Gradient-Only ULD Methods in Langevin Dynamics

• The paper presents a gradient-only third-order ULD discretization method that improves convergence rates under robust smoothness conditions.
• The method leverages stochastic gradient evaluations to bypass expensive Hessian computations, ensuring efficient high-dimensional sampling.
• Empirical studies in Bayesian inference demonstrate reduced gradient calls and lower sampling errors compared to first and second-order schemes.

A gradient-only method for underdamped Langevin dynamics (ULD) refers to the class of MCMC algorithms or optimization procedures for sampling from continuous distributions that use only gradient information of the potential function rather than relying on higher-order derivatives (such as the Hessian or third-order tensors). These methods are designed to efficiently approximate the transition dynamics of the ULD SDE, achieving theoretically optimal or improved convergence rates under various smoothness and convexity assumptions. Gradient-only approaches are particularly significant when higher-order information is expensive or infeasible to compute, and when adaptive or robust numerical schemes are required in stochastic or high-dimensional contexts.

1. Mathematical Framework for Gradient-Only ULD

Fundamental ULD dynamics are governed by the SDE: $$dx_t = v_t\,dt, \qquad dv_t = -\gamma v_t\,dt - \nabla f(x_t)\,dt + \sigma\,dW_t$$ where $f$ is the log-density potential, $\gamma$ is the friction parameter, and $W_t$ is a standard Brownian motion. Gradient-only methods for ULD seek numerical schemes that discretize the above dynamics without explicit computation of higher-order derivatives. Notably, the target distribution for $x$ is proportional to $\exp(-f(x))$, and proper discretization ensures convergence in 2-Wasserstein distance to this density.

Recent methods, such as the shifted ODE (Foster et al., 2021) and QUICSORT (Scott et al., 22 Aug 2025), maintain strong theoretical guarantees and practical accessibility by expressing all updates in terms of gradient evaluations.

2. High-Order Gradient-Only Discretization Schemes

Classical gradient-only schemes (e.g., Euler-Maruyama, Strang splitting) offer first or second-order convergence. However, recent developments have pushed the order further under additional smoothness assumptions:

• When both the gradient and Hessian of $f$ are Lipschitz continuous, the gradient-only QUICSORT and shifted ODE algorithms attain second-order complexity $\mathcal{O}(\sqrt{d}/\varepsilon)$ or $\mathcal{O}(\sqrt{d}/\sqrt{\varepsilon})$ steps for 2-Wasserstein error $\varepsilon$ (Scott et al., 22 Aug 2025, Foster et al., 2021).
• Critically, if the third derivative of $f$ is also Lipschitz, the QUICSORT method achieves third-order complexity: $\mathcal{O}(\sqrt{d}/\varepsilon^{1/3})$ steps for target error $\varepsilon$ (Scott et al., 22 Aug 2025).

This improved rate is realized via a carefully constructed piecewise linear expansion of Brownian motion, introducing intermediate Gaussian random variables (notably $H_n$ and $K_n$) into the discretization so that the error terms arising in the stochastic Taylor expansion are matched without explicit third-order derivative computations.

Method	Smoothness Assumptions	Complexity to $\varepsilon$ error
Euler-Maruyama	Gradient Lipschitz	$\mathcal{O}(\sqrt{d}/\varepsilon)$
Strang splitting	Gradient/Hessian Lipschitz	$\mathcal{O}(\sqrt{d}/\sqrt{\varepsilon})$
QUICSORT	Third derivative Lipschitz	$\mathcal{O}(\sqrt{d}/\varepsilon^{1/3})$

3. Implementation Principles

Gradient-only schemes are formulated to avoid Hessian-vector products or third-order tensor contractions. Given a step size $h$, a typical update in QUICSORT or the shifted ODE method takes the form:

• Draw standard Gaussian vectors $W_n, H_n, K_n$.
• Compute position and velocity updates using weighted averages of prior positions, velocities, and gradients, incorporating terms like $12K_n$ to reproduce higher-order expansion effects.
• Employ an ODE integrator (third-order Runge-Kutta or fourth-order splitting) using only $\nabla f$, possibly at multiple intermediate points per step.

All operations are evaluated on batch gradient information, making the method scalable and implementable in frameworks where Hessians are unavailable. Error estimation for step size adjustment can be assessed via strong error estimators comparing successive step sizes or batch splits.

4. Theoretical Guarantees and Error Bounds

Under strong log-concavity and smoothness (gradient, Hessian, and third derivative all Lipschitz), gradient-only third-order ULD methods yield the following non-asymptotic error bounds for the sampled marginal distribution $\hat{p}_n$: $W_2(\hat{p}_n, p^*) \leq \varepsilon$ using $\mathcal{O}(\sqrt{d}/\varepsilon^{1/3})$ steps, where $p^*(x) \propto \exp(-f(x))$. These bounds significantly improve over earlier schemes in high-accuracy regimes and are established via Taylor expansions, stochastic contractivity, and coupling arguments [Bernoulli 25(4A):2854–2882, 2019; (Scott et al., 22 Aug 2025)].

Furthermore, ergodicity, moment bounds, and stationary distribution invariance are maintained under the gradient-only formulation, ensuring the absence of numerical instability or mode bias (see foundational results in Pavliotis, 2014).

5. Practical Performance and Empirical Validation

Bayesian logistic regression experiments demonstrate that QUICSORT performs on par or better than existing underdamped Langevin MCMC algorithms and popular samplers such as NUTS. Comparative studies reveal that when third-order smoothness holds, third-order gradient-only ULD discretization yields both lower sampling error and reduced number of gradient evaluations per target error threshold, especially in moderately high-dimensional inference tasks.

Empirical validations span a variety of real-world datasets, indicating that the increased per-step computational cost (two to four gradient calls per step versus one for first-order schemes) is offset by the reduction in total steps required for a given sampling accuracy.

6. Context and Impact in Stochastic Sampling

Gradient-only methods for ULD are essential when the potential function is complex, high-dimensional, or computationally intensive, where Hessian computation is prohibitive. The advance to third-order convergence guarantees (given third-order smoothness) expands the design space for practitioners seeking efficient, high-accuracy MCMC samplers for Bayesian inference, uncertainty quantification, and related applications in machine learning and computational statistics.

The approach unifies ideas from stochastic process theory [Pavliotis, 2014], high-order Langevin discretization for accurate sampling [Bussi & Parrinello, 2007], and modern contractivity analysis [Bernoulli 25(4A):2854–2882, 2019], placing gradient-only third-order discretization on firm theoretical and practical ground.

7. Future Directions

A plausible implication is the extension of gradient-only third-order methods to non-strongly log-concave or non-smooth regimes via adaptive step-sizing, geometric integration, or robust regularization. The augmentation of stochastic integrators with additional gradient-only correction terms may further enable high-accuracy sampling in large-scale, non-Euclidean, or multi-modal contexts, with promising applicability to large dataset Bayesian neural networks and complex posterior inference.

The field is likely to investigate optimal trade-offs between per-step computational cost and overall complexity, and quantify the behavior of these methods under non-asymptotic, non-smooth, or noisy-gradient environments.