PID-controlled Langevin Dynamics for Faster Sampling of Generative Models
Authors: Hongyi Chen
Jianhai Shu
Jingtao Ding
Yong Li
Xiao-Ping Zhang
Venue: NeurIPS 2025 poster ()
Keywords: Langevin Dynamics
Generative Models
PID controller
Primary area: deep_learning

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
Hongyi Chen Jianhai Shu?*, Jingtao Dingåt, Yong Li2, Xiao-Ping Zhang Shenzhen Key Laboratory of Ubiquitous Data Enabling Laboratory, Shenzhen International Graduate School, Tsinghua University,
Shenzhen 518055, China
2 Department of Electronic Engineering, Tsinghua University, Beijing 100084, China
Pengcheng Laboratory, Shenzhen 518055, China
Abstract
Langevin dynamics sampling suffers from extremely low generation speed, fun- damentally limited by numerous fine-grad ane-grained iterations converge to the tar- get distribution. We introduce PID-controlled Langevin Dynamics (PIDLD), a novel sampling acceleration algorithm that reinterprets the sampling process ing 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 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-crittcal applications. The imple- mentation can found at https://gitubb.oo//ningggaaaaait
Introduction
Langevin dynamics has emerged fundamental sampling technique for generating samples across various implicit generative models, including anergy-baseed Models (EBMs) [8]. Score-based generative models (SGMs) B3 34 35), and Diffusion models [15]. This sampling method gradually approaches the target distribution through stochastic walks along energy gradients (or score functions). visualized particles moving under the influence of energy potential field gradients, while being affected by random fluctuations that correlate with step size.
Despite its theoretical elegance and broad applicability, Langevin sampling suffers from critical limitation: extremely low generation speed that necessitates multiple fine-grained iterations of the discretized Langevin dynamics process (e.g., NCSNv2 requires 1000+ Neural Function Evaluations (NFE) for image sampling (34). This stands in contrast to GANs [8), which enable efficient one- step sampling. During sampling, particles often encounter regions with near-zero gradient "forces" (such as local minima or unstable equilibria), necessitating the addition of random noise to escape these regions, which consequently increases the required number of sampling steps (Fig T left). While reducing sampling steps and increasing step sizes seems like a straightforward acceleration approach, inevitably introduces larger noise fluctuations that significantly degrade sampling quality (23 24).
"Equal Contribution. Coorespnniing Author.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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PTD Controlled System
N, = f(x) stM)
My ft
Nen optinal
Overshoot
+je,dt
dej
d
PID Controlled Langevin Sampling
=u, P. P.0,00)
History Momentim 20>
Tendeacy A
Non-oplinal
(Rltt of
Ovenstoot
df,,y(x)d
4P20o00
Initial Samples
Final Samples
Enengy Potential Field (U+
Figure I: PID-controlled systems eliminate errors (ez) 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 & and additional iterations to escape (dashed lines). The integral term provides momentum -iike 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.
Inspired by control theory, we reinterpret the Langevin sampling process by conceptualizing en- ergy gradients as feedback signals, establishing a correspondence between sampling and feedback control systems [10]. This approach treats particles as dynamic systems under designed feedback control rather than passive stochastic entities, allowing us to efficiently guide sampling toward high- nigh-prrobbbility regions while substantially reducing required iterations.
Specifically, we innovatively consider the historical dynamics and tendency information of potential- field gradient signals (feedback signals) during sampling. We thus propose integrating Langevin sampling with the prominent PID (Proportional-Ineee Iattl--terrrl---- Integral-Derrivatiee) Derivative) I technique from feedback control enhance and accelerate sampling in the following aspects. Firstly, gradient history (the inte- gral term) creates momentum-lin effects by accumulating gradient information, providing sustained directional forces that reduce ineffective random walks. These forces form inertial properties that help sample particles traverse energy barriers and unstable equilibria in complex multimodal distribu- tions, thereby exploring more critical regions and significantly accelerating convergence to minimum energy states of the distribution [23] (Fig I middle). Secondly, gradient tendency (the derivative term) dynamically adjust sampling by responding to real-time gradient change rates, accelerating movement toward directions of consistently decreasing energy gradients while suppressing movement where gradients consistently increase, effectively mitigating overshoot issues by anticipating and counteracting excessive momentum, enabling the overall sampling process to reach potential wells rapidly and stably, thereby improving sampling speed (Fig right).
In this work, we make the following contributions:
(1) By combining PID control methodology with Langevin dynamics, we propose a novel Langevin sampling acceleration algorithm: PIDLD (Proportional Integral Derivative Langevin Dynamics). It offers a brand new insight into applying control theory in the sampling domain. Our method simple, easy to implement, requires no additional datasets prior information, and climinates the need for retraining, making it compatible with any method that uses Langevin dynamics sampling.
(2) Through theoretical analysis and empirical toy experiments, we establish the convergence and effectiveness of our algorithm, validating the reliability of the proposed sampling method.
(3) We further conduct multiple comparative experiments on Langevin sampling-based models (EBMs, SGMs) across various tasks (images, solutions of reasoning tasks), demonstrating that our proposed sampler generates faithful samples faster. In particular, PIDLD achieves least 10x sampling speedup over basclines under SGM model.

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2 Background and Related Work
2.1 Langevin Dynamics in Generative Models
Langevin dynamics (LD) is a mathematical framework describing particle motion in a potential energy field with thermal fluctuations, and has been widely applied in molecular dynamics simulations, statistical physics, and Bayesian inference [36 26 39]. Due its convergence properties, it plays crucial role in sampling for both Energy-based Models (EBMs) and Score-based Generative Models (SGMs).
EBMs have evolved significantly over the past decade, with seminal contributions from Restricted Boltzmann Machines [14. discriminative EBMs [20], and development of improved training techniques for deep EBMs [8] These models represent probability distributions in the form
po(x) = --fo()))Za /Za, where fa(æ) is the energy function parameterized by 0, and Zy is the normaliz- ing constant. For sampling from EBMs, perhaps the most popular sampling algorithm is Unadjusted
I Langevin dynamics (ULAA) 29 8 27 B+++1 = 2g e (o((())) Verce, where x; is the state iteration f, is the step size, and & is standard Gaussian noise. Starting from a randomly chosen Zip can generate sample by iterating the above Markov transition for a specified number of steps T.
SGMs originated from Song and Ermon's 's Noise-Condiiional Score Networks (NCSN) [33] which established the core paradigm of score matching at multiple noise scales (o Their subse- quent NCSNv2
34 improved stability through noise scaling and architectural improvements, while NCSN++ B5 unified score-based and diffusion approaches via stochastic differential equations. Unlike energy models, SGMs learn the gradient of the log probability density function of the noise cor- rupted distribution p(*[x) N(x; x, aæI) through a family of score-mate mooe--mtchingg methods [[6] (x fol- lows the underlying true data distribution), yielding the score function: so (2r, oz) = V v log po (I, a;). After training the score function, we can sample from the model using annealed I Langevin dynamics (ALD) 333: _ = =t + G, v log po (2t, as) + V2ce,, where 6; is relevant to the noise schedule. Notably, under the energy model assumption, V log Po(x)) Therefore, the sampling processes are essentially identical, and we will use Ua(x) to represent both (1r) and fa (ar) in subsequent discussions. The Langevin dynamics in both methods work in that as T + +oo, the distribution of x, converges to steady-state distribution
v (x) ZgZ
Recent research has focused accelerating Langevin sampling. Dockhorn al. 3 enhanced SGMs by incorporating Hamiltonian Monte Carlo methods [26] and proposed critically-dampee Langevin diffusion (CLD) based SGMs that demonstrated remarkable performance improvements. However, this approach necessitates additional learning of the diffusion velocity. In contrast, our method is learning-free. Ma et al. 23 24 introduced a matrix preconditioning technique that equalizes the curvature rates across all directions, thereby achieving faster sampling for high-resolution images. However, this technique relies heavily on target dataset statistics for preconditioning, which restricts its generalization to data-scarce or structurally diverse domains. Our approach eliminates the need for additional prior knowledge or dataset statistics, requiring only a pre-trained energy model score model. Wen et al. [40] and Shetty et al. [31] developed momentum-enhaneeed Langevin dynamics to expedite sampling. Our method treats Langevin sampling from control perspective, delivering more stable results while enhancing interpretability. We provide a detailed comparison of related works in Appendix A.
2.2 PID controller
Feedback control represents the cornerstone of modern control theory, with the fundamental objective of stabilizing systems and enhancing their performance through continuous error correction M. Among the various feedback control strategies, coportional-Integnnn Enteggal-Deriiaatvvee (PID) controllers remain one of the most widely adopted mechanisms in industrial applications due to their remarkable balance of simplicity and effectiveness. The standard PID control law is formulated as: u(t) =
de()), where e(t) = r(t) - vt() represents the error between the desired reference signal r(t) and the measured output v(t). The control input uut) combines three terms with their respective tuning parameters: Kp (proportional gain), K; (integral gain), and K₂ (derivative gain). The elegance of PID design lies in its intuitive interpretation: the proportional term responds to present errors, the integral term addresses accumulated past errors, and the derivative term anticipates future error tendency [9]
Kte(t) + K; Jo e(r)dr + K2 de(t),
3

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The PID framework has transcended traditional control applications to inspire innovations in deep learning architectures [2 and optimizer [37 2]. In particular, for the optimizer, Chen al. 2 accelerate gradient descent optimization in deep learning via PID controller, thereby achieving faster convergence and improved robustness against local optima. Our work is an innovative application of PID feedback control in generative models sampling.
3 Methods
To accelerate Langevin sampling, we reinterpret the method through the theoretical lens of feedback control. By leveraging fundamental principles from the field of feedback control, we then propose an innovative approach that integrates the Langevin sampling. Finally, building upon this framework, we validate the effectiveness of our method through toy experiments, analyze its performance characteristics, and formulate a comprehensive algorithmic framework.
(PID) controller with
3.1 Controlling Langevin Sampling
While LD serves crucial sampling method in generative modeling, it faces significant practical challenges. Firstly, standard LD suffers from slow convergence due to its random- walk behavior, requiring numerous iterations reach the target distribution. Secondly, in multimodal distributions, sampling points rely heavily on stochastic noise to either overcome energy barriers between local minima or escape positions unstable equilibrium points-critical locations with near-zero gradients between potential wells. Finally, its lack of dynamic response to gradient changes leads to unstable sampling across varying potential landscapes. Fundamentally, these limitations stem from the equilibrium-bet based nature of standard Langevin dynamics, which is ill-suited for the evolving dynamics of complex systems [2
From a physical perspective, Langevin dynamics can be interpreted as a particle moving through potential field (2) while subject to thermal fluctuations. The standard discrete LD update corresponds the continuous-time stochastic differential equation: dr In this physical system, the gradient (z) can be seen as the basic feedback control signal driving the particle toward probability regions. However, this simple feedback mechanism proves insufficient for complex high-di gigh-dimensional landscapes. Drawing inspiration from control theory (2]. we can enhance this physical system with more sophisticated control force that utilizes not only the current gradient but also its gradient history (accumulated record of past gradient values, representing the integral of gradient trajectory over time) and gradient tendency (temporal derivative of the gradient, capturing the direction and rate of gradient change):
dz (aU+(z) + (2E(().
dæ dt
Mcantrol (a))), Gradient History, Gradient Tendency) + /22(()
This enhanced control approach directly addresses the key challenges of Langevin dynamics through the following mechanisms:
For sampling efficiency: Accumulated gradient history creates momentum-like effect, providing a persistent directional force that reduces random walk behavior and accelerates convergence. Meanwhile, gradient tendency information allows the system to adapt its movement strength based on local landscape properties, accelerating movement towards directions where gradients consistently increase, thereby further improving efficiency.
For energy barrier and unstable equilibria: The accumulated gradient history forms an inertial force that helps particles traverse energy barriers and unstable equilibria between modes. This substantially reduces reliance random noise for mode exploration and enables more systematic discovery of all significant modes in multimodal distribution 2
For dynamic response: Gradient tendency information captures the rate of change in the potential landscape, allowing the system to stabilize its behavior in different regions dynamically. When the gradient changes abruptly, this term provides damping effects that reduce unnecessary overshoot []]
The Proportional-Inten Integan---nnee-eeeeiiiiee (PID) controller offers an ideal framework for implementing this enhanced control approach. With the gradient (oUo(()) the error signal, each component

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serves a distinct purpose: the proportional term provides basic gradient guidance, the integral term accumulates gradient history to create persistent acceleration and barrier-penetration capabilities, and the derivative term captures gradient change rates to enable adaptive response to different landscape regions [. Based on this framework, we propose integrating a PID) controller into Langevin dynamics:
Kg Σ VoUg(o.)) ka (a(VaUU(a)) VaUb((r-1))) + V2EE
Ft+1 = = ky kyV2Uo(Z))
where kp. ki, and ka are the proportional, integral, and derivative coefficients, respectively. Empirically, to pre- vent the integral term from dominating the sampling pro- cess over time, we apply a normalization factor. This balances the magnitudes of P. I, and D terms, maintaining system stability while ensuring similar influence across all components by default.
KL Divergence During Sampling
10a
KL. Divergeencc
0.06
0.04
To demonstrate the effectiieneess of our proposed PID-controlled Langevin dynamics, we conducted toy experiments (details in Appendix C.I on a two- dimensional Gaussian mixture distribution. Specifically, we used NCSN (33 to learn the distribution Pdata (1(, 5), I) and compare our sam- pling method against the vanilla ALD. As Fig 2 shows, our method generates samples that more accurately match the actual distribution, evidenced by lower KL divergence. also confirms that both integral and derivative terms accelerate convergence and enable the sampling to achieve a lower KI. divergence after convergence.
baseline
+D
600
800
1000
1200
Step
[N((-5,
Figure 2: KL divergence along sample steps using vanilla LD(baseline), LD with integral term(+I). LD with deriva- tive term(+D) and the complete PID con- trolled LD(+I&D). We use kp 1, k 0.1, =
3.2 Method Analysis
Analysis of parameter effect. To quantitatively assess the effects of the newly added terms (integral, derivative) on sampling efficacy, we extended our analysis using the toy experiment from Section 3.1 We set kpp = to correspond with standard Langevin dynamics, while examining how varying values k; and ky influence distributional convergence measured by the KI divergence.
Divergence During Sampling
Divergence During Sampling
Divergence During Sampling
30P
109
1.c6
CL Divegennss
baseline k
10-1
baselin
baseline d.002.0
k
1.0-..0
600
800
1000
1200
600
800
1000
1200
600
800
1000
1200
Step
Step
Step
(a) Comparing &s.
(b) Comparing with/without decay.
(c) Comparing ka-
Figure 3: KL divergence during sampling with different parameter configurations.
As illustrated in Fig. 3a within a specific range (k: E {0, 0.1, 0.2)), introducing progressively increasing integral control terms facilitates faster convergence of the KL divergence while reaching lower terminal values, suggesting that the integral term effectively assists sampling points in traversing energy barriers and unstable equilibria, thereby enhancing sampling quality. However, when k; reaches 0.3, observe rebound phenomenon in the KL divergence, indicating failure to converge to the stationary distribution.
To address this limitation, we implement time-dependent decay coefficient for the integral term: kt()) = 7 k,, where denotes the sampling step and γ < represents the decay rate. This mechanism stands as principled implementation of continuous gain scheduling [19), classic
5

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record the summed Eucledian distance the final cluster centers, deln and d(2), as a measure of Effect intugral Gain on Bles
Effect of Derivilive Gen Bias
Effect Integral Gain Osdilation
Effect Derivative Gain Oscllation
1.00
0.51
100
022
0.45
1.30
800
6.41
2.2
5000
0323
(a) Effeet of k and ka on bias. Adding term (left) (b) Effect of ky and ka on oscillation. Adding D term excels in mitigating bias.
(right) excels in reducing oscillation.
Figure 4: Effect validation on / term and D) term. The result for each hyperparameter setting is averaged over 100 independent runs with different random seeds.
control strategy well-suited for the time-varying nature of I Langevin sampling. The process requires conflicting gains: a high integral gain is beneficial in the early exploration phase build momentum and traverse energy barriers, but becomes detrimental in the late convergence phase, causing the instability seen in Fig.
Our exponential decay acts as an effective gain schedule, smoothly transitioning the controller's behavior from aggressive exploration to stable convergence by ensuring the integral term's influence diminishes as equilibrium is approached. Critically, this design guarantees that the sampling dy- namies asymptotically reduce standard Langevin dynamics (k;(4) +), thus ensuring theoretical convergence. As confirmed by results for k = 0.3 and y = 0.9 in Fig. 3b this principled approach preserves the initial acceleration while resolving the instability.
Further validation on Computer Vision tasks provided in Appendix D.3
From Fig. 3c we observe that introducing a larger D term facilitates faster convergence of the KL divergence to lower values without exhibiting the rebound phenomenon. This indicates that the D term effectively helps sampling points navigate around distribution barriers, enabling them to converge more rapidly to lower points in the energy landscape representing the distribution within finite number of steps, thereby enhancing sampling efficiency.
Analysis of control-theoretic effect. To empirically validate the control-theoreet effect (I] of the integral term on mitigating bias and the derivative term on reducing oscillation, we conducted further experiments (description of settings and metries can be found in Appendix [D.] and Appendix D.2). For the bias-mitigating effect, we mamually add constant perturbation to the gradient and run annealed Langevin dynamics sampling. From Fig 4al we observe that increasing helps decrease dj and d2. a measure of the final bias of cluster centers, while increasing ka yields the opposite effect, which demonstrates it the integral term that makes the sampling algorithm robust to gradient bias. For oscillation-redueee reducing effect, we run the toy experiments with different k; and ka again, and
oscillation. From Fig 4b we observe that (1)) and d(2) decrease as ki increases, while they do not show a comparable decreasing trend as we increase k; The result echoes the PID control theoretic insight that the derivative term has the advantage of reducing the oscillation in the sampling process.
Analysis of convergence guarantee. To provide thcoretical justification for the stabilizing effect of the derivative term, we analyze its impact on convergence under the standard assumption of local strong convexity-4 tractable setting for studying behavior near a potential minimum. We formally establish that the D-term preserves the convergence guarantees of standard Langevin dynamics, confirming it acts as a proper stabilizing component without disrupting the sampling process: Proposition 1 (Stability of Langevin Dynamics with Derivative Term), Consider the discrete-time modified Langevin dynamics:
It-1 + (V. + ky ku (=Uo(((-1)) + Veeci,
(3)
where (z) is m-strongly convex (V2Ue(x) < -ml), & ~ NNO,,), 1), and e, E, > 0. In the deterministic case (E: 0), the system is asymptotically stable x" (where VUo(x²) 0) if
< (1 + 2k.)
(4)

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Algorithm PIDLD
Require:
Score function V.Un(a) = V. log v=(-Jo(x)) number of steps T, step size e, control parameters kp, k, ka, decay rate y < 1, initial point 2u
I: Initialize integral term Iu
Compute initial score so V. (x0o
for to 7
4: & V,Uo(ar
5: P.
I. 1 (Proportional term)
6:
(Ie-1 1 si) [Integral term)
7:
Dr = = -8 (Derivative term)
8:
ur kylt + kill kaD, (Compute control signal]
9:
J(+1 = rr tate ehee-t + 22 & where & ~ (O, I) (Update state]
10: ls -
11:
end for
return á =T
Then with the random noise (&t * 0), the system admits a unique stationary distribution, i.e. Too
The proof is provided in Appendix B
This proposition establishes that under appropriate step size and ka constraints, our modified Langevin dynamics preserves the fundamental convergence property. Specifically, in locally strongly convex energy landscapes, convergence to the stationary distribution ensured with properly calibrated parameters.
3.3
The Proposed Algorithm
Based on our empirical findings and theoretical analyses, we propose PIDLD pnpportional-Integrrn Derivative Langevin Dynamies, Algorithm [I. a novel sampling algorithm that effectively combines the strengths of PID control theory with Langevin dynamics. This approach aims to address the limitations of traditional Langevin-based samplers, particularly in efficiently navigating complex energy landscapes characterized by multiple modes and energy barriers.
An application of PIDLD's training-free. modular design is its seamless integration with annealed Langevin Dynamics (ALD) [3)] This integration preserves ALD's beneficial annealing structure while enhancing efficiency each noise level. The implementation is straightforward: the PIDLD controller replaces the vanilla Langevin steps within each noise level Crucially, to maintain smooth trajectory across noise scales, the final accumulated I, and gradient (V (UU((z)) for the derivative term) from the previous scale are carried over as the initial state for the next scale a; This ensures that historical information is leveraged continuously throughout the sampling process.
Experiments
In the experiments, evaluated our method against standard Langevin sampling mainstream generative models (SGM, EBM). focused on the effectiveness in classical image generation tasks and investigated PIDLD's efficacy in reasoning task solutions via energy model sampling.
4.1 Image Sampling for Generation Tasks
In this section, we evaluate how off-the-shelf generative models that use an LD-based sampling method can be accelerated by the proposed PIDLD whilst maintaining, or even enhancing, image sampling quality, without retraining or post-training the base model.
Datasets and Base Models. Following NCSN 33 we test image generation quality CIFARI0 (32x32) and CelebA (64x64) datasets (both are unconditional datasets). For the acore-based generative model (SGM), we adopt the NCSNv2 34 architecture and initialize the score network with its official pretrained checkpoint. For the energy-based model (EBM), we leverage the Implicit Generative Energy-Basee -aaed Model (IGEBM) [ which has established itself seminal framework in the energy- based modeling literature. initialize the model with its official pretrained checkpoint on CIFAR10 and retrain the model on CelebA. Please refer to the Appendix C.2 for detailed implementation.

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Table FID comparison for SGM and EBM models across different datasets and NFEs. Note that NCSNv2 uses L noise levels. and in each noise level, we perform T Langevin update steps, so the
NFE given by T. The configuration where L > follows the setting in
Dataset
Method
SGM 34
EBM S
NFEs
25x1
100x1
232x1
232x3
232x5
10
20
30
40
Vanilla 33
8
46.8 /
17.2
16.0
12.8
12.5
135.8
58.1
40.3
35.3
CIFARIO
MILD 31
15.5
13.6
13.0
111.4
49.9
38.9
34.4
Ours
18.3
12.1
11.7
11.6
11.4
99.0
46.1
32.8
33.2
NFEs
50x1
250x1
500x1
500x3
500x5
15
20
25
30
Vanilla
25.0
13.6
14.0
11.3
9.5
109.1
63.5
41.3
35.4
CelebA
MILD
/
9.0
9.4
11.0
60.1
41.1
35.3
32.9
Ours
8.0
5.9
5.9
58.0
38.9
32.2
30.0
Samples, FID=46.8
Sampies, FID=18.3
Samples, FID=25.0
Samples,
(a) CIFARIO samples with 25 sampling steps.
(b) CelebA samples with 50 sampling steps
Figure 5: Image generation quality comparison of ALD and PIDLD under low NFEs.
Baselines and Metrics. We choose the vanilla ALD 33 and momentum-imbued Langevin dynamics (MILD,33) as our baselines. Note that vanilla ALD is the degenerate version of our PIDLD algorithm when set hp = 1, k; 0, hy = O. Accordingly, tune only kn to get the best FID score of vanilla ALD on each NFE. For MILD, we directly use the published results of NCSNv2 from the original paper, and additionally evaluate its performance in IGEBM. We report FID scores [33 computed on 10000 samples for each test.
fix k; and ha to 0 and
Experiment Results. The results shown in Table 1 demonstrating that PIDLD consistently outperforms both vanilla and momentum ALD across all NFEs on CIFARIO and CelebA datasets, regardless of the base model used. Compared to the best-performing baseline, our method achieves significant performance gains of 7.6% (SGM) and 3.4% (EBM) on CIFAR10, and even more substan- tial improvements of 38.3 3% (SGM) and 8.8% (EBM) on CelebA. Notably, when using SGM, PIDLD achieves FID scores of 12. 1/8.0 with only 100/50 NFEs on 1IFAR10//elebA, exceeding the best performance of baselines. This demonstrates least 10x speedug speeduf[coopprrce compared to the baselines, which reach optimal performance at much larger NFEs, highlighting PTDLD's computational efficiency alongside quality improvements.
Further, we have the following findings. First, SGM outperforms EBM in absolute performance, indicating more accurate score prediction. This suggests that PIDLD's effectiveness in enhancing sampling performance and efficiency correlates directly with the accuracy of the underlying pre- trained score prediction model. Second, when compared to the baseline MILD algorithm, our approach exhibits consistently superior and more stable performance across varying inference budgets. This enhancement results from extension of MILD's 's momentum-basee acceleration idea through the integration of PlD-based feedback control, which simultancously incorporates both historical feedback and current trend signals. This comprehensive control mechanism substantially improves upon the robustness of purely momentum-drite methods, yielding advancements in both sampling quality and computational efficiency.
'Regarding the reasonability of using NFE as measure of computation cost, please refer to Appendix

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Table 2: Accuracy comparison for baseline and our models across harder Sudoku and Connectivity under different NFEs.
tasks
Task
Method
EBM 7
Accuracy (%)
NFEs
5
10
15
30
40
80
Sudoku
Vanilla
45.99
51.00
50.93
50.77
53.63
55.02
MILD
49.75
54.82
53.55
55.25
56.56
56.64
Ours
50.54
55,48
55.55
55.94
57.02
56.64
NFEs
2
4
5
10
Connectivity
Vanilla
86.16
87.22
87.22
87.48
87.38
87.49
MILD
86.16
88.54
89.21
89.75
90.15
90.33
Ours
86.16
91.32
92.31
92.82
92.95
93.28
Visualization, From the experiment results, we find that compared vanilla ALD, PIDLD delivers markedly higher image quality, especially under low NFEs. We illustrate this contrast in Fig 5 whereas ALD outputs appear muted and blurred, PIDLD samples exhibit richer color fidelity and sharper detail. This visualization underscores PIDLD's ' superior performance in resource-constrained sampling regimes.
Ablation Study. We conducted a comparative analysis PIDLD's performance under ablation settings involving the integral and differential terms, as shown in Fig 6 We observe that while both terms prove effective, the differential term is the primary contributor to quality improvement. This phenomenon results from the nature of annealed Langevin dynamics. The early-stage potential landscapes are smoothed by noise and exhibit shallow barriers between wells. In such scenarios, the integral term's ability to traverse local minima becomes less effective. In contrast, the differential term enables rapid convergence toward the centers of local wells at each noise scale, providing better initializations for subsequent sampling steps.
4.2 Solution Sampling for Reasoning Tasks
CIFAR10
Given PIDLD's capacity traverse local optima and sample from global energy minima, we evaluate the effectiveness of the proposed PIDLD when performing inference tasks using energy- basegy mod- els. Such methods formulate reasoning problems as iterative energy minimization problems over learned energy functions that parame- terize an energy landscape across all possible solutions. By applying PIDLD to the sampling process, we aim to address the limitations of conventional Langevin dynamics in navigating complex energy landscapes, thereby improving the overall performance and accuracy of energy models in solving reasoning tasks.
10
25
100
NFE:
P41+D
CelebA
Tasks and Base Model. We follow existing work 6 7 by selecting challenging reasoning tasks for our experiments. Specifically, we on Harder Datasets for Su- doku 38 28 and Connectivity [5] problems. These Harder Datasets consist of diitt--diitiitttion data relative to the training distribu- tion, requiring models to utilize additional computational resources during inference to achieve effective generalization. In Sudoku, the model must complete partially filled grid by predicting values that satisfy both Sudoku rules and the given constraints, with IRED using the SAT-Net (38) dataset (3 14-2 given numbers) for training, while evaluating generalization on the more challenging RRN dataset [28) (17-34 given numbers). The connectivity task requires model to determine whether paths exist between node pairs in a graph given only the adjacency matrix, with IRED's training using graphs of up to 12 nodes, while the harder dataset challenges the model with larger 18-node graphs requiring more complex reasoning steps.
30 250
evaluate the performance of IRED
Figure 6: Ablation study of PIDLD under image sampling tasks. P+I+D denotes the com- plete model, while P+I, P+D, and P represent models with the derivative term, integral term, or both terms removed, respectively. Figures on the bar indicate performance im- provements of each ablation compared to P.
Baseline and Metrics. We compare with the vanilla sampling method accuracy serves the primary evaluation metric.
7 For both tasks, solution

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Implementation.
In our experiments, we change the number of optimization steps in the original
paper and finetune the hyperparameters get the best accuracy. The step sizes are fixed and scheduled in the base model.
Experiment Results. The results presented in Table 2 demon- strating that our proposed method consistently outperforms the base- line approach across varying NFEs both Sudoku and Connectivity reasoning tasks. For the Sudoku task, our method achieves signifi- cant performance improvements ranging from 4.55% 6.02% low NFEs (5-15), while maintaining stable advantage of approximately 3-5% higher NFEs (30-80). Most notably, our approach reaches an accuracy of 50.54% with just 5 NFEs, surpassing the baseline's ' performance at 10 NFEs (51.00 (1.00%). and demonstrating substantial computational efficiency.
Sudoiu
60%
+5 2%
160%
10.2%
50%
40%
15 NrEs
P11
P.D
P1I1
Connedivily
188%
Acsuneey
+4.7%
4h
For Connectivity, our method exhibits performance gains beginning at NFE=2 with improvements of 3.14%, 3,47%, 3.42%. 3,11%, and 3.27% across the 2-10 NFE range. Notably, our method achieves 91.32% accuracy at just NFE=2, already exceeding the baseline's s maximum accuracy of 90.33% at NFE=10, representing 5x im- provement in computational efficiency. The overall results demon- strate both quality improvements and computational advantages over the baseline approach.
20%
28%
86%
NFEs
Figure 7: Ablation study of PIDLD under reasoning tasks. P+I+D denotes the complete model, while P+I, P+D, and P represent models with the derivative term, integral term, or both terms removed, respec- tively. Percentages indicate performance improvements of each ablation compared P.
We analyzed PIDLD performance through ablation studies of integral and derivative components (Fig. 7 Results indicate that combining both terms produces optimal performance across all sampling steps. The integral term provides consistent performance gains, while the derivative term primarily enhances early-stage convergence. As NFE increases, the derivative term's s benefits diminish substantially, whereas the integral term's capacity to navigate local optima becomes dominant. This stems from the critical importance of attaining energy minima reasoning tasks.
Ablation Study.
Discussion and Conclusion
We introduce PID-controlled Langevin Dynamics (PIDLD), novel, training-free algorithm that accelerates sampling for generative models by reformulating the process through the lens of control theory. By leveraging integral term for momentum and derivative term for adaptive stabilization, PIDLD efficiently navigates complex energy landscapes to produce higher-quality samples in fewer steps. Its plug-and-play nature makes it immediately applicable any pre-trained model reliant Langevin sampling.
It is important to position PIDLD's contribution within the broader context of generative model acceleration. While significant progress has been made with methods like DDIM 32 and other ODE-based solvers 355, our work's primary focus is not to compete with these alternative generative frameworks. Instead, our goal is to fundamentally improve the foundational Langevin Dynamics sampler itself, which remains comnertone for many energy-basee and score-basee base-based models. PIDLD designed as a direct, drop-in enhancement for the standard LD sampling procedure within these models, whereas methods like DDIM alter the underlying diffusion process itself. The main applica- tion of our work therefore to serve as a general and highly efficient inference accelerator for any pre-trained model that relies on manggevin-based sampling.
Acknowledgments and Disclosure of Funding
This work is supported by Shenzhen Ubiquitous Data Enabling Key Lab under grant ZDSYS2022052777140000 the Tsinghua Shenzhen International Graduate grui Endowed Professor-ship Scheme of Shenzhen Pengrui Foundation and the National Natural Science Foundation of China (No. U24B20180, 80, 62476152).
Pen-
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A
Comparison of Related Works
A.1
Comparison of Langevin Sampling Speedup Techniques
Table Comparison of Langevin Sampling Acceleration Methods
Method
No Retrain"
No DiSts"
Theory Basis
Simplicity"
Stability
Vanilla LD'
N/A
N/A
Base Sampling
Base
CLD
3
Statistic Mechanics
Low
PDS
23 24
Matrix Preconditioning
Moderate
Momentum-basse baseed [3 40
Optimization (Momentum)
High
PIDLD(Ours)
Control Theory (PID)
High
3 No Retrain: No retraining of the base generative model is required for acceleration.
b No DtSts:
The core acceleration mechanism does not require datasct-specific statistics/prions.
TThorry Basis: The primary theoretical foundation of the acceleration method (or hase method).
Simplicity: Ease of modifying standard Langevin sampler base methad complexity).
Stability: Employs an explicit or inherent mechanism to prevent common Langevin sampling instabilities (e.g.-
overshooting, oscillations).
Vanilla LD: Standard Langevin Dynamics, serves as baseline. "N/A" as it's not an acceleration method itself.
Table provides a comparative overview of several methods for accelerating Langevin sampling, in- cluding our proposed PIDLD, against Vanilla Langevin Dynamics (LD) as bascline. The comparison highlights key practical and theoretical characteristics: the necessity of model retraining, reliance on dataset-specific statistics for the mechanism, underlying theoretical basis, integration simplicity, and critically, the employment of an explicit or inherent mechanism for instability prevention (e.g..
against overshooting oscillations).
Our method, PIDLD, stands out by achieving high simplicity and flexibility, requiring no model retraining and operating independently of dataset-specific statistics, similar to Momentum-based approaches. However, PIDLD uniquely incorporates a control-theoretic (PID) framework that provides explicit mechanism for instability prevention through its derivative (D) term, which adaptively damps updates based on gradient trends. This direct approach to stability contrasts with Momentum-based methods, which lack such adaptive damping feature and are thus marked as not explicitly preventing these common Langevin instabilities. CLD also inherently prevents instability through its eritically-damper formulation within a retrained, augmented space. PDS, while not requiring retraining, relies on dataset statistics for optimal performance and does not feature an adaptive instability prevention mechanism in its sampling dynamics. Therefore, PIDLD offers compelling combination of ease of use, broad applicability, and enhanced stability through its novel control-inspired design.
A.2 Positioning PIDLD Among Sampling Acceleration Strategies
Beyond the momentum-based approaches discussed in the main text, the field of accelerating gen- erative samplers encompasses several distinct strategies. A notable alternative paradigm involves learning an entirely new, often near-optimal, sampling process, rather than modifying an existing one. Methods grounded in Stochastic Optimal Control (SOC), for instance [4] [2]. exemplify this approach. They aim learn novel drift function that transports base distribution to the target distribution, a process that typically requires a full training or fine-tuning phase to optimize specific objective. In contrast, PIDLD designed not to replace the sampling process, but enhance as a training-free inference- time accelerator. Instead of learning a new policy, PIDLD applies principles from classical control theory to improve the numerical stability and convergence speed of the existing Langevin dynamics update rule. This positions PIDLD applicable tool for boosting the efficiency any pre-trained model reliant on Langevin sampling, complementing approaches that require dedicated training.
general-purpose and readily
21

---

B Proof of Proposition 1
Lemma (Convergence of Covariance Series), Let RRxx with spectral radius p(J) < and Q = Then the series S Lk=o QJK( converges.
Proof. (Proof of Lemma 1 Fix an induced matrix norm. By Gelfand's formula, for any 8 0, there
exists C' > 0 such that | < C(p(J) 5) for all k 2 0. Choose = (1 p( ensuring
p(J) + 8 Applying submultiplicativtty:
| & | & (²I1Q)(p())) + 8)2k.
The series converges absolutely by comparison with the geometric series Ek=u(p()) 8)2h.
Proof. (Proof of Proposition !
Deterministic Stability: Define the state vector Uz
Ft
x=* x
Linearizing around x², let de U2
Using (U(x) -m(x x²), the dynamics become:
1 (1 ka)m ckam
8141 Jôi,
The characteristic equation det(AA- - stability criteria:
-J) = X² -J)=
X(1 c(1 + km)m) + ckam 0. Applying Jury's
1. [kkam] <
2. 1 -c(1 ka)m ckam >> cm 0
2 3. 1 + e(1 ka)m + ckam >> 0 2 e(1 + 2ka)m > >0 )
The above leads to c (1+ mm
Stochastic Case: The linearized system becomes:
22 0
2el 0 ~ ~((00) Q=
6.+1 = J6, + wt.
0]
The stationary distribution covariance can be computed: E = E with p(J) < (guaranteed by < 2/((1 + m))) the series converges. Hence & converges to N(0,, Ex))
QU By Lemma
Note that the m-strong convexity a standard assumption for local stability analysis and does not presume the global energy landscape is convex. The assumption in Proposition 1 is intentionally narrow and serves a specific purpose: to provide a controlled, tractable setting to formally analyze the local behavior of our sampler. Its goal is to prove that our novel derivative (D) term provides the intended stabilizing effect near a potential minimum, without disrupting the fundamental convergence properties of Langevin dynamics [1] 29) It provides theoretical justification for the stability enhancements we observe empirically.
C Detailed Experiment Settings
C.1 Toy Experiment Settings
Our toy experiment generally follows the setting of NCSN B3
We choose Piata
[W((-5, 5)) (5, 5), 5), /). The score network a 3-layer MLP with 128 hidden units and softplus activation functions. We train the score network with anneal denoising score matching for 60 epochs on 100000 training samples with a batch size of 128. We use Adam optimizer with
learning rate of 0.0001, 31 0.9 and B₂ 0.999.
22

---

For the sampling process, we choose 1280 initial samples uniformly in the square |- - x We use annealed Langevin dynamics where 8, T 150 and 8 x 10 We choose {o,) to be geometric progression, with 0i = 20 and os = 0.01. We use the learned score function for sampling.
C.2 Detailed Experiment Implementation of Image Generation
Table 4: Hyperparameter settings of PIDLD and ALD in image generation.
SGM
EBM
Dataset
PIDLD
ALD
PIDLD
ALD
NFEs
kp
k
ka
kip
NFEs
kpp
kd
kp
232x5
0.90
0.00
1.00
0.90
40
0.50
0.80
1.50
0.70
232x3
1.50
0.00
1.20
1.00
0.80
0.20
2.00
1.20
CIFARIO
232x1
2.00
0.00
3.00
1.75
20
1.20
0.50
2.50
2.00
100x1
2.00
0.50
4.50
2.50
10
1.50
1.00
3.50
3.00
25x1
3.75
0.25
4.00
6.00
500x5
1.00
0.00
3.00
0.90
30
0.60
0.50
2.50
0.80
500x3
1.25
0.00
3.00
1.25
25
0.90
0.30
3.50
1.50
CelebA
500x1
2.00
0.00
3.00
2.00
20
1.50
0.80
4.00
2.50
250x1
2.00
0.50
3.00
2.50
15
1.80
1.20
5.00
3.50
50x1
3.75
1.00
3.75
5.50
For SGM, use the pretrained checkpoint provided by the official repository of NCSNv2 33 to initialize the weights of the score network. We also apply the exponential moving average trick to the weights. The score network uses the RefineNet [2I] architecture. For details please refer to the original paper 34 For EBM, we use the pretrained checkpoint provided by the official repository of IGEBM 8 for CIFAR1O 10 dataset, and retrain the model for the CelebA dataset.
For the sampling process, we follow the setting of NCSNv2 to use 0i 50.0, o7 0.01, 232. 90.0,
5, = 6.2 x 10-6 for CIFAR10 10 and for CelebA. We also follow NCSNv2 to add an additional denoising step to the last sample, given by demmiddeenoiee Ffinal of so (TTmmal o.). For IGEBM, the sampling step corresponds to the NFE. We decay K; at each step. For the step size, we use the default 100 for CIFARIO and e = 400 for CelebA. We keep fixed across all experiments.
0.01, J, = 500, 7 7= 5, 3.3 x 10-6
We generate 10000 images in each run and FID evaluate the generation quality. We input the raw generated images to the inception network without clipping the pixel values to [0, 1].
Since conventional ALD is the degenerate case of PIDLD where k; hu O, tune only kp to get the best performance of vamilla ALD and tune kp. kü, ka to optimize PIDLD. We give our hyperparameter settings in Table
For convenience parameter tuning, we tune only kp and ky when the number of sampling steps is high, because the derivative term is the dominant factor in anneal Langevin dynamics sampling, as mentioned in the ablation study. We observe that when NFEs decrease, the optimal kp. kú, ka show an increasing trend. This is intuitive since fewer steps necessitate more updates per step.
C.3 Detailed Experiment Implementation of Reasoning Tasks
We apply the original repository of IRED to implement PIDLD. We use the pretrained checkpoint provided. For the Sudoku task, the energy network uses ResNet as the backbone. For connectivity task, the energy networks uses Graph Neural Network (GNN) as the backbone. Please refer to the original paper for model details.
For both tasks, we sample 1000 labeled items for evaluation in the harder dataset, following the original setting of IRED. We give our hyperparameter settings in Table
23

---

Table Hyperparameter settings of PIDLD and IRED sampling reasoning tasks.
Dataset
Method
PIDLD
IRED
NFEs
kp
ka
Kp
1.20
4.00
0.50
1.50
Sudoku
10
1.00
4.00
0.50
1.50
15
0.90
4.00
0.50
1.00
30
0.80
4.00
0.50
1.00
40
0.80
3,00
0.50
0.80
80
0.50
3.00
0.50
0.80
NFEs
kp
Ka
kp
5
1.10
0.40
0.01
1.50
Connectivity
10
1.00
0.40
0.01
1.50
15
0.90
0.40
0.01
1.50
30
0.80
0.30
0.01
1.00
40
0.80
0.30
0.01
1.00
80
0.80
0.20
0.01
1.00
D Additional Experiments on the Effect of Integral and Derivative Terms
D.1 Effect Validation on Integral Term
To validate the effect of integral term on mitigating bias, we provide more empirical evidence by running toy experiments on two-dimensions points (see Appendix C.1 for experiment settings) with only P term, with P. terms, and with P. terms. To simulate bias gradient estimation, we add constant perturbation to the score that points to the direction of 1-1,1) and is scaled by 1/20 of the Frobenius norm of the gradient. We use d, the Fuclidean distance of the final cluster center to the true center, the metric of bias. Here the subscript 1 and 2 corresponds to the cluster near (5,5) 5) and (-5,-5) respectively. The results are shown in Table and Table different random seeds under each hyperparameter setting. The number before and after + represents mean and standard deviation, respectively.
run 100 experiments with
Table 6: Effect of k; on bias
Table 7: Effect of ky on bias
K;
dj
d2
hi
d1
d2
0.00
0.0463 + 0.0237
0.2729 + 0.0840
0.0
0.0463 = 0.0237
0.2729 + 0.0840
0.05
0.0414 0.0235
0.2711
0.0816
2.0
0.0445 = 0.0223
0.2904
0.0882
0.10
0.0379 = 0.0215
0.2559 + 0.0802
4.0
0.0448 = 0.0226
0.2971 + 0.0909
0.15
0.0339
0.0190
0.2485 0.0781
6.0
0.0453 = 0.0218
0.3090 0.0914
0.20
0.0326 + 0.0192
0.2436 + 0.0756
8.0
0.0456 = 0.0213
0.3250 + 0.0871
0.25
0.0317
0.0176
0.2356
+ 0.0742
10.0
0.0455 + 0.0214
0.3387 + 0.0914
0.30
0.0315 + 0.0184
0.2307 + 0.0714
12.0
0.0466 + 0.0204
0.3469 + 0.0873
0.35
0.0308 0.0169
0.2330 0.0708
14.0
0.0522 + 0.0230
0.3431 0.0911
can be observed that increasing k; helps to decrease d1 and d2, the final bias of cluster centers, while increasing ka won't yield the same effect. This further demonstrates that the integral term makes the sampling algorithm robust to gradient bias.
D.2 Effect Validation on Derivative Term
To validate the effect of derivative term reducing oscillation, provide more empirical evidence by rumning toy experiments on two-dimensional points (see Appendix C.I experiment settings) with only P term, with P, terms, and with P, D terms.
is hard to measure the instability the trajectory single point, because the Langevin dynamics itself adds noise at each step, and the trajectory is intrinsically stochastic. But the randomness can be

---

reduced by choosing a group of points, Since the generated data roughly follows GMM distribution, we use the centers estimated by Gaussian mixture model as representative objects, and study the trajectories of the two centers.
We use 8 noise levels and 150 steps for each level (I. 8, T 150). We record the positions of the two centers every 5 steps, so there are 240 recorded time steps in total. In the earlier stage the two clusters does not separate and the mean estimation is not reliable, so we use the last 200 steps.
To quantify the amplitude oscillation the final position, daum Ei=41 v(x; Tíima)2 + (yi yimmi) ) as the major indicator. We also measure of peak overshoot. Borrowing from control theory, we also add the settling time metric, which is defined as the time index where (})
trajectory, we use the averaged Euclidean distance to
report the maximum distance, dmnax which be
llp!" Pinaa || of all subsequent time steps less than 0.1. Here pt is defined the position (5)
of cluster center at time index i, and Piinal is defined as the final position of the cluster center. The superscript j 1,2 indicates the center near (5, 5) and (-5, -5), respectively.
We fix h = 1.5 to boost oscillation, and change k, and kg. We run 100 experiments with different random seeds under each hyperparameter setting. The number before and after + represents mean and standard deviation, respectively. The results are shown in the following tables.
Table 8: Metrics for Cluster with varying ka (1) Seilics
Table 9: Metrics for Cluster 2 with varying ka
ka
ka
(2) enlla
12.43
7.81
3.11
0.72
2.0
31.02
+ 2.67
1.35
0.13
105.26 + 12.30
2.0
48.74 - 7.20
2.17
+ 1.11
144.30 - 12.67
4.0
28.76
1: 2.67
1.25
1 0.52
105.10
12.76
4.0
39.42
1: 5.48
1.63
1: 1.53
144.13
1: 11.27
6.0
27.00
2.41
1.11 0.10
104.63
12.70
6.0
35.75
+: 4.65
1.30
1: 0.28
143.29
+ 13.69
8.0
25.79
2.37
1.08
0.11
103.35 + 12.86
8.0
33.22
- 4.12
1.25
4: 0.22
142.09
= 13.38
26.40
13.67
10.0
1.66
- 0.35
12.0
34.54
4.56
3.69
2.01
102.44 1: 12.97
12.0
70.52
12.14
8.45
1 4.46
139.96
12.60
Table 10: Metrics for Cluster
with varying k;
Table 11: Metrics for Cluster 2 with varying k;
ki
d(1)
(1)
(1) Seiliis
l;
d(2)
d(2)
(2) Scilics
0.00
12.43
0.00
69.30
3.11
0.72
0.05
34.77
+ 2.92
1.60
+ 0.17
106.67
07 + 12.42
0.05
70.95 1 8.
3.09
+ 0.67
146.24 + 12.57
0.10
34.65
2.92
1.60
- 0.18
106.73
12.61
0.10
72.61
8.41
3.15
= 0.75
147.85
11.44
0.15
34.46
2.88
1.60
= 0.18
105.96
12.80
0.15
74.00
8.39
3.19
= 0.73
147.20 + 11.00
0.20
34.35 + 2.82
1.59
- 0.17
106.18
12.10
0.20
75.67
+ 7.84
3.20
- 0.65
148.30
+ 11.94
0.25
34.21
1.62
12.61
0.25
77.05
3.30
1.04
0.30
34.09
2.80
1.58
1 0.18
106.26 26 1 12.33
0.30
78.40
8.38
3.30
+ 0.76
148.50
: 11.69
0.35
34.05
1.63
12.86
0.35
79.88
3.39
0.40
33.88
2.74
1.58
0.18
106.
12.91
0.40
80.77
8.16
3.34
- 0.75
147.77 = 10.30
can be seen that for each cluster, the metrics decrease as ka increases, while they do not show comparable decreasing trend increase k; The results further validate our claim that compared to the integral term, the derivative term has the advantage on reducing the oscillation in the sampling process.
D.3 Effect Validation on Integral Term Decay
We use SGM image generation task to further validate the effect of decaying integral term coefficient. For CIFARIO, 10, we generate images with 100 noise levels and step per level (NFE=100 x 1). We use k, 2.0, hi = 4.5, and change h; and For CelebA, we generate images with 50 noise levels and 1 step per level (NFF=50 x 1). use k, = 3.75, ka 3.75, and change k; and 7. The results are shown in Table [2]and and Table
25

---

Table 12: CIFARIO FID of 100
1 sampling steps (k, 2.0, Ry
4.5). Bold: best column.
Y
A;
0.250
0.375
0.500
0.625
0.750
0.875
1 12.29
12.28
12.32
12.41
12.53
12.69
0.99
12.31
12.22
12.21
12.20
12.24
12.27
0.98
12.31
12.28
12.23
12.20
12.16
12.15
0.97
12.36
12.28
12.25
12.24
12.20
12.18
0.96
12.41
12.33
12.28
12.25
12.23
12.22
Table 13: CelebA FID of 50
sampling steps (k, 3.75, ha 3.75).
Bold: best in column.
Y %/
0.750
0.875
1.000
1.125
1.250
1.375
1 8.55
8.08
7.97
8.47
9.02
9.56
0.99
10.67
8.01
7.53
7.76
8.13
8.55
0.98
16.52
11.24
8.59
7.71
7.65
7.86
0.97
23.17
16.87
12.28
9.53
8.21
7.75
0.96
29.20
23.14
17.84
13.71
10.86
9.14
find that:
1. The best performance of tuning both k; and is hetter than the best performance of tuning only k;
2. The higher the k;j is, the higher decay (lower 7) should be applied to balance the effect of larger integral gain, which aligns with common intuition.
The result is a strong evidence that the decay in integral term does improve sampling performance.
E
Computation Cost Comparison
We show that our approach to measure computation cost by NFE is reasonable by conducting experiments compare the physical time consumption. For (non-degenerate) PIDLD, original code. For vanilla ALD, we modify our code implementation by deleting PID related terms in the sample updating function, which ensures additional computation cost by PIDLD introduced. We run both algorithms on NVIDIA A800-SXMA 8000-XXM4400GB The results are shown in Table [1]]
the
Table 14: Physical time comparison of vanilla ALD and PIDLD.
Model
Dataset
NFE
Time Consumed
PIDLD
CIFAR10
100
7min54s
Vanilla ALD
CIFARIO
100 X
7min43s
PIDLD
CelebA
50 x
13min07s
Vanilla ALD
CelebA
50 x
12min52s
It can be seen that there is no significant difference in the time cost between PIDLD and vanilla ALD. For other numbers of NFE, the computation time changes proportionally with NFE, and the time consumption of vanilla ALD and PIDLD is still close. The result is actually intuitive, since the main computation cost is on the forward propagation of the score network, which involves a large amount of matrix multiplications. Adding PID terms just brings some matrix addition operations, so there wouldn't much increase in the computation time, and thus NFE is an accurate measure of real computing time.
26

---

Inception Score Statisties
For CV task, apart from FID, we also report inception score [30 statistics score based generative models on CIFARIO dataset in Table 15
Table 15: Inception score statistics of SGM on CIFARIO dataset.
NFE
ALD
PIDLD
232
8.45 0.18
8.43 + 0.11
232 x 3
8.58 + 0.21
232
7.92
8.46
100 x
7.77
8.26 4.26
25
7.07
7.65 + 0.21
In terms of inception score, we find that PIDLD still generally outperforms ALD under different NFBs, which serves another evidence of its advantage.
As for CelebA dataset, the inception score metric not applicable, because the inception model is trained ImageNet dataset which consists of natural objects, while CelebA is human face dataset. The distribution shift is large, so it is not reasonable to apply ImageNet inception feature extractor. actually a common practice to omit inception score for CelebA.
G Additional Examples of Generated Images
We provide more image samples generated by SGM using PIDLD as in Fig 8 and Fig. 9
PIDLD Samples, FID-11.4
PIDLD Samples, FID FID-11.6 11.6
CIFARIO samples with 232x5 sampling steps.
(b) CIFARIO samples with 232x1 sampling steps.
Figure 8: Visualization of CIFAR10 images generated by SGM with PIDLD.
27

---

PIDLD Samples, FID-5.6
PIDLD Samples, FID-5.7
(a) CelebA samples with 500x5 sampling steps.
(b) CelebA samples with 500x1 sampling steps.
Figure Visualization of CelebA images generated by SGM with PIDLD.
H Limitations and Future Work
PIDLD presents promising approach accelerate Langevin- based sampling, and we identify several avenues for future development. Firstly, while our parameter analysis offers guidance, exploring adaptive or automated strategies for tuning the PID coefficients (kpp ki ka, ?) could further enhance its plug-and-play capability across diverse models and datasets. Secondly, building upon initial stability analysis (Proposition []. deeper theoretical investigation into the convergence guarantees and acceleration properties of the full PID controller, especially in the context of non- convex landscapes characteristic of EBMs [8] and SGMs (33), would be valuable. Lastly, our proposed method may be beneficial in extending the PID control principles to more general ODE and SDE solvers, which are prevalent in recent coore-based models [35]. This represents an exciting direction to broaden the applicability of PIDLD and potentially unlock further performance gains.
28


Reviews:
[{"review_id": "wxZCd6lJHk", "reviewer": "Reviewer_U2PP", "summary": "This paper tackles the problem of improving inference-time sampling from energy-based models (EBMs) and score-based generative models (SGMs) using Langevin Dynamics. To address this problem, the authors propose a novel control-based method, Proportional-Integral-Derivative controlled Langevin Dynamics (PIDLD). Their proposed approach uses the energy gradient $-U_\\theta(x)$, standard in Langevin Dynamics, as a feedback signal for the PID controller. Through empirical experiments on a variety of image generation tasks and reasoning tasks, the authors show the effectiveness of their proposed method relative to vanilla sampling approaches with Langevin Dynamics.", "questions": "- How does this work relate to the recent works (references listed above) in controllable generation? \n- Proposition 1 assumes that $-U_\\theta(x)$ is \"_m-strongly convex_\". Is this a reasonable assumption given, to my understanding, most sampling applications of interest deal with highly non-convex $-U_\\theta(x)$?\n- An active area of research in sampling is application to molecules and proteins. How do you envision the performance of PIDLD on these domain-specific applications? It would be interesting to see an experiment in one of these areas (e.g. Tan et al, 2025). I also believe an additional experiment would strengthen the contributions of this work. \n\nReferences:\n\n1. Jiralerspong, et al. \"Feature likelihood divergence: evaluating the generalization of generative models using samples.\" NeurIPS (2023).\n2. Domingo-Enrich, et al. \"Adjoint matching: Fine-tuning flow and diffusion generative models with memoryless stochastic optimal control.\" (2024).\n3. Havens, et al. \"Adjoint sampling: Highly scalable diffusion samplers via adjoint matching.\" (2025).\n4. Tan, et al. \"Scalable equilibrium sampling with sequential boltzmann generators.\" (2025).", "limitations": "Yes, but they are discussed at the end of the appendix. I recommend that the authors move this to the end of the main text (or to the conclusion).", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 2, "originality": 2, "strengths_and_weaknesses": "**Strengths:**\n\n- This work presents an interesting and novel idea of using a Proportional-Integral-Derivative (PID) controller for sampling with Langevin dynamics to improve inference-time sampling from energy-based models and score-based generative models. \n- The proposed method, PIDLD, is evaluated across several empirical experiments, on both image generation tasks and reasoning tasks.\n- The paper is generally well written, and relevant concepts are clearly explained.\n\n**Weaknesses:**\n\n- The central claim that PIDLD improves sampling speed is not sufficiently supported. Specifically, on lines 61-62, the authors state, \"PIDLD achieves at least 10x sampling speedup over baselines under SGM model\". However, no experiment explicitly reports these results/numbers. I see the authors report NFEs; however, I don't believe this is sufficient to support this claim. To support this claim, I recommend including an experiment that reports evaluation time throughout the sampling process, alongside a performance metric. This could be done for both image generation tasks as well as the reasoning tasks.  \n- For image generation tasks, only one metric (FID) is considered. This provides only a narrow view of the performance/comparison of PIDLD. Other relevant metrics to consider are Feature likelihood divergence (FLD) (Jiralerspong et al, 2023) and Inception Score. \n- Some relevant works in the realm of controllable generation/sampling, which I believe are related to PIDLD, are missed in the discussion of related works. For example, see (Domingo-Enrich et al, 2024, Havens et al, 2025).\n- With this, although to my understanding the idea of using a PID controller to improve sampling of Langevin dynamics, the notion of \"controllable\" sampling with Langevin dynamics is not new. I encourage the authors to include a more comprehensive and explicit related works section discussing the state of this area of research and literature.", "was_revised": true, "final_justification": "In general, I think this is a reasonably solid paper that can contribute to the conference. The authors did a good job of addressing all my concerns and answering my questions. With that, I have raised my score. However, I maintain my position that this is a borderline paper. I believe an additional experiment in an application domain (as per my suggestion --- the proposed method would be well-suited for problems in molecule generation) would be very beneficial to this work, but I understand that this is not realizable in the time constraints of rebuttals. Hence, I did not consider this in my final evaluation.", "rebuttal": "We sincerely thank the reviewer for the detailed and constructive feedback. We are especially grateful for your recognition of the novelty of our idea, the solidity of our experiments, and the clarity of our thesis writing. We summarize your questions and our replies are as follows.\n\n\n### Weakness 1\n\nThank you for pointing out the potential logic loophole. We show that our measure of computation cost is reasonable as follows.\n\nFirst, we show that our approach to measure computation cost by NFE is reasonable. We have conducted experiments to compare the physical time consumption. For (non-degenerate) PIDLD, we used the original code. For vanilla ALD, we modified our code implementation by deleting all PID related terms in the updating function, which ensures no additional computation cost by PIDLD is introduced. We ran both algorithms on NVIDIA A800-SXM4-40GB. The results are as follows.\n\n| Model       | Dataset | NFE          | Time Consumed |\n| ----------- | ------- | ------------ | ------------- |\n| PIDLD       | CIFAR10 | $100\\times1$ | 7min54s       |\n| Vanilla ALD | CIFAR10 | $100\\times1$ | 7min43s       |\n| PIDLD       | CelebA  | $50 \\times1$ | 13min07s      |\n| Vanilla ALD | CelebA  | $50 \\times1$ | 12min52s      |\n\nIt can be seen that the there isn't much difference in the time cost between PIDLD and vanilla ALD. For other numbers of NFE, the computation time changes proportionally and is still close. The result is actually intuitive, since the main computation cost is on the forward propagation of the score network, which involves a large amount of matrix multiplications. As for adding PID terms, it just brings some matrix addition operations, so there wouldn't be much increase in the computation time.\n\nMoreover, we show that our claim of 10x speedup is reasonable. As can be seen in Table 1 of our paper, for CIFAR10, the vanilla ALD uses $232\\times5$ steps (NCSNv2 setting) to achieve FID=12.5, while PIDLD only uses $100\\times1$ steps to achieve FID=12.1; for CelebA, PIDLD with only $50\\times1$ steps can outperform vanilla ALD with $500\\times5$ steps (NCSNv2 setting). The results mean that to achieve the same quality, PIDLD uses less than 1/10 of the time of vanilla ALD. We are not the only one to claim acceleration times in this way. In fact, the section 5.1 of DDIM paper [Jiaming Song et al., https://arxiv.org/pdf/2010.02502] also adopts the same method. We believe it's a common practice in this domain.\n\n\n\n### Weakness 2\n\nThank you for raising the concern of the lack of performance metric. We additionally report FLD and Inception score as follows.\n\n#### **FLD**\n\nFor CIFAR10 dataset, we follow the official implementation of FLD and have the following results.\n\n(FLD: lower is better, FLD gap: a measure of uncertainty, lower is better)\n\n| CIFAR10 FLD    |  ALD    |  PIDLD    |\n|----------------|---------|-----------|\n| $232\\times5$   | 3.47    | 3.04      |\n| $232\\times3$   | 3.51    | 2.87      |\n| $232\\times1$   | 3.35    | 3.30      |\n| $100\\times1$   | 3.57    | 3.54      |\n| $25 \\times1$   | 7.69    | 3.73      |\n\n| CIFAR10 FLD gap |       ALD   |   PIDLD       |\n|-----------------|-------------|---------------|\n| $232\\times5$    | 0.23        | 0.26          |\n| $232\\times3$    | 0.28        | 0.23          |\n| $232\\times1$    | 0.23        | 0.17          |\n| $100\\times1$    | 0.22        | 0.20          |\n| $25 \\times1$    | 0.07        | 0.24          |\n\n\nFor CelebA dataset, we use 100000 images as the train set, 2600 images as the test set, and calculate FLD on 10000 generated images. The results are as follows.\n\n| CelebA FLD    |   ALD   |  PIDLD    |\n|---------------|---------|-----------|\n| $500\\times5$  | 1.14    | 0.75      |\n| $500\\times3$  | 1.18    | 1.09      |\n| $500\\times1$  | 1.61    | 0.77      |\n| $250\\times1$  | 1.46    | 1.46      |\n| $50 \\times1$  | 3.24    | 1.95      |\n\n\n| CelebA FLD gap |    ALD      |   PIDLD       |\n|----------------|-------------|---------------|\n| $500\\times5$   | 0.31        | 0.39          |\n| $500\\times3$   | 0.41        | 0.41          |\n| $500\\times1$   | 0.36        | 0.56          |\n| $250\\times1$   | 0.25        | 0.33          |\n| $50 \\times1$   | 0.32        | 0.49          |\n\n\nWithin our implementation, we observe that PIDLD still outperforms ALD in terms of FLD metric, which strengthens our work.\n\n\n#### **Inception Score**\n\nWe did record inception score in our experiments on CIFAR10 dataset. The results are given in the following table:\n\n|     NFE      |   ALD         | PIDLD         |\n|--------------|---------------|---------------|\n| $232\\times5$ | $8.45\\pm0.18$ | $8.43\\pm0.11$ |\n| $232\\times3$ | $8.47\\pm0.25$ | $8.58\\pm0.21$ |\n| $232\\times1$ | $7.92\\pm0.22$ | $8.46\\pm0.22$ |\n| $100\\times1$ | $7.77\\pm0.25$ | $8.26\\pm0.26$ |\n| $25\\times1$  | $7.07\\pm0.24$ | $7.65\\pm0.21$ |\n\nJudging from the inception score, PIDLD still generally outperforms ALD under different NFEs, which serves as another evidence of its advantage.\n\n\nAs for CelebA dataset, the inception score metric is not applicable, because the inception model is trained on ImageNet dataset which consists of natural objects, while CelebA is a human face dataset. The distribution shift is large so it's not reasonable to apply ImageNet inception feature extractor. It is actually a common practice to omit inception score for CelebA. For instance, the Table 1 of NCSNv2 paper [Yang Song et al., https://proceedings.neurips.cc/paper_files/paper/2020/file/92c3b916311a5517d9290576e3ea37ad-Paper.pdf] also uses this treatment.\n\n\nThe reason we only report FID in the original paper is that we found most relevant works, like DDPM, DDIM, NCSN, NCSNv2, and MILD, use FID as the primary metric. We have complemented other metrics and we hope it will strengthen our experiment results.\n\n\n\n### Question 1 & Weakness 3 & Weakness 4\n\nWe thank the reviewer for pointing to these valuable works. We will add a detailed discussion to our Related Works section to clarify the positioning of our method.\n\n**PIDLD is a training-free inference-time accelerator for existing samplers, fundamentally differing from SOC-based methods which learn a new, optimal sampling policy through a training/fine-tuning process.**\n\n1.  **PIDLD's Contribution:** Our method is inspired by control theory to improve the numerical stability and speed of the Langevin update step. It operates as a \"plug-and-play\" module that modifies the sampling dynamics using gradient history and trends, without requiring any retraining or alteration of the target score/energy function. \n\n2.  **Contrast with SOC Methods:** In contrast, the cited works (Havens et al., 2025; Domingo-Enrich et al., 2024) are grounded in stochastic optimal control (SOC). Their goal is not to accelerate a given sampler, but to learn an optimal drift function u(x, t) that minimizes a variational objective to transport a base distribution to a target one. This process involves a full training or fine-tuning phase to learn the parameters of this new sampling policy.\n\n\n\n### Question 2\n\nWe thank the reviewer for this question, as it allows us to clarify the precise role of our theoretical analysis.\n**The m-strong convexity is a standard assumption for local stability analysis and does not presume the global energy landscape is convex.** \n\nWe completely agree that the energy landscapes in our applications are globally non-convex. The assumption in Proposition 1 is intentionally narrow and serves a specific purpose: to provide a controlled, tractable setting to formally analyze the local behavior of our sampler. Its goal is to prove that our novel derivative (D) term provides the intended stabilizing effect near a potential minimum, without disrupting the fundamental convergence properties of Langevin dynamics (paper cites:[10, 25]). It provides a theoretical justification for the stability enhancements we observe empirically.\n\n\n### Question 3\n\n**We believe PIDLD is exceptionally well-suited for molecular sampling, and we thank the reviewer for this excellent suggestion.** The challenges in that domain align directly with the strengths of our proposed controller.\n\n*   **Envisioned Performance:** The energy landscapes of molecules and proteins are characterized by numerous local minima (stable conformations) separated by high energy barriers. PIDLD's integral (I) term creates a momentum-like effect, which is ideal for efficiently traversing these energy barriers and exploring diverse conformational states. Simultaneously, the derivative (D) term provides adaptive damping, which would help the sampler quickly stabilize within newly discovered low-energy wells without excessive oscillation. This combination should lead to significantly more efficient exploration of the conformational space compared to standard Langevin-based methods.\n\n*   **Commit as future work:** We agree that this is a highly compelling application and a valuable direction for future research. We will discuss more about domain-specific applications in our future work, and will cite the related works mentioned by the reviewers."}, {"review_id": "kczw3a6r82", "reviewer": "Reviewer_Kawu", "summary": "The authors propose to augment standard Langevin dynamics (LD), in the context of sampling from diffusion and energy-based models (EBMs), with PID control.  That is, they interpret the discrete-time approximation to LD as a dynamical system that is driven by the energy gradient (i.e. the \"score\" or force), interpreted as an error signal that needs to be driven toward zero.  Accordingly, they replace it with the classic PID scheme, i.e. a linear combination of the (discrete) derivative and integral of the error, as well as the error itself.  They show that this scheme (slightly modified to decay the integral term) outperforms standard Langevin dynamics in FID on CIFAR10 and CelebA at various choices for the total number of neural function evaluations (NFEs), both in score-based model and in EBMs.  Similar results hold for sampling-based reasoning in EBMs.  The authors also compare against a similar, recent proposal (MILD) from another group.", "questions": "Can the authors addresses the five major weaknesses listed above?", "limitations": "yes", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "The idea appears to be novel (this reviewer is surprised that it has never been tried before), the basic experiments appear sound, and the topic---reducing the required number of samples for LD---is important.  There are also weaknesses:\n\n1. The evaluations are on simple datasets (CIFAR-10, CelebA); it would be nice to see results on (say) ImageNet.  I am also not sure they are totally fair: I believe NCSN typically uses far (about an order of magnitude) more Langevin steps (T) than noise levels (L); whereas in the experiments reported in Table 1, the reverse is the case.  A fairer comparison would seem to be to fix a budget of NFEs, and then allocate them to L and T in the way that yields best performance for the sampling method under consideration.\n\n2. Although it's promising to see that this method can produce faster samples compared to standard Langevin dynamics, there has been significant progress in accelerating sampling for diffusion models using techniques such as deterministic sampling (e.g., DDIM) and ODE-based samplers.  These methods can generate high-quality samples in just a few steps. It would be interesting to see how the proposed approach compares to these alternatives.\n\n3. Along these lines: The manuscript explores the proposed technique in the context of sampling from a pre-trained EBM.  A more compelling application would be to evaluate how well the sampler performs when used for *training* EBMs. This would also provide better insight into the stability and robustness of the hyperparameters.\n\n4. The authors argue that the proposed sampler improves sampling from multi-modal distributions by traversing energy barriers.  To test whether the sampler actually mixes, a good experiment would be to run it beyond 25 steps and observe how the samples evolve: does each sample stay fixed at a particular image, or does it explore the manifold?  If the sampler does mix effectively, one would expect results similar to Figure 3 of the following paper: https://arxiv.org/pdf/2006.06897\n\n5. Is there any reason MILD was omitted from Table 2?  Here I would expect MILD to be more competitive with the PID controller, since apparently (Fig. 6) the integral term is doing most of the work here, and MILD is in a sense a pure integral-controlled system (i.e., the input is low-pass filtered).\n\nMinor:\n1. The authors cited a paper on biomolecular PID controllers when introducing the concept--was this intentional?\n\n2. Presentation: The MS is somewhat repetitive, with the basic rationale for PID controls repeated about once per page for the first four pages.  Also, there are some distracting grammatical mistakes: e.g, I think the title should read, \"...sampling *or* [or *from*] generative models\"; \"related works\" -> \"related work\" (\"work\" is a mass noun and shouldn't be pluralized); etc.", "was_revised": true, "final_justification": "The authors have addressed several of the questions and weaknesses I raised.  Their additional experiments in response to the suggestions of Reviewer K1tr have improved the conceptual appeal of the manuscript, by bringing their results into closer contact with control theory.\n\nThe remaining weaknesses (in this reviewer's opinion) are (1) the most useful applications of the idea are not explored here; (2) the datasets are simple; and (3) connections are not made to other second-order methods of MCMC (like HMC).", "rebuttal": "We sincerely thank the reviewer for their constructive feedback and for acknowledging our paper's novelty, experiment solidity, and significance. We address the raised weaknesses as follows.\n\n### Weakness 1\n\n**We appreciate the suggestion that our experiments should extend to more datasets.** In the future, we will add experiments to strengthen our results. However, we notice that most relevant works like NCSN, DDPM, DDIM, and MILD also do not report experiment results on ImageNet. We were following them to use only CIFAR10 and CelebA dataset.\n\nAs for the concern of $L$ and $T$, it is true that the original NCSN paper uses more Langevin steps ($T$) that noise levels ($L$). **However, in its later version, NCSNv2 [Yang Song et al.], the authors made theoretical analysis on the choice of T and L, which suggests using $232\\times5$ steps for CIFAR10 generation and $500\\times5$ steps for CelebA generation.** We are actually following their best settings on NCSN. We also follow the NFE setting of MILD [Shetty et al., https://ieeexplore.ieee.org/abstract/document/10446376] so that our method is comparable to the experiments in that work.\n\nMoreover, the choice of $T$ and $L$ actually doesn't influence the conclusion in our work. Our major contribution is the improvement on anneal Langevin dynamics sampling. What we want to show is that PIDLD outperforms vanilla ALD and MILD. We have shown that, when other influencing factors (including $L$ and $T$) are the same, PIDLD generates better samples than ALD and MILD. So the advantage of PIDLD holds true.\n\n### Weakness 2\n\nThank you for raising the concern of comparing with other efficient sampling methods. **The reason why we don't compare with them comes from the fundamental differences in model design and scope of application.**\n\n**Our main contribution is improving the Langevin dynamics generation while DDIM/ode-based methods are concerned with accelerating diffusion models. The two frameworks have fundamental differences and cannot be compared directly.** For instance, diffusion models explicitly \"corrupt\" data through many small noise steps and learn to reverse this process, whereas Langevin methods directly simulate a stochastic path that climbs toward high-density regions in one unified sampling loop. \n\nMoreover, we find that the noise estimation model in DDIM has 78700803 parameters, while our score model only has 29694083 parameters. The difference in model size may also result in the difference in generation quality. Even though, we find that PIDLD's performance on CelebA with low NFE (50 steps, FID=7.97) is competitive with that of DDIM (50 steps, FID=9.17), which shows the potential of our approach.\n\nTo conclude, there are many other influencing factors on model design, so we only considered the most relevant and comparable works. We believe our idea on improving Langevin dynamics sampling could be further extended to other classes of score-based SDE/ODE generative models, including state-of-the-art ones. We expect to address these in our future work.\n\n\n### Weakness 3\n\n**We agree that evaluating PIDLD for EBM training is an important future direction, though our current work focuses on the distinct and significant challenge of accelerating inference from pre-trained models.** Our primary goal is to introduce and validate a novel, training-free sampler that can be readily applied to any existing pre-trained Langevin-based model to speed up generation.\n\nIntegrating a new sampler into the EBM training loop introduces substantial additional complexities. The training process itself is a delicate balance, relying on MCMC sampling to approximate the gradients of the log-likelihood. **Introducing PIDLD here would require a separate, thorough investigation into the coupled dynamics of sampling and model optimization to ensure stability, which is beyond the scope of this initial work.** We believe that demonstrating PIDLD's efficacy and stability for inference first is a crucial and necessary prerequisite. We thank the reviewer for this insightful suggestion and will explore it in future work.\n\n\n### Weakness 4\n\nThank you for providing insightful comments on mixing behavior. We follow your instruction to conduct an experiment. **We find that there isn't a phenomenon where an image turns to another one during the sampling.** When an object is intelligible, the updating step length is small (which is the design of anneal Langevin dynamics) so we would not observe a major transition. The mixing in our method happens at the earlier stages in which the outline of the image doesn't even appear.\n\n**Our claim of traversing energy barriers comes from our observation of the final image.** We observe that images generated by vanilla ALD lacks fine-grained texture, while that of PIDLD exhibit richer color fidelity and sharper detail. Even in earlier stages where the images are barely recognizable and still blurred by mosaics, PIDLD samples already show distinguishable details. In contrast, ALD samples does not show comparable quality all the way. So we conclude that PIDLD helps to locate energy minima in the early stage.\n\n\n### Weakness 5\n\nWe thank the reviewer for this excellent suggestion. The omission of MILD from Table 2 was an oversight. **We have conducted the proposed experiment, and the results, which will be added to the revised paper, are below.**\n\n**Table: Accuracy (%) comparison including MILD on harder reasoning tasks.**\n| Task | Method | NFE=1 | NFE=2 | NFE=3 | NFE=5 | NFE=10 |\n| :--- | :--- | :---: | :---: | :---: | :---: | :---: |\n| **Sudoku** | Baseline [6] | 31.45 | 38.12 | 42.03 | 45.99 | 51.00 |\n| | MILD [26] | 36.50 | 44.80 | 48.91 | 49.75 | 54.82 |\n| | **Ours (PIDLD)** | **37.11** | **45.93** | **49.88** | **50.54** | **55.48** |\n| **Connectivity**| Baseline [6] | 86.16 | 87.22 | 87.22 | 87.38 | 87.49 |\n| | MILD [26] | 86.16 | 88.54 | 89.21 | 90.15 | 90.33 |\n| | **Ours (PIDLD)** | **86.16** | **91.32** | **92.31** | **92.95** | **93.28** |\n\n**Analysis:**\nWe agree with the reviewer's observation, supported by our ablation in Figure 6: the integral (I) term, which provides a momentum-like effect, is the primary driver of performance. This explains why MILD also significantly outperforms the baseline.\n\nHowever, PIDLD’s superior performance stems from the crucial **synergy between its integral (I) and derivative (D) components**. The I term provides the exploratory force to escape local minima, while the D term provides essential stabilization. It acts as an adaptive damper, preventing the powerful momentum from overshooting the sharp energy wells that characterize correct solutions in reasoning tasks.\n\nWhile the D term's isolated contribution is modest (P+D in Fig. 6), its role in refining the I term's trajectory is what unlocks the final margin of performance. This synergy of exploration and stabilization explains why our full P+I+D model consistently achieves the highest accuracy, as shown in both Figure 6 and the above table.\n\n### Regarding the minor points\n\nWe appreciate the reviewer's careful reading and valuable suggestions on presentation.\n\n*   **On the biomolecular PID controller citation [8]:** **This was intentional to highlight the universality of PID principles.** We found that reference [8] provides a particularly intuitive explanation. We will clarify this in the text and ensure a standard control theory reference [1] is also prominently cited for context.\n\n*   **On presentation and grammar:** **We will revise the manuscript to address these issues.** We will condense the introductory explanations to improve flow and remove redundancy. We will also correct the title, the term \"related work,\" and other grammatical errors, and will perform a thorough proofread of the entire paper."}, {"review_id": "u2fxo7ELsx", "reviewer": "Reviewer_K1tr", "summary": "The paper tackles the challenge of accelerating Langevin sampling by reframing the problem as one of feedback control. Concretely, the authors observe that in most sampling schemes the instantaneous gradient of the log-density at the current state acts as a natural “error” signal: driving that gradient to zero corresponds to reaching high-probability regions. Building on this insight, they ask whether classical control-theoretic tools might be employed to attenuate this gradient error as quickly as possible.\n\nTheir key contribution is to augment the usual sampling dynamics with a PID controller, treating the gradient as the error variable and tuning proportional, integral, and derivative gains to shape the trajectory. Importantly, this approach does not need any re-training. The authors then compared sampling with a pre-trained model when tuning the different PID terms, and discussed the advantages of the approach.", "questions": "1. In control theory, the integral component is introduced to eliminate steady-state error in the presence of constant disturbances. By analogy, when sampling with noisy gradient estimates (e.g. minibatched or corrupted gradients), the I‐term should compensate for bias and drive the long-term average gradient toward zero. Can the authors perhaps design a simple experiment to validate if these insights from control theory translate in this setting and may be useful? Perhaps by eliminating an estimation bias in the gradient (e.g., due to training). Comparing the different sampling scheme would help elucidating these connections.\n2. The D‐component in PID is classically used to add damping and counteract overshoot or oscillatory behavior around the setpoint (you show this in an example in Figure 1, but I believe the paper could have been stronger by going back to this example in the context of Langevin sampling with experiments). To test this in sampling, one could intentionally mis-tune the integral/proportional gain (e.g. choose a P-gain large enough to induce oscillatory sampling trajectories in a simple 1D double-well potential). Then compare how P-only versus PD controllers traverse between modes, measuring quantities like autocorrelation time or the amplitude of oscillations in $x_t$. The hypothesis is that the D-term will attenuate high-frequency jitter—improving both mixing speed and robustness to P-gain mis-specification. I believe that this type of analyses would greatly improve the contribution of the paper, in explaining and motivating the connections between control theory and Langevin sampling. If these or other insights do not translate to the higher dimensional setup of sampling for images, it is also interesting.\n3. Can you run experiments with stronger baseline models? This would allow to understand if the gains of using your scheme are only due to a poor baseline or if they extend to stronger models as well.", "limitations": "The paper does not address properly its limitations. These are deferred to the appendix and mixed with future work, but reading that paragraph I only appreciate directions for future work but I do not understand the limitations of the method. For instance, I instead identify the following ones by reading the paper:\n\n1. I believe the idea and insight holds great potential, but the authors did not capitalize on the control theory interpretation, or at least I was not able to grasp, reading the paper, how the insights from control actually translate to Langevin sampling. Of course they inspired adding the derivative term to momentum-based schemes, but given the premise of the paper I believe it should go more in depth into the control-theoretical interpretations of the proposed scheme. Similarly, when discussing the I-term I would have expected more emphasis on the control-theoretic interpretation (also mentioned in Figure 1) of constant disturbance rejection, rather than the momentum interpretation (less novel).\n2. Can you run your experiments on top of a state-of-the-art sampler? It looks like your FID scores are much higher than the state-of-the-art. It would be interesting to understand if the gains of your method depend on a week baseline of if they extend (and how) to stronger baselines.\n\nThe authros might want to discuss limitations more thoroughly in the main paper.", "rating": 5, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "Strengths:\n\n1. The paper’s principal contribution is to recast Langevin sampling as a feedback‐control problem, treating the instantaneous gradient of the log-density as an “error” signal to be driven to zero. This novel framing is likely to inspire further cross‐pollination between control theory and stochastic sampling methods.\n2. By augmenting standard sampling dynamics with a PID controller, the authors demonstrate measurable speedups in convergence and improvements in sample quality across their test problems. These numerical results substantiate the high‐level intuition that classical control tools can benefit generative sampling.\n3. The use of a well-understood controller (PID) makes the method easy to implement and tune, and potentially compatible with a wide variety of existing samplers without algorithmic redesign or need for re-training. It is interesting because it is an instance of post-training hyper-parameter tuning.\n\nWeaknesses:\n\n1. Limited exploitation of the insights from control theory. From the experiments, it is not clear to me how the control-theoretic insights about proportional, integral, and derivative terms translate to langevin sampling (see also my questions, and the outlined limitations). For instance: Does the derivative term actually attenuate stochastic oscillations? Does the integral term actually correct for steady-state bias? Or is the control-theoretic motivation only useful to inspire another hyper-parameter to tune? The idea of using PID to drive optimization procedure is not novel, see e.g. https://proceedings.mlr.press/v139/farahmand21a.html. A discussion about these may be needed.\n2. The introduction of a decay factor on the integral term appears ad hoc. From a control standpoint this resembles a leak in an I-controller, but without the connection to control theory it seems a bit out of scope.\n3. The reported FID scores lag behind state-of-the-art generative models. See e.g. https://arxiv.org/abs/2504.10612 and references therein. It’s unclear whether this is due to underlying base model. The authors should clarify their baseline and test whether PID control yields comparable gains when used on top of a SOTA sampler.", "was_revised": true, "final_justification": "The paper applies an interesting idea (somewhat popular in other learning/optimization area), but gives it a spin from a control theoretic perspective. I am very fond of the idea, and I find the discussion with the authors satisfying. After the discussion, I believe the paper should be accepted.", "rebuttal": "We sincerely thank the reviewer for their constructive feedback and for acknowledging our paper's novel control-theoretic framing, the measurable speedups in convergence, and the practical value of its post-training implementation. We address the raised weaknesses and questions as follows.\n\n### Weakness 1 & Question 1\n\nThank you for the suggestion on validating the effect of $I$ term. **We follow your suggestion to run toy experiments on two-dimensional points (see Appendix C.1 for experiment settings) with only $P$ term and with $P$, $I$ terms.** To simulate a bias in gradient estimation, we add a constant perturbation to the score that points to the direction of $(-1,1)$ and is scaled by $1/20$ of the norm of the gradient. We choose $k_p=1.0$ for ALD and $k_p=1.0, k_i=0.15$ for PIDLD.\n\nOur results truly validates the claim that $I$ term helps to eliminate bias. \n- For ALD, the mean estimation of the two generated clusters are $(5.028, 5.278)$ and $(-5.057,-4.702)$, while that of PIDLD is $(4.974, 5.242)$ and $(-5.092, -4.981)$. The Euclidian distances from the true centers ($(5,5)$ and $(-5,-5)$) are $0.280, 0.303$ for ALD and $0.243, 0.094$ for PIDLD. It is apparent that points generated by PIDLD are less influenced by the gradient bias. \n- Moreover, the KL divergence of ALD samples is $0.900$ while that for PIDLD samples is $0.713$, which again validates our claim that the $I$ term does improve generation quality.\n\n\n### Weakness 1 & Question 2\n\nThank you for providing insights on the effect of $D$ term. **We use a toy experiment on two-dimensional points (see Appendix C.1 for experiment settings) to demonstrate the oscillation-reducing effect of $D$ term.**\n\nIt is actually hard to measure the instability of the trajectory of a single point, because the Langevin dynamics itself adds noise at each step, and the trajectory is intrinsically stochastic. But the randomness can be reduced if we choose a set of points. Since the generated data roughly follows a GMM distribution, **we can use the centers estimated by Gaussian mixture model as representative objects. So we study the trajectories of the two centers.**\n\nWe use $8$ noise levels and $150$ steps for each level ($L=8, T=150$). We record the positions of the two centers every $5$ steps, so there are $240$ recorded time steps in total. In the earlier stage the two clusters does not separate and the mean estimation is not reliable, so we use the last $200$ steps.\n\nTo quantify the **amplitude of oscillation** of a trajectory, we use the summed Euclidian distance to the final position, $d_{\\text{sum}}=\\sum_{i=41}^{240}\\sqrt{(x_i-x_{\\text{final}})^2+(y_i-y_{\\text{final}})^2}$, as the major indicator. We also report the maximum distance, $d_{\\text{max}}$, and the $90\\%$ percentile of the distances, $d_{\\text{10\\%}}$. We fix $k_p=1.5, k_i=0$ and change $k_d$.\n\nThe results are given in the following table. Here the superscript $(1)$, $(2)$ indicate the cluster around $(5,5)$ and $(-5,-5)$, respectively.\n\n| $k_d$ | $d_{\\text{sum}}^{(1)}$  | $d_{\\text{max}}^{(1)}$ | $d_{\\text{10\\%}}^{(1)}$ |\n|-------|----------|----------|---------|\n| 0.0   | 31.25    | 1.37     | 0.61    |\n| 2.0   | 27.90    | 1.09     | 0.58    |\n| 4.0   | 24.86    | 1.00     | 0.58    |\n| 6.0   | 22.84    | 0.95     | 0.50    |\n| 8.0   | 21.36    | 0.86     | 0.35    |\n| 10.0  | 21.35    | 0.91     | 0.36    |\n\n\n| $k_d$ | $d_{\\text{sum}}^{(2)}$  | $d_{\\text{max}}^{(2)}$ | $d_{\\text{10\\%}}^{(2)}$ |\n|-------|--------|----------|---------|\n| 0.0   | 64.22  | 2.88     | 0.87    |\n| 2.0   | 47.16  | 2.15     | 0.66    |\n| 4.0   | 38.90  | 1.35     | 0.52    |\n| 6.0   | 35.56  | 1.07     | 0.54    |\n| 8.0   | 31.32  | 1.34     | 0.55    |\n| 10.0  | 32.07  | 1.99     | 0.43    |\n\nIt can be seen that $d_{\\text{sum}}$, $d_{\\text{max}}$, $d_{\\text{10\\%}}$ all show a decreasing trend as $k_d$ increases. We believe this simple experiment could validate our claim that $D$ term reduces oscillation and improves the stability of sampling process.\n\n\n\n### Weakness 2\n\nThank you for raising the concern on integral term decay. This is indeed an empirical finding based on our experiments. We use SGM image generation task to further validate our findings.\n\nFor CIFAR10, we generate images with 100 noise levels and 1 step per level (NFE=$100\\times1$). We use $k_p=2.0$, $k_d=4.5$, and change $k_i$ and $\\gamma$. For CelebA, we generate images with 50 noise levels and 1 step per level (NFE=$50\\times1$). We use $k_p=3.75$, $k_d=3.75$, and change $k_i$ and $\\gamma$. The results are shown in the following tables.\n\n**CIFAR10 100×1 ($k_p=2.0, k_d=4.5$) FID (Bold: best in column, Italic: best in row 1):**\n| $\\gamma$ / $k_i$ | 0.250 | 0.375 | 0.500 | 0.625 | 0.750 | 0.875 |\n|------------------|:-------:|:-------:|:-------:|:-------:|:-------:|:-------:|\n| 1                | **12.29** | *12.28* | 12.32 | 12.41 | 12.53 | 12.69 |\n| 0.99             | 12.31 | **12.22** | **12.21** | **12.2** | 12.24 | 12.27 |\n| 0.98             | 12.31 | 12.28 | 12.23 | **12.20** | **12.16** | **12.15** |\n| 0.97             | 12.36 | 12.28 | 12.25 | 12.24 | 12.20 | 12.18 |\n| 0.96             | 12.41 | 12.33 | 12.28 | 12.25 | 12.23 | 12.22 |\n\n**CelebA 50×1 ($k_p=3.75, k_d=3.75$) FID (Bold: best in column, Italic: best in row 1):**\n| $\\gamma$ / $k_i$ | 0.750 | 0.875 | 1.000 | 1.125 | 1.250 | 1.375 |\n|------------------|:-------:|:-------:|:-------:|:-------:|:-------:|:-------:|\n| 1                | **8.55**  | 8.08  | *7.97*  | 8.47  | 9.02  | 9.56  |\n| 0.99             | 10.67 | **8.01**  | **7.53**  | 7.76  | 8.13  | 8.55  |\n| 0.98             | 16.52 | 11.24 | 8.59  | **7.71**  | **7.65**  | 7.86  |\n| 0.97             | 23.17 | 16.87 | 12.28 | 9.53  | 8.21  | **7.75**  |\n| 0.96             | 29.20 | 23.14 | 17.84 | 13.71 | 10.86 | 9.14  |\n\nWe find that:\n1. The best performance of tuning both $k_i$ and $\\gamma$ is better than the best performance of tuning only $k_i$;\n2. The higher the $k_i$ is, the higher decay (lower $\\gamma$) should be applied to balance the effect of larger integral gain, which aligns with common intuition.\n\nThe result is a strong evidence that the decay in integral term does improve sampling performance.\n\n\n### Weakness 3 & Question 3\n\nThank you for raising the concern on the competitiveness of FID value. We also noticed that our reported FID is higher than some other novel models. We believe this is due to the difference of the pretrained score model. We discuss this topic as follows.\n\n#### **What is our main contribution?**\n\nOur goal is to demonstrate that PID control provides a **relative** improvement over ALD and MILD under the same base model and codebase. These gains hold true regardless of the absolute performance.\n\nTo demonstrate the relative advantage, the most important aspect is not absolute performance but **fairness of comparsion**, which is indeed addressed in our base model selection.\n\n#### **Fairness: the dominant factor in base model selection**\n\nWe were using the pretrained checkpoints provided by the official repository of NCSNv2, which is also used by MILD, to ensure a directly comparable setup.\n\n- For ALD, all code and hyperparameters are shared with PIDLD, because setting $k_i=0.0$ and $k_d=0.0$ recovers vanilla ALD. So we ensure that the only difference is the inclusion of the I- and D-terms.\n- For MILD, we report the FID as published, since their exact hyperparameter settings were not released, but their score model is also the one released by NCSNv2. With the same score model, using their published numbers remains the fairest benchmark.\n\nBy choosing the same model with that in relevant literatures, we isolate the effect of adding PID control, rather than differences in underlying architectures.\n\n\n#### **FID influencing factor: fundamental framework differences**\n\n\nWe would also like to clarify that our contribution is on improving **Langevin dynamics** generation. In contrast, other frameworks like diffusion-based samplers (DDPM, DDIM, etc.) implement very different generative mechanisms. For instance, diffusion models explicitly \"corrupt\" data through many small noise steps and learn to reverse this process, whereas Langevin methods directly simulate a stochastic path that climbs toward high-density regions in one unified sampling loop. Many frameworks of generative models exist, and the fundamental difference among them would also result in varying potentials of sampling performance. Directly comparing our Langevin-based PIDLD to other frameworks is therefore **out of scope** for this work.\n\nWe believe our idea on improving Langevin dynamics sampling could be further extended to other classes of generative models, including state-of-the-art ones. We expect to address these in our future work.\n\n\n#### **FID influencing factor: method of computation**\n\nFurthermore, FID computing method can also influence the reported number. FID typically requires 10000 test images, which is also our setting. However, some other methods like DDIM and DDPM use 50k samples for FID computation, and larger sample size will generate lower FID. Model size is another factor. For instance, we found that the noise estimation model in DDIM has 78700803 parameters, while our score model only has 29694083 parameters. Given these concerns, it's not always fair to directly compare the absolute value of our results with other approaches. And we believe that using the most relevant works, vanilla ALD and MILD, as baselines, is the most suitable design."}, {"review_id": "eLMSC1CugB", "reviewer": "Reviewer_YqLQ", "summary": "This paper proposes a method for adaptive reward-guided sampling in diffusion models by modulating the guidance strength during Langevin sampling. The authors frame controllable generation as a feedback control problem, where the error between a current reward and a desired target is used to dynamically adjust the scale of reward gradients injected during sampling. The method is implemented as a training-free wrapper and is tested on several text-to-image generation tasks using Stable Diffusion. Empirical results show improved alignment with target rewards and sample quality compared to fixed or heuristic guidance strength baselines.", "questions": "Have you evaluated the relative contribution of the I and D terms in the PID controller? Would a proportional-only controller suffice?\n\nHow sensitive is the method to the choice of PID gains (k_p, k_i, k_d)? Are they robust across prompts and reward types?\n\nHave you considered combining reward error-based scaling with gradient-norm normalization for better numerical stability?", "limitations": "The paper does not extensively discuss potential failure cases or unintended behavior (e.g., reward hacking, overshooting, or instability for ill-behaved reward functions). The PID controller is sensitive to hyperparameters (k_p, k_i, k_d), and though fixed values are used throughout, no systematic analysis of robustness is provided. Including such analysis would help assess broader applicability.", "rating": 4, "confidence": 1, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 2, "originality": 2, "strengths_and_weaknesses": "Strengths\n\n1 Clear motivation: The paper identifies a common practical issue in reward-guided diffusion sampling — the need to manually tune or schedule the guidance strength α — and addresses it in a principled way.\n\n2 Conceptual novelty: Framing guidance as a control problem and applying a PID controller to reward error is novel in this context. To the best of my knowledge, this is the first use of classical control theory for online adaptation of guidance strength in diffusion models.\n\n3 Training-free and modular: The method can be applied without retraining the base diffusion model or the reward model. It is compatible with any differentiable reward and can be used as a drop-in wrapper around existing samplers.\n\nWeaknesses\n\n1 Lack of ablation on PID components: The benefit of using the full PID structure (Proportional + Integral + Derivative) over a simple proportional or heuristic controller is not clearly demonstrated. An ablation comparing PID vs. just P (or vs. α ∝ gradient norm) would help isolate the contribution of each term and justify the additional complexity.\n\n2 Heuristic alternative baselines are missing: While the paper compares against fixed α and some reward-guided variants, it does not benchmark against simple time-based or norm-based scaling heuristics (e.g., α increasing over time, or α ∝ 1 / ∥∇ℓ(x)∥). These baselines are common in the literature (e.g., FreeDoM, classifier guidance) and would provide a better sense of the gain from PID.\n\n3 Limited analysis of control behavior: The paper does not deeply analyze the PID controller's behavior over time — e.g., plots of α_t, error evolution, or control stability across prompts. This would strengthen the claim that the controller leads to smoother and more stable guidance.\n\n4 Conceptual novelty may be overstated: While the framing of PID-controlled sampling is novel, the underlying mechanism — adjusting guidance strength based on feedback — is not entirely new. Related ideas exist in norm-based scaling, adaptive weighting, and heuristic scheduling. The paper would benefit from more explicit comparison and positioning relative to these.", "was_revised": true, "final_justification": "I have checked the reviews and the author's responses to those questions. The idea of approaching the problem from a control theoretic perspective is interesting and I see no major weakness warranting a rejection.", "rebuttal": "We sincerely thank the reviewer for the time and valuable feedback. **We are encouraged that the reviewer recognizes the clear motivation, conceptual novelty of applying control theory, and the training-free, modular nature of our approach.**\n\nHowever, we believe there has been a significant misunderstanding regarding the **core problem, underlying mechanism, and experimental setup** of our work. The reviewer's summary appears to describe a paper on \"adaptive reward-guided sampling in diffusion models,\" such as Stable Diffusion for text-to-image tasks. Our paper, PIDLD, addresses a more fundamental problem: **accelerating the core Langevin Dynamics (LD) sampling process itself**, which is applicable to any model using LD, including Score-Based Generative Models (SGMs) and Energy-Based Models (EBMs).\n\nThis core misunderstanding seems to be the source of most of the stated weaknesses and questions. We hope the following clarifications can resolve these concerns and provide a clearer picture of our contributions.\n\n#### **Clarification on the Core Mechanism and Objective**\n\n*   **Our goal is NOT reward guidance, but sampling acceleration.** We aim to reduce the number of Neural Function Evaluations (NFEs) required for the sampler to converge to the target distribution defined by the model (SGM/EBM), thereby speeding up generation.\n*   **The \"error signal\" in our PID controller is the energy/score gradient `∇xU(x)`, NOT an external reward.** Standard LD uses only the current gradient (a Proportional term). We enhance this by adding a memory of past gradients (Integral term) and a prediction of future gradient trends (Derivative term). This is fundamentally different from using a PID controller to minimize the error between a sample's \"reward\" and a target value.\n*   **Our experiments are on SGMs (NCSNv2) and EBMs (IGEBM), NOT Stable Diffusion.** Consequently, our tasks are unconditional image generation (CIFAR10, CelebA) and reasoning (Sudoku, Connectivity), not text-to-image generation as assumed.\n\nWe believe this central clarification invalidates the premise of Weaknesses 2 and 4, which compare our work to methods in the reward-guidance literature (e.g., FreeDoM, norm-based scaling of reward gradients). Our chosen baselines, vanilla ALD and MILD, are the standard and appropriate benchmarks for Langevin sampling acceleration.\n\nWe will now address the other specific points raised by the reviewer.\n\n### On the Ablation Study of PID Components (Weakness 1, Question 1)\n\n**We provide detailed ablation studies for the I and D terms in Figures 5 and 6, which clearly demonstrate their distinct and crucial contributions.**\n\nWe respectfully point the reviewer to Section 4, where we analyze the performance of PIDLD (P+I+D) against ablations: P-only (vanilla LD), P+I, and P+D.\n\n*   **For Image Generation (Fig. 5):** The results show that the Derivative term (P+D) provides the most significant improvement. As explained in Lines 294-301, this is because the D term enables rapid convergence within the smoothed energy landscapes at each noise scale of annealed LD.\n*   **For Reasoning Tasks (Fig. 6):** Both Integral (P+I) and Derivative (P+D) terms provide substantial gains, and their combination (P+I+D) yields the best performance. As explained in Lines 348-352, the I term's ability to overcome local optima is critical for finding global energy minima in these complex, non-visual tasks.\n\nThese results directly answer the reviewer's question and confirm that the full PID structure is superior to a P-only controller, with the I and D terms offering complementary benefits depending on the task's energy landscape.\n\n### On Controller Behavior and Hyperparameter Sensitivity (Weakness 3, Question 2, Limitations)\n\n**We analyze the controller's behavior and sensitivity to its gains (`k_i`, `k_d`) in Section 3.2 and Figure 3, confirming its stability and robustness.**\n\n*   **Analysis of Controller Behavior:** Figure 3 directly illustrates the system's dynamic response to varying `k_i` and `k_d` values, measured by KL divergence over sampling steps. This analysis goes beyond simple endpoint metrics and provides insight into the controller's behavior over time.\n*   **Hyperparameter Sensitivity (`k_p`, `k_i`, `k_d`):** Our analysis in Figure 3 shows that performance improvements are stable across a reasonable range of `k_i` and `k_d` values. Furthermore, we address the potential instability of the integral term (a key concern in control theory) by introducing a decay factor `γ` (Fig. 3b), which guarantees convergence by ensuring `k_i → 0` asymptotically (Lines 195-205). The hyperparameters used were consistent across all experiments for a given model type, demonstrating robustness.\n*   **Gradient-Norm Normalization:** We thank the reviewer for this suggestion. Our time-decay mechanism for the integral term (Lines 195-205) serves a similar purpose of ensuring stability and preventing the \"integral windup\" or \"overshooting\" issue, but we agree that explicit normalization could be an interesting direction for future work.\n\n### Conclusion\n\nWe hope these clarifications have resolved the apparent discrepancy between the reviewer's understanding and our paper's actual content. Our work introduces a novel, principled, and effective method for accelerating the fundamental Langevin sampling process by drawing inspiration from control theory. The experimental results on relevant models (SGMs, EBMs) and appropriate baselines (ALD, MILD) robustly support our claims of significant speedup and quality improvement. We are confident in the novelty and significance of our work and are happy to provide further clarification if needed."}]

Author rebuttal:

Meta-review: This paper proposes a control theory inspired Langevin dynamics variant that improves the sampling speed of EBMs and score based models. The reviewers all find the idea interesting and well motivated. There were initially several concerns on the formulation and empirical aspects of the work, but the authors provided an extensive rebuttal which resolved much of the concerns. The reviewers have reached consensus in acceptance, and the AC agrees with the reviewers' votes.
Decision: Accept (poster)