PID-controlled Langevin Dynamics for Faster Sampling on Generative Models

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.

However, 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).

This 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.

Clarification on the Core Mechanism and Objective

• 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.
• 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.
• 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.

We 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.

We will now address the other specific points raised by the reviewer.

On the Ablation Study of PID Components (Weakness 1, Question 1)

We provide detailed ablation studies for the I and D terms in Figures 5 and 6, which clearly demonstrate their distinct and crucial contributions.

We 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.

• 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.
• 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.

These 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.

On Controller Behavior and Hyperparameter Sensitivity (Weakness 3, Question 2, Limitations)

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.

• 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.
• 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.
• 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.

Conclusion

We 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.

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.

Weakness 1

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.

As 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.

Moreover, 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.

Weakness 2

Thank 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.

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.

Moreover, 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.

To 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.

Weakness 3

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.

Integrating 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.

Weakness 4

Thank 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.

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.

Weakness 5

We 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.

Table: Accuracy (%) comparison including MILD on harder reasoning tasks.

Analysis: We 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.

However, 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.

While 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.

Regarding the minor points

We appreciate the reviewer's careful reading and valuable suggestions on presentation.

• 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.

• 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.

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.

Weakness 1

Thank you for pointing out the potential logic loophole. We show that our measure of computation cost is reasonable as follows.

First, 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.

It 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.

Moreover, 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.

Weakness 2

Thank you for raising the concern of the lack of performance metric. We additionally report FLD and Inception score as follows.

FLD

For CIFAR10 dataset, we follow the official implementation of FLD and have the following results.

(FLD: lower is better, FLD gap: a measure of uncertainty, lower is better)

For 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.

Within our implementation, we observe that PIDLD still outperforms ALD in terms of FLD metric, which strengthens our work.

Inception Score

We did record inception score in our experiments on CIFAR10 dataset. The results are given in the following table:

Judging from the inception score, PIDLD still generally outperforms ALD under different NFEs, which serves as another evidence of its advantage.

As 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.

The 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.

Question 1 & Weakness 3 & Weakness 4

We 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.

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.

1. 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.

2. 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.

Question 2

We thank the reviewer for this question, as it allows us to clarify the precise role of our theoretical analysis. The m-strong convexity is a standard assumption for local stability analysis and does not presume the global energy landscape is convex.

We 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.

Question 3

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.

• 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.

• 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.

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.

Weakness 1 & Question 1

Thank 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.

Our results truly validates the claim that $I$ term helps to eliminate bias.

• 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.
• 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.

Weakness 1 & Question 2

Thank 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.

It 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.

We 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.

To 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 $k_d$.

The results are given in the following table. Here the superscript $(1)$, $(2)$ indicate the cluster around $(5,5)$ and $(-5,-5)$, respectively.

It 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.

Weakness 2

Thank 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.

For 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.

CIFAR10 100×1 ($k_p=2.0, k_d=4.5$) FID (Bold: best in column, Italic: best in row 1):

CelebA 50×1 ($k_p=3.75, k_d=3.75$) FID (Bold: best in column, Italic: best in row 1):

We find that:

1. The best performance of tuning both $k_i$ and $\gamma$ is better than the best performance of tuning only $k_i$;
2. 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.

The result is a strong evidence that the decay in integral term does improve sampling performance.

Weakness 3 & Question 3

Thank 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.

What is our main contribution?

Our 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.

To 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.

Fairness: the dominant factor in base model selection

We were using the pretrained checkpoints provided by the official repository of NCSNv2, which is also used by MILD, to ensure a directly comparable setup.

• 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.
• 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.

By choosing the same model with that in relevant literatures, we isolate the effect of adding PID control, rather than differences in underlying architectures.

FID influencing factor: fundamental framework differences

We 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.

We 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.

FID influencing factor: method of computation

Furthermore, 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.

Response to W2: Control-Theoretic Grounding for the Integral Term Decay

We sincerely thank the reviewer for this insightful follow-up. The decay factor on our integral term is a practical implementation of Gain Scheduling[A], a classic and well-studied adaptive control strategy for time-varying systems. This provides the formal connection to control literature that you requested.

The rationale is that Langevin sampling is inherently a time-varying process with conflicting control requirements at different stages:

1. Early Stage (Exploration): The system is far from equilibrium and requires a high integral gain ($k_i$) to build momentum for rapidly traversing energy barriers.
2. Late Stage (Convergence): As the system approaches the target distribution, this same high integral gain becomes detrimental, causing overshoot and instability (as empirically observed in our Fig. 3a). Here, a low integral gain is needed to ensure stable convergence.

Our exponential decay, ki(t) = γ^t * ki, serves as a simple yet effective gain schedule. It uses the sampling step t as the scheduling variable to dynamically reduce the integral gain, smoothly transitioning the controller's behavior from aggressive exploration to stable final convergence.

We will explicitly introduce and discuss the concept of Gain Scheduling in the revised manuscript to formally ground our method's design. Thank you again for pushing us to clarify this crucial theoretical link.

[A] Khalil, Hassan K., and Jessy W. Grizzle. Nonlinear systems. Vol. 3. Upper Saddle River, NJ: Prentice hall, 2002.

Response to W3 & Q3: Applying PIDLD to a Stronger Baseline

Thank you for clarifying your point and for the excellent suggestion. We agree that validating our method on a stronger, state-of-the-art baseline is crucial for demonstrating the generalizability of its benefits.

You are correct that applying PIDLD to the model proposed in the paper you cited (arXiv:2504.10612) should be feasible, as it also utilizes Langevin-based sampling.

We are actively working on integrating our PIDLD sampler with their publicly available code and pretrained models. We will report the results of this experiment as soon as they become available during the rebuttal period. We fully commit to including these new results on this stronger baseline in the final version of the manuscript, regardless of the outcome, to provide a more comprehensive evaluation of our method's performance.

We appreciate you pushing us on this front, as it will undoubtedly strengthen the paper.

We thank the reviewer for their valuable follow-up and the insightful suggestions, which have helped us further strengthen our paper.

Response to W1 & Q1: Statistical Validation of the Integral Term's Role

Our new, more rigorous experiments statistically confirm that the integral (I) term systematically reduces gradient estimation bias, directly validating the analogy to its role in control theory.

To provide statistical evidence, we conducted experiments on the 2D toy problem with a constant gradient bias(the same as in the rebuttal), averaging results over 4 random seeds for each hyperparameter setting. We measured the Euclidean distance from the final estimated cluster centers to the true centers as a proxy for the remaining bias.

1. The I-Term Effectively Reduces Bias: As shown in the table below, there is a clear and consistent trend: increasing the integral gain ($k_i$) systematically reduces the final bias (distances $d_1, d_2$).

2. The D-Term Does Not Reduce Bias: To isolate the effect, we performed a comparative study. Increasing the derivative gain ($k_d$) does not reduce the bias; in fact, it slightly exacerbates it for one cluster. This control experiment confirms the distinct roles of the terms.

These results provide evidence that the I-term's function of correcting long-term, steady-state error (i.e., bias) from control theory directly translates to the Langevin sampling setting. We will incorporate this experiments and discussions in the final manuscript.