein geodesics, and geodesic convexity, then states gradient-flow existence, uniqueness, and Fokker–Planck connections.
- An absolutely continuous curve in P2(Rd) has Wasserstein distance controlled by an integrable speed function.
- For any fixed measure, its Wasserstein distance to an absolutely continuous curve is itself absolutely continuous and differentiable almost everywhere.
- A constant-speed geodesic joins two measures with distance proportional to |t−s| and is represented by interpolating an optimal transport plan.
- Geodesic convexity requires the functional to satisfy a quadratic interpolation inequality along an appropriate constant-speed geodesic.
- For convex lower-semicontinuous V, the functional F̃ is geodesically convex, and its unique gradient flow solves the Fokker–Planck equation.
- The flow is absolutely continuous with respect to Lebesgue measure for positive times; the V = 0 case yields a stated specialized result.
B On the second order moment of logconcave measures
This section bounds the effect of an algorithm’s initial condition on its W2 distance from π under assumptions A1(0) and A7. Proposition 32 combines bounds involving the potential U, the entropy H(π), the normalizing constant Z, and the minimizer x⋆.
- Assumption A7 introduces constants η > 0 and Mη ≥ 0 outside a ball centered at the origin.
- The section addresses the W2 distance between the algorithms’ initial condition and π.
- Proposition 32 assumes A1(0) and A7 to provide the relevant bound.
- The proof bounds an integral involving U(x) + log(Z) + H(π) and a term involving H(π), log(Z), and U(x⋆).
- The proof concludes by combining the two auxiliary bounds in equation (50).