PID-Controlled Langevin Dynamics

Updated 23 November 2025

PID-controlled Langevin Dynamics is a feedback-augmented sampling algorithm that integrates proportional, integral, and derivative components to accelerate convergence in high-dimensional, nonconvex energy landscapes.
It decomposes the gradient into P, I, and D terms with time-decayed gains, ensuring robust and efficient navigation towards the target distribution without modifying underlying energy models.
PIDLD achieves an order-of-magnitude reduction in sampling steps with negligible computational overhead, making it highly effective for image generation and reasoning tasks in generative modeling.

PID-controlled Langevin Dynamics (PIDLD) is a feedback-augmented sampling algorithm that accelerates Langevin dynamics-based sampling in generative modeling by introducing proportional-integral-derivative (PID) control concepts into the update mechanism. The approach unifies gradient-based “forces” and their temporal feedback, enabling faster, more robust convergence to the target distribution in high-dimensional, possibly nonconvex energy landscapes. PIDLD enables an order-of-magnitude reduction in the number of steps needed for high-fidelity sample generation, with negligible computational and memory overhead and no modification to underlying energy or score models (Chen et al., 16 Nov 2025).

1. Langevin Dynamics and Sampling Framework

Langevin dynamics (LD) provides a foundation for sampling from target distributions $\pi(x) \propto \exp\left( U_\theta(x) \right)$ , particularly with unadjusted Langevin algorithms (ULA) commonly used in both score-based generative models (SGMs) and energy-based models (EBMs). The standard LD iteratively updates sample states according to

$x_{t+1} = x_t + \epsilon\,\nabla_x U_{\theta}(x_t) + \sqrt{2\,\epsilon}\,\xi_t,\quad \xi_t\sim\mathcal{N}(0,I)$

where $U_\theta(x)$ is the (unnormalized) energy function or score potential, $\epsilon$ is the discretization step size, and the additive noise term $\xi_t$ injects stochasticity. Convergence to the target distribution relies on the careful balance of time discretization, noise, and the energy gradient; however, practical application is often bottlenecked by the large number of fine-grained iterations required for adequate mixing (Chen et al., 16 Nov 2025).

2. PIDLD: Mathematical Formulation and Update Rule

PIDLD generalizes the update mechanism to incorporate feedback on historical and predicted evolution of the gradient signal, following the principles of proportional-integral-derivative (PID) control. At each iteration $t$ , the score is denoted $s_t = \nabla_x U_{\theta}(x_t)$ . The update decomposes the control signal into

$P_t = s_t, \quad I_t = \frac{1}{t+1}\sum_{s=0}^{t}s_s, \quad D_t = s_t - s_{t-1},$

where $P_t$ , $I_t$ , and $D_t$ are the proportional, integral, and derivative components, respectively. The complete PIDLD sampling step is given by

$x_{t+1} = x_t + \epsilon \Bigl( k_p\,P_t + k_i(t)\,I_t + k_d\,D_t \Bigr) + \sqrt{2\,\epsilon}\,\xi_t$

with $k_p$ (proportional gain), $k_i(t)$ (time-decayed integral gain, $k_i(t) = \gamma^t k_i$ , $\gamma<1$ ), and $k_d$ (derivative gain). The normalizing factor $1/(t+1)$ in $I_t$ ensures scale comparability between terms. Notably, the algorithm requires no changes to the underlying energy model and can be applied to any LD-based sampler without re-training or new data (Chen et al., 16 Nov 2025).

3. Stability and Theoretical Properties

Linearization around a local minimum $x^*$ of an $m$ -strongly-convex potential leads to the spectral stability condition: $\epsilon < \frac{1}{(1+2\,k_d)\,m}$ which is sufficient to ensure, in the presence of noise, the existence of a unique stationary distribution and asymptotic stability of the process. The dynamics extend naturally to annealed Langevin dynamics (ALD); one forwards the integral and derivative states between consecutive noise levels. Convergence guarantees for global, nonconvex sampling remain open, but numerical experiments indicate that PIDLD enables much faster convergence in KL divergence and other measures relative to vanilla LD for both unimodal and multimodal situations (Chen et al., 16 Nov 2025).

4. Algorithmic Implementation

Implementation requires only storing previous gradients and updating the integral, proportional, and derivative terms at each step; the dominant cost remains the energy/skipping forward pass. Key steps are:

Initialize $I_0 \leftarrow 0$ , $s_0 = \nabla_x U_\theta(x_0)$ .
For $t=0, \dots, T-1$ $t = 0, \dots, T - 1$ :
- $s_t \leftarrow \nabla_x U_\theta(x_t)$
- $P_t \leftarrow s_t$
- $I_t \leftarrow (I_{t-1}\,t + s_t)/(t+1)$
- $D_t \leftarrow s_t - s_{t-1}$
- $u_t \leftarrow k_pP_t + k_i\,I_t + k_dD_t$
- $x_{t+1} \leftarrow x_t + \epsilon\,u_t + \sqrt{2\epsilon}\,\xi_t$
- $k_i \leftarrow \gamma\,k_i$
Return $x_T$ .

Memory and compute overhead is $O(d)$ per step and vanishes relative to model size and evaluation cost (Chen et al., 16 Nov 2025).

5. Empirical Performance and Applications

PIDLD has been extensively benchmarked on CIFAR-10 and CelebA image generation (NCSNv2 SGM, IGEBM) and reasoning tasks (Sudoku, Connectivity using IRED EBM). FID statistics at various network function evaluations (NFE) demonstrate $5$– $10\times$ reduction in sampling steps required to match standard LD at a given quality threshold:

Dataset/Model	LD (min steps, FID)	PIDLD (min steps, FID)	Speedup
CIFAR-10 SGM	232, 12.5	25, 18.3	%%%%32 $k_i(t)$ 33%%%%
CelebA SGM	500, 9.5	50, 8.0	%%%%34 $k_i(t)$ 35%%%%
CIFAR-10 EBM	40, 35.3	10, 99.0	%%%%36 $\xi_t$ 37%%%%

For reasoning, accuracy is improved or matched at substantially reduced step counts (e.g., Sudoku 5–80 steps: vanilla LD 46% $\rightarrow$ 55%, PIDLD 50.5% $\rightarrow$ 56.6%; Connectivity 1–10 steps: vanilla 86.2% $\rightarrow$ 87.5%, PIDLD 86.2% $\rightarrow$ 93.3%). Ablation studies show that for image generation in shallow landscapes, P+D contributes most of the gain, while P+I is more effective for complex, highly multimodal tasks (Chen et al., 16 Nov 2025).

6. Hyperparameter Tuning and Practical Concerns

Selection of gains $(k_p, k_i, k_d)$ is crucial and typically involves increasing all gains as NFE is decreased; $\gamma$ (integral decay) balances exploration and late-phase stabilization, commonly chosen from $[0.95, 0.99]$ . Empirical grids are reported for a range of tasks. PIDLD incurs negligible additional wall-clock cost per NFE (within $1$– $2\%$ compared to standard LD), and the relative memory/computational overhead vanishes at scale. The method is fully “drop-in” and model-agnostic, requiring only existing pre-trained energy/score models (Chen et al., 16 Nov 2025).

7. PID Control in Langevin Systems: Connections and Variants

PIDLD generalizes prior control-inspired modifications of Langevin dynamics by employing full proportional, integral, and derivative feedback at the gradient level. In contrast, previous works such as pairwise adaptive Langevin (PAdL) thermostats implement an integral feedback on a friction coefficient to match instantaneous kinetic observables to target values, with P and D dynamic extension in related PNHL schemes (Leimkuhler et al., 2016). While PAdL and PNHL aim at accurate thermostatting and stability in particle dynamics (with superconvergence of ergodic averages and adaptive friction), PIDLD’s feedback is applied directly to the energy gradient in state space, enabling its use for accelerated convergence in high-dimensional generative models. Both schemes reflect the broader utility of control theory for correcting, accelerating, and stabilizing stochastic dynamics, but address distinct system observables and optimization landscapes (Chen et al., 16 Nov 2025, Leimkuhler et al., 2016).

PID-controlled Langevin Dynamics synthesizes control-theoretic and stochastic sampling concepts to achieve high-velocity, high-quality sample generation from energy models. By leveraging PID feedback on gradients, it provides a principled approach to navigate high-dimensional, nonconvex energy landscapes with efficiency and robustness unattainable by classical vanilla Langevin dynamics.