Modified RLMC Algorithms for Non-Lipschitz Drifts

• Modified RLMC algorithms are advanced variants of RLMC sampling methods that incorporate projection operators to tame non-globally Lipschitz, superlinear drifts.
• They ensure controlled discretization error and offer non-asymptotic, dimension-explicit error bounds, making them ideal for high-dimensional, non-log-concave distributions.
• By projecting iterates to prevent divergence, these methods enable reliable sampling from complex potentials, addressing limitations of classical Euler–Maruyama schemes.

Modified RLMC Algorithms are advanced variants of randomized Langevin Monte Carlo sampling methods designed to address the limitations of classical RLMC in scenarios where the drift (the gradient of the log-density) exhibits non-globally Lipschitz and superlinear growth. These modifications introduce projection (taming) operators for the drift component, enabling provable non-asymptotic error bounds for sampling from high-dimensional, non-log-concave distributions. The approach ensures controlled discretization error and avoids divergence that may arise under conventional Euler–Maruyama or RLMC schemes with unbounded drifts.

1. Classical RLMC and Its Limitations

Classical RLMC (Randomized Langevin Monte Carlo) is a time-discretized stochastic process for sampling from a target distribution $\pi(dx) \propto \exp(-U(x))dx$ over $\mathbb{R}^d$, based on the Langevin SDE $dX_t = -\nabla U(X_t)dt + \sqrt{2}dW_t$. The method typically relies on the following assumptions:

• The potential $U(x)$ is convex (log-concave target).
• The gradient $\nabla U$ is globally Lipschitz.

Under these conditions, the RLMC algorithm generates iterates based on

$X_{n+1} = X_n - h \nabla U(X_n) + \sqrt{2h} \xi_n,$

where $\xi_n$ are i.i.d. standard normal vectors and $h$ is the stepsize. A randomized version may also include a random time step via an auxiliary variable $\tau_n \sim \mathrm{Uniform}(0,1)$ to achieve better mixing properties.

However, if $\nabla U$ is not globally Lipschitz, or $U$ is non-log-concave (e.g., multimodal, double-well, or with superlinear growth), classical RLMC and the associated explicit Euler discretization may diverge in finite time or yield uncontrolled discretization error (Wang et al., 30 Sep 2025).

2. Modified RLMC: Projected and Tamed Drift Schemes

The modified RLMC algorithms introduce a projection operator $T^h$ that applies to the drift (gradient) component. This operator enforces boundedness on iterates for large values, effectively "taming" the drift when it becomes superlinear. The modified algorithm consists of two primary steps for each iteration:

Predictor block:

$$\bar{Y}_{n+1}^\tau = \bar{Y}_n + F(T^h(\bar{Y}_n)) \cdot \tau_{n+1}h + \sqrt{2} \Delta W_{n+1}^\tau,$$

where $F(x) = -\nabla U(x)$, and $T^h$ acts as a truncation or projection, defined (for polynomial growth $\gamma$) as

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

Corrector block:

$$\bar{Y}_{n+1} = T^h(\bar{Y}_n) + F(T^h(\bar{Y}^\tau_{n+1})) h + \sqrt{2} \Delta W_{n+1}.$$

This projection ensures that iterates cannot escape to regions where the drift is unbounded, maintaining stability and enabling controlled local errors in the discretization.

3. Non-Asymptotic Error Analysis and Uniform-in-Time Bounds

A significant advance in the analysis of modified RLMC algorithms is the derivation of non-asymptotic, dimension-explicit error bounds for the resulting sample distribution under relaxed regularity conditions. Assuming the drift $F$ satisfies a polynomial growth condition: $$|F(x) - F(y)| \leq L_2 (1 + |x|^\gamma + |y|^\gamma) |x-y| \quad \forall x,y \in \mathbb{R}^d,$$ and under log-Sobolev and dissipativity assumptions, the modified pRLMC algorithm admits the following uniform-in-time Wasserstein-2 distance error bound: $$\mathcal{W}_2(\nu_{\bar{Y}_n}, \pi) \leq \mathcal{C}_1 d^{(11\gamma+2)/4} h + \mathcal{C}_2 \sqrt{d} \exp(-\lambda_1 n h),$$ where $h$ is a sufficiently small stepsize, and $\mathcal{C}_1, \mathcal{C}_2, \lambda_1$ are constants independent of $d$ (Wang et al., 30 Sep 2025). The bound is sharp in $h$ and dimension $d$, matching classical rates obtained under global Lipschitzness for the mixing term.

The projection $T^h$ is designed so the local error does not accumulate uncontrollably with nonlinear drifts, while also allowing sampling from measures with superlinear potentials, considerably broadening the applicability of RLMC.

4. Comparison with Traditional and Coordinate-wise RLMC Methods

While the modified RLMC (pRLMC) approach harnesses the projection for stability under superlinearity, other alternatives in the literature (such as Random Coordinate LMC (Ding et al., 2020)) achieve scalability by updating only random coordinates at each iteration. These coordinate-wise methods are most effective for log-concave cases when the gradient and Hessian are regular, leading to computational savings proportional to $\sqrt{d}$, but do not natively address non-Lipschitz drifts or provide the error guarantees established for modified RLMC (Wang et al., 30 Sep 2025).

5. Parameterization and Practical Implementation

The modified RLMC framework is defined in terms of several parameters:

• The stepsize $h$ must satisfy $h \leq \min\{1, 1/(2L), 1/\mu, 1/d^\gamma\}$, where $\mu$ reflects dissipativity.
• The projection operator $T^h$ depends on the degree $\gamma$ of polynomial growth and dimension $d$, ensuring that bounding and dimension-adaptation are automatic.
• Initialization with appropriate moments is assumed: $\mathbb{E}[|x_0|^{2p}] \leq \sigma_p d^p$.

Performance trade-offs involve balancing the stepsize $h$ (smaller is better for accuracy, at the cost of longer run times), the polynomial growth parameter $\gamma$, and computational cost per iteration (projection operations and drift evaluations may require additional computation when compared to classical RLMC).

6. Significance for High-Dimensional Bayesian Sampling and Inference

The practical import of modified RLMC algorithms is their capability to sample efficiently from high-dimensional, non-log-concave distributions as encountered in Bayesian inference, probabilistic machine learning, and scientific computing. The uniform-in-time Wasserstein bounds and explicit control of discretization errors provide guarantees that are absent in conventional unmodified schemes when the drift is unbounded.

This approach unlocks provably convergent and robust sampling for models such as multimodal distributions with double-well potentials, distributions with polynomial tails, and other complex systems where superlinear gradient growth precludes the viability of classical Euler–Maruyama or standard RLMC methods.

7. Summary Table: Classical vs Modified RLMC

Algorithm	Drift Requirement	Error Bound in $\mathcal{W}_2$	Projection/Taming	Sampling Range
Classical RLMC	Globally Lipschitz	$O(\sqrt{d}h)$	None	Log-concave only
Modified RLMC (pRLMC)	Polynomial growth	$O(d^{(11\gamma+2)/4}h)$, $O(\sqrt{d})e^{-\lambda_1 n h}$	$T^h$ before drift	Non-log-concave, superlinear
Coord. RLMC (Ding et al., 2020)	Globally Lipschitz/Hessian	$O(\sqrt{d}h)$ (under regularity)	None	Log-concave (scalable update)

The modified RLMC framework thus constitutes a rigorous and flexible methodology for sampling from distributions previously inaccessible to classical methods when faced with high nonlinearity and dimensionality. Its explicit non-asymptotic error analysis guarantees robust performance and convergence in practice (Wang et al., 30 Sep 2025).