Randomized Langevin Monte Carlo

• Randomized Langevin Monte Carlo (RLMC) is a sampling method that integrates stochastic time-stepping into Langevin dynamics to effectively explore high-dimensional, complex distributions.
• It achieves uniform-in-time, non-asymptotic error bounds by using randomized gradient evaluations and coupling techniques under log-Sobolev conditions.
• Modified RLMC variants use projection techniques to stabilize sampling in non-globally Lipschitz settings, proving useful in Bayesian inference, uncertainty quantification, and machine learning.

Randomized Langevin Monte Carlo (RLMC) refers to a class of Markov chain Monte Carlo (MCMC) algorithms designed to efficiently sample from complex, high-dimensional probability distributions by incorporating stochastic elements into the discretization of Langevin diffusions. RLMC generalizes and often improves upon the classical Langevin Monte Carlo (LMC), especially in regimes where the target measure is non-log-concave, non-globally Lipschitz, or possesses superlinear drift, and/or when variance reduction or computational scalability is critical.

1. Algorithmic Foundations and Formulation

The canonical Langevin diffusion dynamics for sampling from a target density $\pi(dx) \propto \exp(-U(x))dx$ are governed by the stochastic differential equation:

$dX_t = -\nabla U(X_t)dt + \sqrt{2}dW_t,$

with $U$ the potential and $W_t$ a standard $d$-dimensional Wiener process. Standard LMC employs an Euler–Maruyama discretization:

$$x_{n+1} = x_n - h\nabla U(x_n) + \sqrt{2h} \xi_n, \qquad \xi_n \sim N(0,I_d),$$

with step size $h > 0$.

RLMC introduces an additional level of randomization—most notably, randomized time-stepping for the drift evaluation—yielding increased robustness and often improved error bounds under weaker regularity assumptions. A prototypical RLMC scheme, analyzed in (Wang et al., 30 Sep 2025), employs a two-stage update per iteration: $$\begin{align*} Y_{n+1}^\tau &= Y_n - \nabla U(Y_n) \cdot \tau_{n+1} h + \sqrt{2}\,\Delta W_{n+1}^\tau,\ Y_{n+1} &= Y_n - \nabla U(Y_{n+1}^\tau)\cdot h + \sqrt{2}\,\Delta W_{n+1}, \end{align*}$$ where $\tau_{n+1}\sim \mathrm{Uniform}(0,1)$, $\Delta W_{n+1}^\tau$ is a Brownian increment over $[t_n, t_n + \tau_{n+1}h]$, and $Y_{n+1}$ is the final iterate. The key distinction is the use of a randomly located drift evaluation between $x_n$ and $x_{n+1}$, as opposed to the fixed step in classical LMC.

Modified RLMC variants include further stabilizations, such as projection (“taming”) operators to prevent divergence under non-globally Lipschitz drifts.

2. Non-asymptotic Uniform-in-Time Error Bounds

A principal theoretical advance is the establishment of uniform-in-time, non-asymptotic error bounds in $2$-Wasserstein distance: $$\mathcal{W}_2(\nu_n, \pi) \leq C_1 \sqrt{d}\,h + C_2 \sqrt{d}\,e^{-\lambda n h},$$ as shown in (Wang et al., 30 Sep 2025). Here $\nu_n$ is the law of the $n$-th RLMC iterate, and the constants $C_1,C_2$ and $\lambda$ are independent of $n$ and the dimension $d$ (beyond the $\sqrt{d}$ scaling), under the following assumptions:

• $U$ is $L$-smooth: $|\nabla U(x) - \nabla U(y)| \leq L|x-y|$ for all $x,y$.
• The target $\pi$ satisfies a log-Sobolev inequality (LSI).

Notably, neither convexity nor global dissipativity is required under LSI; the result applies to nonconvex, non-log-concave, and multimodal settings if an LSI holds. The error has two terms: the bias (scaling as $O(\sqrt{d} h)$), and the contraction (scaling down exponentially with the number of steps). This is in contrast to standard LMC, for which such uniform-in-time guarantees under mere gradient Lipschitzness are not established for general nonconvex potentials.

For non-globally Lipschitz potentials (e.g., those with superlinear drift), RLMC alone may diverge. The modified RLMC (sometimes called pRLMC) projects iterates onto balls with radius depending on $h$ and $d$ via an operator

$$T^h(x) := \min\left\{1,\, C_0 d^{1/(2\gamma+2)} h^{-1/(2\gamma+2)} |x|^{-1}\right\} x,$$

where $U$ grows no faster than $C_0 |x|^{2\gamma+2}$ for large $|x|$ and the error bound becomes

$$\mathcal{W}_2(\nu_n, \pi) \leq C\ d^{(11\gamma+2)/4} h + C' d^{(11\gamma+2)/4} e^{-\lambda n h}.$$

Although dimension dependence worsens, this is the first provable uniform-in-time, non-asymptotic guarantee for randomized discretizations under superlinear drift growth (Wang et al., 30 Sep 2025).

3. Algorithmic Comparisons and Theoretical Context

Method	Target Condition	Error Bound	Uniform-in-Time	Notes
Classical LMC	Log-concave, Lipschitz	$O(\sqrt{d}h)$	No (nonconvex)	Worst-case loss under nonconvexity
RLMC	LSI, Lipschitz	$O(\sqrt{d}h)$	Yes	Applies to nonconvex, LSI targets
pRLMC (modified RLMC)	LSI, Superlinear	$O(d^{(11\gamma+2)/4} h)$	Yes	For superlinear drifts

This table summarizes the applicability and theoretical guarantees for LMC and RLMC variants (Wang et al., 30 Sep 2025).

The uniform-in-time control is achieved via coupling, moment bounds, and a careful use of the log-Sobolev inequality, avoiding reliance on strongly log-concave assumptions or global marked dissipativity.

4. Relation to Randomization, Variants, and Non-classical Regimes

Randomization in RLMC refers to the randomized location of the gradient evaluation, which improves the stability and error decay. This connects closely to the family of midpoint/randomized discretization methods, such as the randomized midpoint discretization for kinetic Langevin dynamics (Yu et al., 2023), as well as regime-switching methods (Wang et al., 31 Aug 2025) in which step sizes (or friction coefficients) are chosen randomly according to a prescribed stochastic rule (e.g., finite-state Markov process). The “randomization” in RLMC should not be confused with random coordinate descent, which is aimed at computational reduction per iteration, nor with injected gradient noise.

In the context of non-smooth or non-convex potentials, RLMC and its modifications using smoothing, variance-reduction (via randomization or coordinate averaging), or step-size modulation can extend mixing guarantees to cases otherwise beyond the reach of standard LMC (see also (Chatterji et al., 2019, Doan et al., 2020, Ding et al., 2020)).

5. Practical Implications and Applications

RLMC and its variants are relevant in high-dimensional sampling for:

• Bayesian inference (complex posteriors, multimodal targets, heavy-tailed priors) in large-scale settings.
• Scientific computing tasks such as uncertainty quantification, where posterior distributions are often nonconvex but satisfy a log-Sobolev inequality.
• Machine learning, especially in unsupervised learning and generative modeling where the target is complex, non-globally convex, but log-Sobolev or Poincaré inequalities can be verified or plausibly assumed.

The non-asymptotic, dimension-explicit, and uniform-in-time error bounds in the 2-Wasserstein distance provide actionable quantitative estimates for tuning the step size $h$ and the number of steps required to guarantee a prescribed accuracy $\epsilon$, with

$$n \gtrsim \frac{1}{\lambda h} \log\frac{\sqrt{d}}{\epsilon},$$

and $h$ scaling as $O(\epsilon / \sqrt{d})$ (for Lipschitz drift), where $\lambda$ is the contraction rate for the chain.

In regimes of non-globally Lipschitz drift, projected RLMC still ensures stability, though with polynomially worse dependence on $d$ and on the drift growth exponent $\gamma$.

6. Extensions, Limitations, and Future Research

Potential extensions involve:

• Analysis for drift functions with “locally” but not globally controlled growth, via adaptive or local taming in the projection.
• Extension to underdamped Langevin dynamics with randomized or regime-switching numerical schemes (see (Wang et al., 31 Aug 2025)).
• Smoothing-based randomization in non-smooth/Hölder targets coupled with bias-variance control (see (Chatterji et al., 2019, Doan et al., 2020)).
• The design of randomized or high-order algorithms that further reduce bias, e.g., randomized Runge-Kutta integrators (Bou-Rabee et al., 2023) or splitting methods for UBU/BUB schemes (Chada et al., 2023).

While RLMC achieves convergence under much weaker regularity conditions than classical LMC, the dependence on dimension can be severe in extreme non-Lipschitz settings, and the explicit constants may become prohibitive for very high dimensions or extremely heavy-tailed (i.e., small log-Sobolev constant) potentials.

7. Connections to Broader Literature

The RLMC framework is situated among a spectrum of recent advances:

• High-order Langevin (Itô–Taylor) discretizations for superlinear drifts (Sabanis et al., 2018), conferring higher-order convergence in Wasserstein and total variation distances.
• Random smoothing and variance reduction for non-smooth potentials (Doan et al., 2020, Chatterji et al., 2019), improving mixing under weak smoothness.
• Random coordinate and regime-switching variants (Ding et al., 2020, Wang et al., 31 Aug 2025), enhancing computational efficiency per iteration.
• Rigorous uniform-in-time convergence beyond strong convexity for both LMC and RLMC under general functional inequalities (Chewi et al., 2021).

In conclusion, RLMC and its higher-order or variance-reduced modifications significantly expand the scope of tractable MCMC for high-dimensional and non-classical sampling problems, providing uniform-in-time, non-asymptotic guarantees under log-Sobolev (or even weaker) conditions, with explicit dimension- and accuracy-dependence—meeting the theoretical and applied demands of contemporary computational statistics and machine learning.