Underdamped Langevin MCMC

• Underdamped Langevin MCMC is a momentum-augmented sampler that lifts sampling into phase space, providing accelerated convergence in strongly log-concave settings.
• It utilizes a second-order dynamic combining Hamiltonian transport, friction, and Gaussian forcing to improve convergence rates and reduce dimensional dependency compared to overdamped methods.
• Advanced discretization techniques such as OBABO, randomized midpoint, and shifted ODE methods help maintain hypocoercivity and control bias for more efficient sampling.

to=arxiv_search.search 玩北京赛车 ＿久久json {"query":"underdamped Langevin Monte Carlo strongly log-concave randomized midpoint KL trace H underdamped Langevin MCMC", "max_results": 10} to=arxiv_search.search ’wini 大发快三和值json {"query":"title:Underdamped Langevin MCMC OR kinetic Langevin Monte Carlo sampling", "max_results": 10} Underdamped Langevin MCMC, also called kinetic Langevin Monte Carlo or underdamped Langevin Monte Carlo, denotes a family of momentum-augmented samplers for targets of the form $\pi(x)\propto e^{-U(x)}$. The method lifts sampling from position space to phase space by introducing an auxiliary momentum or velocity variable, then simulates a nonreversible diffusion whose invariant law has the desired $x$-marginal. In the smooth strongly log-concave regime, classical nonasymptotic analyses show that this second-order construction can improve the dependence on dimension and accuracy relative to overdamped Langevin, reaching $\widetilde{\mathcal O}(\sqrt d/\varepsilon)$ rather than $\mathcal O(d/\varepsilon^2)$ under matching assumptions (Cheng et al., 2017).

1. Phase-space formulation and invariant structure

A standard formulation of underdamped Langevin dynamics evolves $(q_t,p_t)\in\mathbb R^n\times\mathbb R^n$ according to

$$dq_t=M^{-1}p_t\,dt,\qquad dp_t=-\nabla U(q_t)\,dt-\Gamma M^{-1}p_t\,dt+\sqrt{2\Gamma}\,dW_t,$$

with mass matrix $M\in\mathbb S_{++}^n$ and friction matrix $\Gamma\in\mathbb S_{++}^n$. Its invariant Gibbs law is

$$\tilde\pi(dq,dp)=Z^{-1}\exp\!\left(-U(q)-\frac12 p^\top M^{-1}p\right)\,dq\,dp,$$

so the position marginal is precisely the target $\pi(dq)\propto e^{-U(q)}\,dq$ (Chak et al., 2021). In the equivalent kinetic notation used widely in the sampling literature,

$x$0

the invariant law is $x$1, again yielding the target in the position marginal (Cheng et al., 2017).

This lifted construction separates three roles that are fused in overdamped Langevin: Hamiltonian transport moves the state coherently through phase space, friction damps momentum, and Gaussian forcing restores the Gibbs law. Small friction makes the process closer to Hamiltonian transport; large friction drives it toward overdamped behavior (Chak et al., 2021). The overdamped limit is not merely heuristic: for the stochastic exponential Euler discretization, if $x$2 and the step size is accelerated according to $x$3, then the underdamped scheme converges to overdamped LMC behavior as $x$4 (Kim et al., 4 Oct 2025).

The phase-space perspective also clarifies why underdamped Langevin occupies an intermediate position between overdamped Langevin and HMC. Like HMC, it uses momentum and nonreversible transport; unlike standard HMC, it retains continuous friction and stochastic forcing. Like Langevin samplers, it has an explicit stochastic differential equation and an invariant Gibbs law. Much of the literature on underdamped Langevin MCMC can therefore be read as the study of how this lifted Gibbs structure survives discretization, preconditioning, adaptivity, and approximation.

2. Hypocoercivity, acceleration, and continuous-time convergence

A central interpretation of underdamped Langevin MCMC is that it is the sampling analog of accelerated gradient descent. Overdamped Langevin is the Wasserstein gradient flow of $x$5; underdamped Langevin augments this with momentum and realizes an accelerated dynamics on the space of probability measures. The key technical point is that entropy dissipation acts only in the momentum variable, so raw KL is not coercive enough. The resulting convergence theory is therefore hypocoercive: one proves decay of a modified Lyapunov functional containing KL together with cross terms coupling position and momentum derivatives (Ma et al., 2019).

In the smooth strongly log-concave setting, a classical continuous-time analysis uses transformed coordinates $x$6 rather than $x$7 directly. For $x$8 and $x$9, there exists a coupling such that

$\widetilde{\mathcal O}(\sqrt d/\varepsilon)$0

which yields exponential contraction in $\widetilde{\mathcal O}(\sqrt d/\varepsilon)$1 after comparison between the transformed and original coordinates (Cheng et al., 2017). This is the continuous-time mechanism behind the improved $\widetilde{\mathcal O}(\sqrt d/\varepsilon)$2-type complexity of the sampler.

The same perspective extends beyond global strong convexity. For targets $\widetilde{\mathcal O}(\sqrt d/\varepsilon)$3 where $\widetilde{\mathcal O}(\sqrt d/\varepsilon)$4 is globally $\widetilde{\mathcal O}(\sqrt d/\varepsilon)$5-smooth, has a stationary point at the origin, and is $\widetilde{\mathcal O}(\sqrt d/\varepsilon)$6-strongly convex only outside a ball of radius $\widetilde{\mathcal O}(\sqrt d/\varepsilon)$7, underdamped Langevin still admits a nonasymptotic analysis. The resulting complexity is $\widetilde{\mathcal O}(\sqrt d/\varepsilon)$8 in $\widetilde{\mathcal O}(\sqrt d/\varepsilon)$9, whereas the corresponding overdamped guarantee is $\mathcal O(d/\varepsilon^2)$0 (Cheng et al., 2018). The improvement in $\mathcal O(d/\varepsilon^2)$1 and $\mathcal O(d/\varepsilon^2)$2 remains, while the factor $\mathcal O(d/\varepsilon^2)$3 quantifies the cost of non-log-concavity.

These results support, in a mathematically precise sense, the long-standing intuition that momentum accelerates sampling. They also delimit that intuition: the acceleration is established under specific geometric conditions, via specific metrics such as $\mathcal O(d/\varepsilon^2)$4, KL, or $\mathcal O(d/\varepsilon^2)$5, and with proofs that depend heavily on hypocoercive couplings or Lyapunov constructions rather than simple reversibility arguments.

3. Discretization families and numerical integrators

The basic unadjusted discretization freezes the gradient over each step and integrates the resulting linear SDE exactly. In the strongly log-concave analysis of Cheng, Chatterji, Bartlett, and Jordan, each step is an explicit Gaussian transition whose mean and covariance can be written in closed form, and the local discretization error is $\mathcal O(d/\varepsilon^2)$6 in $\mathcal O(d/\varepsilon^2)$7 (Cheng et al., 2017). This “exact-in-step, frozen-gradient” viewpoint remains a reference point for later KLMC and ULMC analyses.

A different line of work studies splitting methods. The standard OBABO scheme alternates Ornstein–Uhlenbeck damping, force kicks, and Hamiltonian transport, and in the high-dimensional convex setting it requires only one gradient computation per iteration after initialization. A notable feature of that analysis is that contraction, regularization, and total-variation convergence are proved for the discrete chain itself, not only by comparison with the continuous diffusion (Monmarché, 2020). Invariant-measure error can then be treated either through the unadjusted invariant law $\mathcal O(d/\varepsilon^2)$8 or through a Metropolized OM(BAB)O variant that preserves $\mathcal O(d/\varepsilon^2)$9 exactly (Monmarché, 2020).

Higher-order underdamped samplers exploit more structure. The shifted ODE method replaces one stochastic step by a carefully chosen random ODE that matches the relevant Brownian integral moments, yielding $(q_t,p_t)\in\mathbb R^n\times\mathbb R^n$0 complexity under standard smoothness and stronger exponents under Lipschitz Hessian or Lipschitz third derivative assumptions (Foster et al., 2021). QUICSORT pushes this further with a gradient-only scheme whose global $(q_t,p_t)\in\mathbb R^n\times\mathbb R^n$1 bias scales like $(q_t,p_t)\in\mathbb R^n\times\mathbb R^n$2, $(q_t,p_t)\in\mathbb R^n\times\mathbb R^n$3, or $(q_t,p_t)\in\mathbb R^n\times\mathbb R^n$4 depending on regularity, leading respectively to $(q_t,p_t)\in\mathbb R^n\times\mathbb R^n$5, $(q_t,p_t)\in\mathbb R^n\times\mathbb R^n$6, and $(q_t,p_t)\in\mathbb R^n\times\mathbb R^n$7 step complexities (Scott et al., 22 Aug 2025).

Adaptive time-rescaling produces a different sort of modification. If a positive monitor function $(q_t,p_t)\in\mathbb R^n\times\mathbb R^n$8 is introduced naively, the invariant law changes; the corrected transformed underdamped SDE therefore adds the drift term $(q_t,p_t)\in\mathbb R^n\times\mathbb R^n$9 so that the original Gibbs measure remains invariant. The associated splitting methods use an implicit midpoint A-step, exact OU O-steps, and no Metropolis correction (Leroy et al., 2024).

The randomized midpoint discretization occupies yet another niche. In the KL framework, its virtue is improved weak local error: in the strongly convex case it yields KL complexity $$dq_t=M^{-1}p_t\,dt,\qquad dp_t=-\nabla U(q_t)\,dt-\Gamma M^{-1}p_t\,dt+\sqrt{2\Gamma}\,dW_t,$$0, whereas standard ULMC gives $$dq_t=M^{-1}p_t\,dt,\qquad dp_t=-\nabla U(q_t)\,dt-\Gamma M^{-1}p_t\,dt+\sqrt{2\Gamma}\,dW_t,$$1 (Zhang et al., 2 Mar 2026).

Scheme	Characteristic update	Distinctive feature
Frozen-gradient Gaussian step	Exact Gaussian transition with frozen force	First classical nonasymptotic $$dq_t=M^{-1}p_t\,dt,\qquad dp_t=-\nabla U(q_t)\,dt-\Gamma M^{-1}p_t\,dt+\sqrt{2\Gamma}\,dW_t,$$2 analysis (Cheng et al., 2017)
OBABO / OM(BAB)O	O-B-A-B-O splitting	Discrete-chain contraction; Metropolized exact variant (Monmarché, 2020)
Shifted ODE / QUICSORT	Random ODE or high-order gradient-only integrator	Exploits higher smoothness without Hessians (Foster et al., 2021, Scott et al., 22 Aug 2025)
Transformed adaptive dynamics	State-dependent monitor plus $$dq_t=M^{-1}p_t\,dt,\qquad dp_t=-\nabla U(q_t)\,dt-\Gamma M^{-1}p_t\,dt+\sqrt{2\Gamma}\,dW_t,$$3	Adaptive stepping without changing the target law (Leroy et al., 2024)
Randomized midpoint	Randomized force evaluation points	Dimension-free KL rates in terms of $$dq_t=M^{-1}p_t\,dt,\qquad dp_t=-\nabla U(q_t)\,dt-\Gamma M^{-1}p_t\,dt+\sqrt{2\Gamma}\,dW_t,$$4 (Zhang et al., 2 Mar 2026)

4. Nonasymptotic rates and complexity regimes

The best-known rate depends strongly on geometry, discretization, and metric. In the classical strongly log-concave setting, the basic underdamped sampler reaches $$dq_t=M^{-1}p_t\,dt,\qquad dp_t=-\nabla U(q_t)\,dt-\Gamma M^{-1}p_t\,dt+\sqrt{2\Gamma}\,dW_t,$$5 in $$dq_t=M^{-1}p_t\,dt,\qquad dp_t=-\nabla U(q_t)\,dt-\Gamma M^{-1}p_t\,dt+\sqrt{2\Gamma}\,dW_t,$$6 steps, with a log-free $$dq_t=M^{-1}p_t\,dt,\qquad dp_t=-\nabla U(q_t)\,dt-\Gamma M^{-1}p_t\,dt+\sqrt{2\Gamma}\,dW_t,$$7 variant under a varying step-size schedule (Cheng et al., 2017). The discrete OBABO chain yields a complementary picture: for the unadjusted chain, the Wasserstein efficiency bounds are of order $$dq_t=M^{-1}p_t\,dt,\qquad dp_t=-\nabla U(q_t)\,dt-\Gamma M^{-1}p_t\,dt+\sqrt{2\Gamma}\,dW_t,$$8 in the general smooth strongly convex case, $$dq_t=M^{-1}p_t\,dt,\qquad dp_t=-\nabla U(q_t)\,dt-\Gamma M^{-1}p_t\,dt+\sqrt{2\Gamma}\,dW_t,$$9 when the Hessian is Lipschitz, and $M\in\mathbb S_{++}^n$0 for separable targets (Monmarché, 2020).

Preconditioning changes the conditioning dependence. For strongly log-concave $M\in\mathbb S_{++}^n$1, a fixed-matrix scaled underdamped diffusion with

$M\in\mathbb S_{++}^n$2

and Hessian-variation parameter $M\in\mathbb S_{++}^n$3 yields a $M\in\mathbb S_{++}^n$4 bound whose iteration complexity scales like

$M\in\mathbb S_{++}^n$5

improving the $M\in\mathbb S_{++}^n$6-dependence of the earlier unscaled theorem (Zajic, 2019). The improvement is geometric: if a single reference Hessian captures most of the target’s curvature, the kinetic dynamics can be globally better conditioned.

A more recent development replaces ambient dimension by an aggregate curvature quantity. Under global convexity with $M\in\mathbb S_{++}^n$7, standard ULMC satisfies

$M\in\mathbb S_{++}^n$8

while the randomized midpoint method attains

$M\in\mathbb S_{++}^n$9

in the strongly convex case (Zhang et al., 2 Mar 2026). These are dimension-free KL guarantees in the sense that $\Gamma\in\mathbb S_{++}^n$0 is replaced by $\Gamma\in\mathbb S_{++}^n$1.

Setting	Representative guarantee	Source
Strongly log-concave ULMC	$\Gamma\in\mathbb S_{++}^n$2 in $\Gamma\in\mathbb S_{++}^n$3 steps	(Cheng et al., 2017)
Nonconvex outside a ball	$\Gamma\in\mathbb S_{++}^n$4 in $\Gamma\in\mathbb S_{++}^n$5	(Cheng et al., 2018)
Discrete OBABO chain	$\Gamma\in\mathbb S_{++}^n$6, $\Gamma\in\mathbb S_{++}^n$7, $\Gamma\in\mathbb S_{++}^n$8 regimes	(Monmarché, 2020)
Scaled underdamped diffusion	$\Gamma\in\mathbb S_{++}^n$9 conditioning dependence in $$\tilde\pi(dq,dp)=Z^{-1}\exp\!\left(-U(q)-\frac12 p^\top M^{-1}p\right)\,dq\,dp,$$0	(Zajic, 2019)
Dimension-free KL	$$\tilde\pi(dq,dp)=Z^{-1}\exp\!\left(-U(q)-\frac12 p^\top M^{-1}p\right)\,dq\,dp,$$1-based ULMC and RMD rates	(Zhang et al., 2 Mar 2026)

The literature therefore does not support a single universal complexity statement. Instead, underdamped Langevin MCMC exhibits a stratified theory: first-order rates under minimal smoothness, higher-order rates under additional regularity, condition-number gains under effective preconditioning, and dimension-free KL behavior when curvature is anisotropic enough that $$\tilde\pi(dq,dp)=Z^{-1}\exp\!\left(-U(q)-\frac12 p^\top M^{-1}p\right)\,dq\,dp,$$2.

5. Tuning, preconditioning, and stochastic variants

The friction parameter is not merely a nuisance constant. For ergodic averages of an observable $$\tilde\pi(dq,dp)=Z^{-1}\exp\!\left(-U(q)-\frac12 p^\top M^{-1}p\right)\,dq\,dp,$$3, the asymptotic variance $$\tilde\pi(dq,dp)=Z^{-1}\exp\!\left(-U(q)-\frac12 p^\top M^{-1}p\right)\,dq\,dp,$$4 is differentiable with respect to the friction matrix, and its directional derivative is

$$\tilde\pi(dq,dp)=Z^{-1}\exp\!\left(-U(q)-\frac12 p^\top M^{-1}p\right)\,dq\,dp,$$5

This yields a principled friction-learning algorithm based on short tangent-process simulations and shows explicitly that the friction minimizing asymptotic variance depends on the observable, not only on generic relaxation heuristics (Chak et al., 2021).

Randomization can also target computational cost. Random Coordinate ULMC updates a single coordinate at each iteration and proves

$$\tilde\pi(dq,dp)=Z^{-1}\exp\!\left(-U(q)-\frac12 p^\top M^{-1}p\right)\,dq\,dp,$$6

with optimal coordinate probabilities $$\tilde\pi(dq,dp)=Z^{-1}\exp\!\left(-U(q)-\frac12 p^\top M^{-1}p\right)\,dq\,dp,$$7. In the paper’s gradient-coordinate cost model, RC-ULMC is always cheaper than classical ULMC and can be much cheaper for highly skewed targets (Ding et al., 2020).

Stochastic-gradient underdamped sampling introduces additional noise-management issues. NOGIN modifies ABOBA by using noisy force kicks together with injected Gaussian noise and a covariance-dependent damping matrix

$$\tilde\pi(dq,dp)=Z^{-1}\exp\!\left(-U(q)-\frac12 p^\top M^{-1}p\right)\,dq\,dp,$$8

Its main theorem gives second-order weak consistency with effective friction $$\tilde\pi(dq,dp)=Z^{-1}\exp\!\left(-U(q)-\frac12 p^\top M^{-1}p\right)\,dq\,dp,$$9, and in the Gaussian target plus Gaussian stochastic-gradient case it preserves the exact target marginal in $\pi(dq)\propto e^{-U(q)}\,dq$0 (Matthews et al., 2018).

Two recent directions push beyond classical kinetic Langevin. GAUL adds a gradient adjustment in the position equation,

$\pi(dq)\propto e^{-U(q)}\,dq$1

and proves that the continuous-time augmented law still has the correct target marginal. For Gaussian targets, the Euler–Maruyama discretization mixes toward a biased target in $\pi(dq)\propto e^{-U(q)}\,dq$2 iterations, compared with $\pi(dq)\propto e^{-U(q)}\,dq$3 for the paper’s EM-underdamped baseline (Zuo et al., 2024). Separately, a refined analysis of the stochastic exponential Euler discretization shows that its contraction and bias bounds extend stably from the underdamped regime to the overdamped regime once the correct time acceleration is applied (Kim et al., 4 Oct 2025).

6. Applications, exactness, and open technical tensions

Underdamped Langevin MCMC is now used as a generic posterior simulator and as a subroutine inside larger decision-making systems. In approximate Thompson sampling for smooth strongly log-concave posteriors, replacing overdamped LMC with ULMC reduces the per-round sampling complexity needed to maintain logarithmic regret from $\pi(dq)\propto e^{-U(q)}\,dq$4 to $\pi(dq)\propto e^{-U(q)}\,dq$5 (Zheng et al., 2024). Application-driven adaptations also appear in signal processing: an annealed underdamped Langevin detector for massive MIMO uses a momentum-augmented splitting scheme with annealed prior scores and achieves improved low-complexity symbol detection, but the authors explicitly describe the resulting method as a hybrid stochastic search / annealed sampling / optimization heuristic rather than as a strict MCMC method targeting a single fixed posterior (Zilberstein et al., 2022).

Constraint handling is another active frontier. Softly constrained underdamped Langevin systems with large or small parameters in confinement forces, masses, or frictions can converge to genuinely constrained dynamics on a spatial or momentum subspace, and the limiting sampler depends on the mechanism by which the soft constraint is imposed. For holonomic constraints, this has direct implications for sampling conditional measures and for deciding whether initial conditions must already satisfy the constraint (Hartmann et al., 2 Apr 2026).

Several technical tensions recur throughout the literature. First, acceleration is criterion-dependent: in the scalar Gaussian case, critical damping is natural from spectral-gap considerations, yet for certain odd or linear observables the asymptotic-variance optimum can be arbitrarily small friction (Chak et al., 2021). Second, exactness and efficiency are often traded against one another. Unadjusted schemes such as OBABO target a biased invariant law $\pi(dq)\propto e^{-U(q)}\,dq$6, while Metropolized variants preserve $\pi(dq)\propto e^{-U(q)}\,dq$7 exactly but reintroduce acceptance-rejection overhead (Monmarché, 2020). Adaptive transformed dynamics preserve the target not by Metropolization but by modifying the SDE itself through the corrective drift $\pi(dq)\propto e^{-U(q)}\,dq$8 (Leroy et al., 2024). Third, many of the sharpest rates rely on strong geometric assumptions—global convexity, Lipschitz Hessian, or even Lipschitz third derivative—and the strongest dimension-free KL theorems remain specific to particular discretizations (Zhang et al., 2 Mar 2026).

Taken together, these developments establish underdamped Langevin MCMC as a broad research area rather than a single algorithm. Its core object is the kinetic Gibbs diffusion on phase space; its central mathematical themes are nonreversibility, hypocoercivity, and bias control under discretization; and its principal practical advantages arise when momentum, geometry, and numerical design are aligned closely enough that the lifted dynamics mix faster than their overdamped counterparts.