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PID-controlled Langevin Dynamics for Faster Sampling of Generative Models

Hongyi Chen, Jianhai Shu, Jingtao Ding, Yong Li, Xiao-Ping Zhang

TL;DR

This work introduces PID-controlled Langevin Dynamics (PIDLD), a training-free method that accelerates Langevin-based sampling by treating energy gradients as feedback signals and incorporating gradient history (integral) and gradient tendency (derivative). It is a drop-in replacement for standard Langevin steps, compatible with any LD-based sampler and preserving convergence with a decaying integral gain. Across image-generation and reasoning tasks, PIDLD delivers higher sample quality with far fewer steps, achieving substantial speedups (e.g., at least 10× under low NFEs) without retraining. By importing control-theoretic principles into stochastic sampling, PIDLD broadens the practical applicability of EBMs and SGMs for efficiency-critical applications.

Abstract

Langevin dynamics sampling suffers from extremely low generation speed, fundamentally limited by numerous fine-grained iterations to converge to the target distribution. We introduce PID-controlled Langevin Dynamics (PIDLD), a novel sampling acceleration algorithm that reinterprets the sampling process using control-theoretic principles. By treating energy gradients as feedback signals, PIDLD combines historical gradients (the integral term) and gradient trends (the derivative term) to efficiently traverse energy landscapes and adaptively stabilize, thereby significantly reducing the number of iterations required to produce high-quality samples. Our approach requires no additional training, datasets, or prior information, making it immediately integrable with any Langevin-based method. Extensive experiments across image generation and reasoning tasks demonstrate that PIDLD achieves higher quality with fewer steps, making Langevin-based generative models more practical for efficiency-critical applications. The implementation can be found at \href{https://github.com/tsinghua-fib-lab/PIDLD}{https://github.com/tsinghua-fib-lab/PIDLD}.

PID-controlled Langevin Dynamics for Faster Sampling of Generative Models

TL;DR

This work introduces PID-controlled Langevin Dynamics (PIDLD), a training-free method that accelerates Langevin-based sampling by treating energy gradients as feedback signals and incorporating gradient history (integral) and gradient tendency (derivative). It is a drop-in replacement for standard Langevin steps, compatible with any LD-based sampler and preserving convergence with a decaying integral gain. Across image-generation and reasoning tasks, PIDLD delivers higher sample quality with far fewer steps, achieving substantial speedups (e.g., at least 10× under low NFEs) without retraining. By importing control-theoretic principles into stochastic sampling, PIDLD broadens the practical applicability of EBMs and SGMs for efficiency-critical applications.

Abstract

Langevin dynamics sampling suffers from extremely low generation speed, fundamentally limited by numerous fine-grained iterations to converge to the target distribution. We introduce PID-controlled Langevin Dynamics (PIDLD), a novel sampling acceleration algorithm that reinterprets the sampling process using control-theoretic principles. By treating energy gradients as feedback signals, PIDLD combines historical gradients (the integral term) and gradient trends (the derivative term) to efficiently traverse energy landscapes and adaptively stabilize, thereby significantly reducing the number of iterations required to produce high-quality samples. Our approach requires no additional training, datasets, or prior information, making it immediately integrable with any Langevin-based method. Extensive experiments across image generation and reasoning tasks demonstrate that PIDLD achieves higher quality with fewer steps, making Langevin-based generative models more practical for efficiency-critical applications. The implementation can be found at \href{https://github.com/tsinghua-fib-lab/PIDLD}{https://github.com/tsinghua-fib-lab/PIDLD}.

Paper Structure

This paper contains 28 sections, 2 theorems, 7 equations, 9 figures, 15 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background and Related Work
  3. Langevin Dynamics in Generative Models
  4. PID controller
  5. Methods
  6. Controlling Langevin Sampling
  7. Method Analysis
  8. The Proposed Algorithm
  9. Experiments
  10. Image Sampling for Generation Tasks
  11. Solution Sampling for Reasoning Tasks
  12. Discussion and Conclusion
  13. Comparison of Related Works
  14. Comparison of Langevin Sampling Speedup Techniques
  15. Positioning PIDLD Among Sampling Acceleration Strategies
  16. ...and 13 more sections

Key Result

Proposition 1

Consider the discrete-time modified Langevin dynamics: where $-U_{\theta}(x)$ is $m$-strongly convex ($\nabla^2 U_{\theta}(x) \leq -mI$), $\xi_t \sim \mathcal{N}(0, I)$, and $\epsilon,k_d > 0$. In the deterministic case ($\xi_t \equiv 0$), the system is asymptotically stable at $x^*$ (where $\nabla U_{\theta}(x^*) = 0$) if Then with the random noise ($\xi_t\neq0$), the system admits a unique sta

Figures (9)

  • Figure 1: PID-controlled systems eliminate errors ($e_t$) at steady-state through the integral term while dampening overshoot via the derivative term. In the Langevin sampling context, we reinterpret PID's value: vanilla Langevin samples can be trapped at non-optimal equilibrium points with zero gradients, requiring noise $\xi_t$ and additional iterations to escape (dashed lines). The integral term provides a momentum-like effect that helps samples quickly escape these suboptimal points, though it introduces overshoot. The derivative term adds awareness of real-time tendencies, accelerating descent while reducing overshoot. Together, these mechanisms enhance both sampling speed and quality.
  • Figure 2: KL divergence along sample steps using vanilla LD(baseline), LD with integral term(+I), LD with derivative term(+D) and the complete PID controlled LD(+I&D). We use $k_p=1,k_i=0.1,k_d=6$.
  • Figure 3: KL divergence during sampling with different parameter configurations.
  • Figure 4: Effect validation on $I$ term and $D$ term. The result for each hyperparameter setting is averaged over 100 independent runs with different random seeds.
  • Figure 5: Image generation quality comparison of ALD and PIDLD under low NFEs.
  • ...and 4 more figures

Theorems & Definitions (4)

  • Proposition 1: Stability of Langevin Dynamics with Derivative Term
  • Lemma 1: Convergence of Covariance Series
  • proof
  • proof