Convex Potential Mirror Langevin Algorithm for Efficient Sampling of Energy-Based Models
Authors: Yang Zitao
Amin Ullah
Shuai Li
Li Fuxin
Jun Li
Venue: NeurIPS 2025 poster ()
Keywords: mirror Langevin dynamics
efficient sampling
energy based models
Primary area: deep_learning
TL;DR: We introduce a novel approach which leverages a convex potential for efficient sampling of EBMs.

Abstract: This paper introduces the Convex Potential Mirror Langevin Algorithm (CPMLA), a novel method to improve sampling efficiency for Energy-Based Models (EBMs). CPMLA uses mirror Langevin dynamics with a convex potential flow as a dynamic mirror map for EBM sampling. This dynamic mirror map enables targeted geometric exploration on the data manifold, accelerating convergence to the target distribution. Theoretical analysis proves that CPMLA achieves exponential convergence with vanishing bias under relaxed log-concave conditions, supporting its efficiency in adapting to complex data distributions. Experiments on benchmarks like CIFAR-10, SVHN, and CelebA demonstrate CPMLA's improved sampling quality and inference efficiency over existing techniques.

Convex Potential Mirror Langevin Algorithm for Efficient Sampling of Energy-Based Models
Zitao Yang', Amin Ullah?, Shuai Li², Li Fuxin', Jun Li'
Fudan University, 'Boeing Research and Technology
3 University of Bonn, *Oregon State University yangzt21@m fudan. edu. en, amin. ullahhboeing. com, lishuai@iali uni bonn. eduu, Lif@oregonstate.co edu, jun li@fudan edu.
Abstract
This paper introduces the Convex Potential Mirror Langevin Algorithm (CPMLA), novel method improve sampling efficiency for Bneeryy-Basee Models (EBMs). CPMLA uses mirror Langevin dynamics with a convex potential flow as dynamic mirror map for EBM sampling. This dynamic mirror map enables targeted geo- metric exploration on the data manifold, accelerating convergence to the target distribution. Theoretical analysis proves that CPMLA achieves exponential con- vergence with vanishing bias under relaxed log-concave conditions, supporting its efficiency in adapting to complex data distributions. Experiments on benchmarks like CIFAR-10, SVHN, and CelebA demonstrate CPMLA's improved sampling quality and inference efficiency over existing techniques.
Introduction
Energy-based models (EBMs) represent class of generative machine learning models designed to capture and synthesize complex data distributions. EBMs define an unnormalized probability distribution via an energy function, assigning low energy values likely data samples (corresponding
to the target distribution) and high energy values to unlikely ones (55] 39 TO]. Known for their
conceptual simplicity and training stability, EBMs have found diverse applications ranging from 3D
object recognition [15] and analysis [56] to image segmentation (24,, super-resolution restoration
[60], machine translation 47), and protein folding 48 52]
critical limitation of energy-based models (EBMs) lies in their reliance on Markov Chain Monte Carlo (MCMC) sampling methods, particularly when operating in high-dimensional data spaces 10 29) MCMC algorithms like Langevin dynamics often get trapped in local energy minima when the underlying data manifold is characterized by multi-modal energy landscapes [21] or exhibits non-Euclidean geometry [59] (21] When sampling from the complex, highly multi-modal energy landscapes characteristic of deep EBMs, these MCMC methods can become computationally intensive and yield biased sampling [38]. These factors hinder the efficient approximation of complex distributions and can lead slow convergence towards the target distribution.
Recent methods address sampling inefficiencies within EBMs. Some strategies refine MCMC initialization 18 10), while others explore gradient approximation techniques 25
29 However, persistent challenges such as non-mixing issues remain unresolved (58). Mirror Langevin algorithms have recently emerged an alternative approach alter sampling geometry via fixed mirror map, i.e., a predefined function. Prior work [1] 30 demonstrates that mirror Langevin algorithms, under certain assumptions, exhibit vanishing bias (bias 0 as the step size 0). This property ensures reliable convergence to the target distribution and improves sampling accuracy. Moreover, mirror Langevin algorithms achieve mixing times independent of the domain's condition number, enabling fast convergence [19 (59). However, fixed miror maps in conventional mirror Langevin algorithms
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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struggle to capture complex data manifolds efficiently, limiting their use for large-scale problems, especially those associated with deep neural networks.
This paper introduces Convex Potential Mirror Langevin Algorithm (CPMLA), a novel approach for sampling EBMs with enhanced efficiency. Unlike conventional mirror Langevin algorithms in Euclidean space, CPMLA employs a learnable, data-driven mirror map that actively infers the intrinsic manifold structure of the data. By parameterizing the mirror map as the gradient of a convex potential function (cf. Brenier's ' theorem [4]). CPMLA dynamically reorients sampling trajectories to align with the oon-Euclidean geometry of the target distribution, enabling adaptive exploration of high-density regions while avoiding metastable states.
We employ cooperative learning strategy that jointly trains the dynamic mirror map and the EBM. First, the dynamic mirror map is learned by optimizing a convex potential flow (CP-Flow) [20] Building on Brenier's theorem for optimal transport [44] this formulation guarantees that CP-Flow
defined as the gradient of a convex potential function 3 -inherenlly captures the geometric structure of the data distribution. Then, the EBM trained by contrasting the energy of real samples with that of those synthesized via CPMLA. Concurrently, synthesized samples are fed back into the CP-Flow training phase. This alternating process aligns the CP-Flow's transport dynamics with the EBM's energy-based density estimation, mitigating sampling bias and accelerating sampling convergence.
We theoretically analyze the convergence of CPMLA. Based on the recent study 221, we prove exponential convergence under relaxed log-concavity assumptions with two improvements. First, specialize our proof for the dynamic mirror map modeled with deep neural networks. Second, beyond the sampling algorithm's 's error, our analysis also incorporates the approximation errors from modeling both the CP-Flow and the EBM with deep neural networks. These improvements broaden the applicability to a wider range of target distributions in various machine learning tasks. To the best of our knowledge, this is the first analysis of mirror Langevin algorithms within the framework of deep neural networks, resulting in exponential convergence with vanishing bias (Theorem 4.5).
We evaluate CPMLA across several benchmark datasets, including CIFAR- 10, SVHN and CelebA. The results demonstrate that CPMLA not only achieves superior sampling quality but also exhibits enhanced inference efficiency compared to existing cooperative algorithms. Specifically, CPMLA achieves an FID score 73% lower than Flow+EBM 13 38), indicating a substantial improvement in visual quality. Additionally, CPMLA not only achieves lower FII) score (20.85 vs. 21.16) than CoopFlow [58] with fewer inference iterations (20 vs. 30) and less time (15.92s vs. 16.84s) as shown in Table but also operates with only 0.9% of the parameter count w.r.t. the flow part as shown in Table 3 underscoring its efficiency in both sampling and inference, CPMLA also excels in specialized tasks like image reconstruction and inpainting, further emphasizing its effectiveness in tackling complex image processing challenges.
Our main contributions are summarized as follows:
We propose novel Convex Potential Mirror Langevin Algorithm (CPMLA) for efficient sampling of EBMs. The efficiency comes from the modification of the sampling geometry through dynamic mirror map modeled with deep neural network.
We provide a theoretical convergence analysis of the proposed CPMLA, specifically focusing on deep neural networks under relaxed assumptions.
We evaluate the efficacy of our proposed algorithm through comprehensive experimental analyses on various benchmark datasets, including CIFAR-10, SVHN, and CelebA. Our experiments demonstrate that CPMLA achieves superior sampling efficiency compared to existing methods. Furthermore, it surpasses alternative approaches in terms of sample quality and the fidelity of image reconstruction and inpainting.
2 Background
2.1 Energy-Based Models
Energy-Basce Models (EBMs) characterize a probability density over data x e IR as follows:
1
po(x) = Z(0) exp (x))]
(1)
2

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Here, fo ::::" Rd + I represents the negative energy function, parameterized by neural network with parameters 0. The term ( = explfo dz is the normalizing constant, which is generally intractable to compute.
Generating samples from px()) involves Markov Chain Monte Carlo (MCMC) methods, with Langevin Monte Carlo (LMC) [50] being a prevalent choice. The LMC update rule is given by:
8 = a'
ae 0
(2)
2
where æ' is the sample at step i, d is the step size, ef ~ NN 1) is Gaussian noise, and the process is often initialized with z drawn from a simple distribution like uniform pu()).
The parameters 0 of the energy function are learned by maximizing the log- oigliikeldood of observed data samples Fz I....,n drawn from the true data distribution Pdaa(z)) The gradient of the log-likelihood objective is:
n E Volo (z.) 7
n Volo (i:a)
Vo log (o = Exps [VoJo(=)] Epg [Volo(>)]
(3)
n2
In this expression, 2; represents samples drawn from the current model distribution po(),, usually obtained via LMC described above. The expectation under pa. which implicitly depends on the intractable (()), is estimated using these generated samples â; Consequently, the learning updates by contrasting the average gradient of the energy function evaluated on real data with the average gradient evaluated on samples generated by the model.
2.2 Convex Potential Flow
A foundational requirement for CPMLA to satisfy the mirror Langevin algorithm is that the dynamic mirror map must be derived from strongly convex potential function via its gradient. To this end, we choose Convex Potential Flow (CP-Flow) (20 for this role precisely because its architecture, based on Input-Convex Neural Networks (ICNNs), guarantees this convexity property. A standard normalizing flow, in contrast, does not generally have convex potential, making it unsuitable for a mirror Langevin framework. As shown below, CP-Flow Iearns a tractable probability density by approximating the optimal transport map between noise distribution and the target data distribution, specifically minimizing the quadratic cost (Monge) problem.
Optimal Transport The Monge problem expected cost as follows:
49 seeks an optimal transport map g minimizing the
Jalx pV)
inf 3-(0))>
Ex-px ((2,
where c(2,, y) is the given cost function.
Theorem 2.1. (Brenier's Theorem (44)) Suppose i and are probability measures with finite second moments, and assume that μ has Lebesgue density px. this case, there exists a convex potential G such that the gradient map g VG (uniquely defined except a null set) provides the
solution to the Monge problem in Equation with square cost function c(z, y) = =|x yl
To approximate the optimal solution for the Monge problem, the convex potential is modeled with several layers of an input-convex neural network (ICNN) Go 3 which is convex w.r.f the input:
G (I) = L (s (zx)) + (x)
(5)
Zk : L (s (z--1)) - Lu(x), 21 == (x)
where & denotes parameters of the neural network, L(x) denotes a linear layer, L (z) denotes linear layer with positive weights, and s is a non-decreasing convex activation function.
To ensure Go is strongly convex, which is required for VG, to be an invertible mirror map, For small positive sealar a, this guarantees
quadratic term added: (I that the Hessian V²G, ž al > 0. This modification ensures that the gradient VG, is bijective and its inverse can be computed efficiently. For brevity, we omit the subscript a and use Go to denote the strongly convex potential hereafter.
3

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Primal space
To
Tg. 1/2
3k
3 Diinal
V62
VG.
VG² (Ix)
Yo
VT
Dual space
Figure Overview of CPMLA sampling. Starting from a noisy sample æi, CPMLA iteratively refines it by alternating between the primal space (interpretable images) and the dual space (geometry encoded by VGg). At each step, 2y is mapped to the dual space, then updated via an EBM energy gradient step and perturbed with geometry mapped back to the primal space as TR+1, yielding progressively sharper samples that efficiently explore the data manifold.
noise scaled by Go Finally, the result is
Like standard flow-based models, CP-Flow is trained by max- imizing the log-likelihood of the model density. This requires computing the log-determinant of the Hessian matrix of the convex potential, log det H, where H (2). For high-dimensional data, forming and storing the full Hessian computationally infeasible. To overcome this, we use matrix-free approach based on Hutchinson's ''s trace estimator, which relies on efficient Hessian- vectoo products (HVPs). The gradient of the log-determinant can be estimated as:
Algorithm CP-Flow training ob- jective
Procedure: Obj z, CG) Sample Rademacher r def hvp(v):
return ar VG,(x)
f stop_ stop_grad r))
0H
return h
log det H = E, V T
H
V
(6)
do
00
where v a random vector with zero mean and unit covariance (e.g., Rademacher). The term H-1 expensive to compute directly. Instead, reframe its calculation quadratic optimization problem, xº = arg ming {+2 Hz v z). which can be solved efficiently using the conjugate gradient (CG) algorithm without instantiating H. This procedure is summarized in Algorithm
3 Algorithms
3.1 Mirror Langevin Algorithm
To generate synthesized examples from a target distribution p(x) with mirror Langevin dynamics [9], we solve the stochastic differential equation:
dy: V d( ( 22V2G(.))
VG (Vt)
where x, and y/ stochastic processes in the primal and dual spaces, respectively, W, is the standard Brownian motion in Rd, and VG is the mirror map. The term VG is the gradient of the convex conjugate G*, which serves as the inverse of the mirror map, i.e., (VG) VG" (Appendix
We use the Alternative Forward Discretization Scheme (MLAxFd) which has exponential convergence and vanishing bias 21 We CP-Flow VG2 (Section 2.2 as the dynamic mirror map. For

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an iteration with step size the update is:
Jk+1/2 VG² (VGo (1%) v/ (xx))
solve dy
2 (v6))
dW. 'dW
22 (VGju)ddW, for yo _ VGa (Ix+1/2)
T++1= (vr)
The * step is derived from the property of convex conjugate [2] (see Appendix F.. The computation of VG in step and VG, in step can be simplified by noting that they are inverses and cancel each other out in successive iterations.
3.2
CPMLA
Our CPMLA facilitates exploration of the underlying data manifold. It achieves this by using MLAAFD in Equation (8) with a CP-Flow dynamic mirror map, which dynamically transforms the sampling geometry based on the metric induced by the V²
Like standard mirror Langevin methods, CPMLA alternates between updates in primal and dual spaces. The alternating sampling process entails transitioning between updating samples in the dual space using a dynamic mirror map for LMC exploration, followed by mapping the sample back to the primal space utilizing the inverse of the mirror map.
Figure lillstratts the sampling process of the proposed CPMLA. Specifically, each CPMLA iteration involves three steps: First, noise examples {yo} are generated from a standard Gaussian distribution (0, in the dual space. And for each sampling step h, noise vector & generated from Gaussian distribution (O, V=Go( Second, starting from {yo T steps of EBM sampling (gradient and SDE steps) are performed in the dual space, yielding {8r} }- Third, the inverse map transforms fyr} back to the primal space, yielding {+}. The synthesized examples {+) are considered as outputs sampled by CPMLA.
Algorithm 2 shows the cooperative learning of EBM and CP-Flow. At each update, we re-initialize the MCMC chain. This is a standard practice methods like Persistent Contrastive Divergence (PCD) to prevent chain collapse and ensure that model gradients estimated from samples of the current model distribution, avoiding feedback from stale samples. First, we update the parameters of the CP-Flow using both original examples {x} and synthesized examples {ø}. Then, we update the parameters of the EBM based on the contrast between {a} and { as in Equation 3 The updates for both 8 and & are performed using the Adam optimizer, with learning rates and other hyperparameters specified in Appendix [ This cooperative mechanism simultaneously improves sampling efficiency and model expressiveness, creating virtuous cycle of mutual enhancement.
In Algorithm 2 initial samples (uo) ~ NN /) are drawn in the dual space, so no initial mapping with VGa is needed. We also use computational trick to avoid the expensive matrix square root of the Hessian: the term (zx)) (where & ~ (0 I)) is statistically equivalent to 2/2 Ex (where & (0, G=)()))) In practice, we approximate V²G, with its diagonal to reduce computational complexity from O(d)) to (dd, enabling efficient high-dimensional sampling.
Theoretical Analysis
This section presents the convergence analysis of CPMLA, the first for mirror Langevin algorithms with deep neural network mirror maps. Our analysis relies on standard properties of neural networks (e.g., bounded gradients via clipping) and set of theoretical assumptions, which are standard in the analysis of Langevin-type algorithms [ framework below.
We provide detailed justifications for their validity in
Assumption 4.1. (3-Mirror Log-Sobolev Inequality, B-Mirror LSI) The target distribution satisfies 3-Mirror L.I with constant w.r.t given mirror map VG, i.e., for every locally lipschitz function h. it holds that π satisfies
/ [hi| dd 2 B
2
/ lo log 2
(h'd)) log (fheds
5

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Algorithm 2 Convex Potential Mirror Langevin Algorithm (CPMLA)
Input: (1) Observed images { Pdaa(x);; (2) Number of Mirror Langevin steps T; (3) Step size in dual space 7)-
Output: Parameters of EBM and CP-Flow (0, }}
Randomly initialize and 0.
repeat
Sample noise examples {vo} ~ NO0 1) in dual space.
for k 0 to T - do
Let Ir = (yv))
Sample noise &a No0 (x))
Mx+1/2 yA V fa(
V/++1 / + v 2nj Ek
end for
Map back to primal space & (yr)
Starting from {+). update & by Algorithm
Given {2) and { update 0 with Equation 3
until converged
Justification: This is foundational assumption about the properties of the target data distribution itself, relative to the geometry induced by the mirror map. While difficult to verify empirically for complex, high-dimensional data distributions, it is standard and necessary assumption in the literature for proving the convergence of Langevin-type algorithms in non-Euclidean spaces [2 Our contribution focuses on the aspects of the algorithm we can control and verify.
Assumption 4.2, (ç-Self-Con (Sseo--Concordance) There exists a constant S 2 O such that the conjugate mirror map VG satisfies that Vy, u, 8, V,
V (w),,, 8, w|| 2 2.. (V)=)')))) 1/2 ()(' 1/2 (0
1/2 (x")"))) (
(10)
Justification:
This assumption bounds the third derivative of the potential function relative to its second derivative, ensuring the geometry does not change too abruptly. We empirically validate this assumption for our trained models on CIFAR-10. As direct computation of the third-order derivative tensor is infeasible, we employ matrix-free validation approach. We estimate the Frobenius norms of the Hessian V²G, (2) and five random directional third derivatives (ar) [ using Hutchinson's
estimator, which relies on efficient Hessian-vector products. We then compute the proxy metric
Spooky G [=)]] ++ Across all training checkpoints, the value of Sproxy consistently remains
small and stable (in the range [10-1, 10-21), providing strong empirical support that this assumption
holds in practice.
Assumption 4.3. (L-Relative Lipschitz) For all æ, it holds that f Rd + IR is differentiable with
[[(()][ S L
(11)
This assumption is satisfied in our framework due to standard deep learning practices. Our potential function G is designed to be strongly convex, meaning Hessian V²G(x) > al for some a > 0. In practice, we use gradient clipping on the EBM, which ensures that |[V is bounded by a constant C. This directly leads ||V [=V Thus, the assumption holds by setting I, = C/va.
Justification:
S (V(E)) < C/va.
Assumption 4.4. (Weaker Rellatvvv Smooth) For all x' E dom(G),
[V f (a' (2') ([ S 7 [[V VG (a') (x))l
(12)
Justification:
Similarly, the gradient of our EBM, VS. is Lipschitz with some constant L, (deter- mined by the network architecture and enforced by weight decay and gradient clipping). The potential G is also smooth. This allows us bound the relative smoothness, and the assumption holds by setting = Ly/a.
Theorem 4.5 (Convergence of CPMLA). Let d be the dimension of the data space. For any mirror map VG, define M := exp (2çD/Va) where D = max,, IVG(u) VG(u)]la is the diameter

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of the image of VG. Under Assumptions 4.1 4.4 for any 8 > O, after k 2 St (M^2d/325) iterations with step size h ( the total variation distance between the sampling distribution Pt. and the data distribution Pdata satisfies:
drv (pt, Pdatt) <
where (2()) hides potylogarithmic factors, i.e. f (t(g) + 3c> O, nog, p E N such that f(n) Z g(n) [log Tog Vn 2 no. 5 = / + 82 + 83, with 81, 82, 83 being small constants related to the convergence error of CPMLA, approximation errors from the EBM and CP-Flow respectively.
This theorem provides framework. It states that if the EBM and the CP-Flow are trained to a certain approximation accuracy (represented by the epsilon terms), then the sampler is guaranteed to be within certain Total Variation distance of the true data distribution. The number of iterations and step sizes are implicitly embedded in the conditions required reach these error bounds.
an-asymptotic bound that characterizes the best achievable error of our
Proof sketch: Lemma from [21] provides the form of shifted drift and covariance of Equation in primal space. Using this lemma, we express CPMLA in primal space weighted Langevin dynamics in differential form with a shifted drift term j. Analyzing the Fokker-Planck equation for the conditional density (zjo zo), bound the KL-divergence between Pt and the target inequality, and the mirror log- Sobolev inequality (As-
using integration by parts, the Cauchy-Schwar sumption 4.1p. This yields differential inequality showing exponential decay of the KL-divergence, with convergence rate governed by the algorithm's 's parameters and target distribution properties. The total variation bound (pt Pdmaa < 8 decomposes into three terms: ō1 measures the distance between the distribution Pt generated after l outer iterations of CPMLA (Algorithm (2 and the stationary distribution Pg- associated with the learned energy function fø-- 52 and 63 represent the fundamental limitation in the expressive power of the chosen model architectures, namely the EBM fe and the CP-flow Go. They reflect how well the model family can intrinsically capture the target data distribution, irrespective of sampling or optimization efficiency.
The theorem states that, incorporating slight assumptions, CPMLA not only achieves exponential convergence but also exhibits vanishing bias, making it more applicable to the practical training scenario where both the energy model and mirror map parameters continuously updated. Details on this proof may be found Appendix H
5 Experiments
We evaluate the proposed CPMLA on diverse tasks. We start with a toy example in Section 5.1 Next, we present image generation results in Section 5.2 Finally, demonstrate CPMLA for image reconstruction and inpainting in Section 5.3
(a) Data
(b) CP-Flow
(c) CPMLA
Comparison between CPMLA and CP-Flow for Fitting Eight Gaussians. CPMLA reaches the same result in just 3 iterations that CP-Flow takes 10 iterations to achieve.
Figure 2:
7

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5.1 Toy Model Study
We first illustrate our approach on a toy example. Specifically, We apply CPMLA to model the eight Gaussians density from [42] and [5]. The results, presented in Figure 2 show that CPMLA efficiently fits these distributions. It demonstrates that on synthetic data, CPMLA provides reliable approximation of the target distribution without introducing bias. Notably, CPMLA matches CP- Flow's 10-iteration result (20 in only iterations, highlighting its superior convergence speed.
5.2 Image Generation
Model type
Models
FID
VAE
VAE
[26
78.41
Autoregressive
PixeICNN
43
65.93
GAN
WGAN-GP 14]
36.40
StyleGAN2-ADA
23
2.92
Score-Based
NCSN 45
25.32
NCSN+ 446
2.20
Flow
Glow 27
45.99
Residual Flow 66
46.37
EBMs
LP-EBM 41
70.15
EBM-SR 39
44.50
EBM-IG
38.20
CoopVAEBM 57
36.20
CoopNets [54
33.61
Flow+EBM
NT-EBM 38
78.12
EBM-FCE 13
37.30
CoopFlow (T=20) [58]
30.74
CoopFlow (T=30) [58
21.16
CPMLA (Ours)
CPMLAprt (T=20)
20.85
CPMLA (T=30)
21.09
Table I: FID scores the CIFAR-10. Our work focuses improving the sampling efficiency and quality for the EBM family of models, making them more competitive, While other classes of generative models like score-based diffusion (e.g.. NCSN++) or flow-matching models may achieve lower (better) FID scores on benchmark datasets, direct comparison is not the primary goal. EBMs offer greater modeling flexibility, they only require specifying unnormalized energy function, unlike models requiring specific architectures or tractable noise processes. Our method helps make this flexibility more practical by closing the sample quality gap. The comparison CoopFlow, strong FBM baseline, demonstrates CPMLA's superior efficiency this context.
We evaluate image synthesis performance on three datasets: CIFAR-10 28 which consists of 50,000 training images and 10,000 test images across 10 categories; SVHN 37 dataset with over 70,000 training images and more than 20,000 test images of house numbers; and CelebA 31. a large dataset of celebrity faces containing over 200,000 images. For fair comparison, all images are resized to 32 x 32 pixels. We present results under two settings. CPMLA: CP-Flow and EBM trained from scratch. CPMLAprt: CP-Flow first pretrained on data, then used to initialize CPMLA training. Pretraining provides a better initialization, potentially leading to higher quality images.
We present qualitative results (Figure 3) and quantitative FID scores (Table FID scores 17 computed based on 50,000 samples. Our models outperform most baselines, achieving lower FID scores compared to standalone normalizing flows and previous EBM+ ffow methods 13 38
The results demonstrate that CPMLA is parfmeter-efficient and effective compared to other coopera- tive and flow-only approaches. In particular, compared to CoopFlow. CPMLA provides distinct

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19
8
2
Si
(a) CIFAR-10
(b) SVHN
(c) CelebA
Figure 3: Generated Samples (32 32 pixels) by CPMLA from CIFAR- 10, SVHN, and CelebA datasets. These images are produced under the CPMLAprt training setting.
EBM part
Flow part
Models
Time (s/1k images)
FID
CoopFlow CPMLA
17.13M
28.78M
CoopFlow (T=30) CPMLAprt (T=20)
16.84 15.92
21.16
17.13M
0.26M
20.85
Table 3: Comparison of the param- eter amount between CoopFlow and CPMLA. CPMLA achieves lower FID) scores to CoopFlow with only 0.9% pa- rameter count w.r. the flow part.
Table 2: Wall-clock time (s/1k images) and FID com- parison on CIFAR-10 (50k samples). CPMLAA achieves lower FID than CoopFlow with fewer LMC iterations and less computation time.
advantage in terms of inference efficiency. (i) As shown in Table CPMLA achieves a lower FID than CoopFlow with fewer LMC iterations and less computation time. Figure 4 further illustrates
how CPMLA's FID improves faster than CoopFlow's 's across sampling steps (?" = 3 to T = 30), highlighting its superior convergence speed. (ii) CPMLA's CP-Flow component uses significantly fewer parameters (0.27M) than CoopFlow's normalizing flow (28. 78M, see Table 3). Remarkably, CPMLA outperforms CoopFlow while using only 0.9 % of its parameters.
(iii) To further analyze computational cost, compare the total training time of CPMLA and CoopFlow on CIFAR- 10 under realistic hardware constraints. When maximizing batch size to fit within 24GB of VRAM, CoopFlow is estimated to require approximately 38 hours for training, whereas CPMLA completes in only 10.5 hours. While direct per-iteration comparison for fixed batch size shows that CPMLA is marginally slower (15.7 s/iter vs. 12.0 s/iter for CoopFlow) due to the more complex CP-Flow architecture, its memory efficiency allows for larger batches, leading to sig- nificantly better overall training throughput. This highlights CPMLA's superior training efficiency in practical, resource-constraine scenarios.
Figure 4: FID comparison from
T = 3 T 30 between CPMLA and CoopFlow on CIFAR- 10 dataset. From an inferior initial- ization, CPMLA demonstrates faster inference speeds than CoopFlow.

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5.3 Image Reconstruction and Inpainting
We evaluate CPMLA for image reconstruction task, with focus on the CIFAR-I0 testing set as illus- trated in reconstructions demonstrates the model's 's capabil- ity. This empirical evidence suggests the CPMLA framework can function effectively for reconstruc- tion.
The high fidelity of
We further demonstrate CPMLA for image inpaint- ing. Let's assume we have an image represented by function
I R$ where Ω is the domain of the image, and [(x,8) y) gives the color coordinates (z) We optimize the ob- jective energy function Equation 1 to measure the difference between the restored region and the original image. To ensure that the restora- tion process does not alter the undamaged parts of the original image, we introduce constraint: u(x,)) y) = (xx M(x.y) 1.
Figure Image inpainting on the CelebA. The first 17 columns exhibit the inpainting results various iterations, while the last two columns visually compare the masked images and the originals. CPMLA faithfully inpaints the masked images.
[ (1(x,)) - u(x, v)) M (x, y)drdy
E(u) =
(13)
Experiments conducted on CelebA, are shown in Figure 5 The first 17 columns show inpainting results over optimization iterations, offering a dynamic view of the reconstruction process. The last two columns visually compare the masked images and the originals. Figure shows CPMLA successfully inpaints masked images from diverse initializations.
6 Limitations and Future works
In our experiments, estimating the Hessian can introduce bias to the optimal point. However, compared to the exact evaluation the inverse Hessian, this a trade-off we must make. While our experimental results demonstrate effectiveness for diverse sampling tasks, the mirror LSI assumption (Assumption 4.1 is rather general, as we cannot ensure that the target distributions of different sampling tasks satisfy this assumption, particularly in EBMs where the target distribution is highly complex. We note that, like other generative models, improvements could potentially be misused (e-g., for deepfakes). For future work, plan explore the deeper connection between sampling and optimization. For instance, can optimization techniques (e.g., adaptive step sizes like Adam, trust regions) accelerate sampling or correct bias? Additionally, higher-order discretizations (e.g., Rutte-Kutta) might improve convergence rates. We aim to investigate these questions and further advance the field of sampling and optimization.
7
Conclusions
This paper presented CPMLA, a sampling algorithm developed for Energy-Based Models (EBMs). The method utilizes Convex Potential Flow (CP-Flow) as a dynamic mirror map. allowing the sam- pling process to adapt to the underlying geometry of the data distribution. This adaptive mechanism facilitates sampling with vanishing bias and contributes to sampling efficiency. Theoretical analysis establishes the algorithm's 's convergence properties within the mirror Langevin dynamics framework. The algorithm demonstrated its applicability and effectiveness in image generation, reconstruction. and inpainting tasks. Experimental results indicated favorable performance concerning computational time and parameter count compared to related methods. In summary, CPMLA provides a principled approach to EBM sampling, integrating theoretical convergence properties with empirical perfor- mance. The method's capacity for adaptive sampling suggests its potential utility for enhancing the application of EBMs in various domains.
10

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Acknowledgements
Zitao Yang and Jun Li are supported by the National Natural Science Foundation of China (No. 72342016). Amin Ullah and Fuxin Li are supported by ONR/NAVSEA contracts N0014-21-1-2052 and N00024-10-D-6318.
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Langevin sampling Many discretizations of Langevin dynamics within Euclidean geometry have been studied in the literature, with non-asymplotic error bounds derived for various metrics like Kullback- Leibler divergence, Total Variation, and Wasserstein distance. The most extensively studied and A Related Work
scenarios include cases where the target distribution is m-strongly log-concave 12 those where relaxed log-concave [51]
Mirror Langevin Dynamics (MLD) extends standard Langevin dynamics by operating in a 'curved' Riemannian geometry defined by a convex potential, rather than the 'flat' Euclidean space. This allows the sampling process to adapt to the underlying geometry of the data distribution, which can lead to faster convergence. When the convex potential is quadratic, ML.D reduces exactly to standard Langevin dynamics. MLD has recently gained attention in the field of non- non-Euclidean geometry sampling due its superior convergence properties in constrained optimization problems. Introduced by [19] as a measure transformation of the classical Langevin dynamics, its convergence under relaxed log-concavity was investigated by [59]. where the authors demonstrated convergence to a Wasserstein ball with non-vanishing bias. I showed vanishing bias under similar conditions as the step size decreases. [8] studied convergence using similar functional inequalities, but without exploring practical applications.
Cooperative learning The cooperative learning concept, first introduced in [53] involves the joint maximum likelihood training of a ConyNet-EBM (55) and top-down generator [16] Similarly, [57
replaced the generator in the original CoopNets with a variational autoencoder (VAE) 26 to improve
inference efficiency. Our learning algorithm draws inspiration from the recent Coopflow approach
58). which collaboratively trains a Langevin flow and a normalizing flow to improve initial samples.
B Optimal Transport
In recent years, there has been increasing interest in applying optimal transport theory to generative modeling, which considers the training process as a task of minimizing the distance between two probability distributions. More specifically, the objective is to transform random distribution into target distribution that closely approximates the underlying data distribution, with the distance between these two distributions often quantified using the Wasserstein distance in the context of optimal transport. The Wasserstein p-distance between two probability measures and metric space M with finite p-moments is
1/r
W Wp(un =
inf d(a (ET())))
(14)
where r(p, v) is the set of all couplings of and v. shows that the Langevin dynamics manifests the gradient flow of the Kullback-Leibler divergence within the probability measure space, characterized by the Wasserstein metric, as elucidated through the Fokker-Planck equation. This observation establishes more substantial linkage between the realms of sampling and optimization; see also by the paper [40].
C Mirror Langevin Algorithm for Constrained Sampling
The mirror Langevin algorithm is powerful technique for sampling from complex distributions, particularly those with constraints or intricate geometries. It leverages the concept of mirror maps, which can adapt to the geometric structure of the target distribution, enabling efficient and accurate sampling. One notable application of the mirror Langevin algorithm is constrained sampling, where the goal is to draw samples from a population while adhering to specific conditions or constraints.
Constrained sampling involves drawing set of samples S from population U U1, U2 .... Un while satisfying predefined constraint conditions expressed as inequalities. The general form of these constraints be written as:
C(x) : (z) < 0, 1,2,..,
(15)
Here, (z) represents the constraint functions, and the goal is ensure that (x) 0 for all
22

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The mirror Langevin algorithm addresses constrained sampling by leveraging mirror maps that can adapt the geometry of the constraints. Specifically, we employ CP-Flow our dynamic mirror map, which utilizes Implicit Convex Neural Networks (ICNNs) to approximate arbitrary convex functions effectively. Given that the derivative a convex function is monotonic, we can ensure that the convergence of potential functions implies the convergence of the associated gradient fields, as stated in the following theorem:
Theorem C.1 (Optimality (Theorem 4 in (0))). Let be the Brenier potential of p and Y P, and let G. be a convergent sequence of differentiable, convex potentials, such that V X - Y in distribution. Then, VG,, converges almost surely to V..
In our context, where represents the unconstrained space and serves as the convex constraint, Theorem C.T guarantees the existence of convex potential whose derivative maps μ v. By leveraging the expressive nature of ICNNSs through CP-Flow, we can adapt to the intrinsic geometry of the constraints, resulting in accelerated convergence during constrained sampling. The mirror Langevin algorithm's 's ability to handle complex constraints makes it valuable tool for various applications beyond constrained sampling, such as sampling from EBMs, Bayesian inference, and more
D Derivative of CP-Flow
20 presents an alternative formulation of the gradient as the solution to a convex optimization problem, eliminating the need to differentiate through the log-determinant estimation process. By adapting the gradient formula from Appendix C in 66 to the context of convex potentials, and utilizing Jacobi's formula* alongside the adjugate representation of the matrix inverse f. we derive the following identity for any invertible matrix H parameterized by A:
θ det H 00
0H I tr det H ir adj(H))
011 00
HH
log det H 00
det H
H
E,
H
00
00
(16)
In the last equality, 20 apply the Hutchinson trace estimator using Rademacher random vector which is an unbiased Monte Carlo gradient estimator.
E Property of Convex Conjugate
G* is the convex conjugate of G. Then
VG(x) = xx(x) = arg - -
G" (x"
(17)
V (x*) = (x*) sug (z, x*) G(x)
Hence
I == VG and x² VG=VG (VG* (x*))
(18)
F Lemma of Convex Conjugate
Lemma F.1. Suppose we have a dualistic structure
= VGGE), VG (E')
(19)
G is the Legendre dual of G. which is defined as
G (E*)
max (E' .6" - ((')) &
(20)
Then the Hessian of G (E") is written as
ag
VVG (E*)
(21)
00
23

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which is the inverse of the Hessian of G(E)
og
VVG(E)
og
The last step is guaranteed by VG = VG
which can be shown from Appendix
E
G Details of Assumptions
have required the relative j--troong convexity of f with respect to G to guarantee convergence. However, work introduces Assumption
Previous investigations into the mirror Langevin algorithm 59
4.1
which relaxes this requirement and permits consideration non-strongly convex distributions.
Assumption 4.1 can be transformed as following. Taking h(x)) =
dp(z) then Vp (m()))
(pI(T p(x) log ((x) 1 TΦx)
/ p(a) V log
p(x) π (σ) [[
dr S 23
de =: (p) (22)
23
The D (pllx) term represents the KL divergence, often serving as measure of the distance between distributions P and æ. On the other hand, the right-hand side term, (p). signifies the weighted Fisher information. As demonstrated by (22). Langevin dynamics can be interpreted as the gradient flow of the KL divergence within the space of probability measures, equipped with the Wasserstein metric through the Fokker-Planck equation. This connection establishes a link between sampling and optimization. In this context, Assumption w.a.iiooon...aa can be perceived as the condition of gradient domination for KL-divergence in the Wasserstein metric.
Assumption 4.2 specifically relates to the interplay between the higher-order derivatives and the lower-order derivatives of the function. When the secondary derivative is small, it implies that the first derivative, which governed by the secondary derivative, also small. This property ensures the solution of continuous dynamics and Hessian stability [59], indicating that the underlying geometry does not undergo rapid changes. Moreover, this property is preserved under Fenchel conjugation (with the same parameter), affine transformation and summation also prevalent in quadratic optimizations, such as the interior point method, where it guarantees the convergence performance O (VC log 2).
(36). The concept of self-concordance is
when G(z) ||a|| we regain the conventional definition of a differentiable function being Lipschitz continuous with parameter This property has been extensively employed in prior research. In the case where G = f and a function f satisfies [F
In Assumption 4.3
S L. is
referred to as a barrier function (35). This property also emerges in the analysis of Newton's 's method in quadratic optimization scenarios.
In formal algorithms, it often necessary to have the y-relative smooth property in order to ensure convergence. -erelttvve smooth is defined by
[['0(=)] V (V' 'v (2') (aa(e))
(23)
(VVI(V VG (z' [[([)))))))---
However, the CPMLA utilizes distinct approach by employing deterministic gradient steps and stochastic steps separately. This allows for the utilization of weaker notion of smoothness assump- tion, namely Assumption 4.4 Unlike the 7-relative smooth, which necessitated Lipschitz continuity across different metrics ~ and could be unavoidable when discretizing the geometry, this definition of relative smoothness only considers the local metric V²G at single point.
H Proof of Theorem 4.5
Proof. We first clarify which parts of this proof are novel contributions and which are standard techniques adapted from prior work. Our primary contribution is the adaptation the convergence proof of Mirror Langevin Dynamics to setting where the mirror map is a learnable neural network
24

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(CP-Flow) and is trained jointly with the target distribution (EBM). The overall structure of the proof, including the use of the Bekker-Plancts equation and the mirror LSI, follows the framework established by [2I]. Our novel steps include explicitly accounting for the approximation errors from both the EBM and CP-Flow (ò2 and 63) and ensuring the proof holds under standard deep learning practices like gradient clipping.
For the proof, we analyze the convergence the sampling distribution Pt to the stationary distribution of the learned EBM, ps*. We therefore assume that pa- satisfies the 3-Mirror LSI (Assumption 4.1 as pø- is trained to be close approximation of the target data distribution x Pdata-
We decompose the total variation distance using the triangle inequality:
drv (pe, Pamaa) S drv (p..po) (po- qo-) + drv (90= PPata).
(24)
where pa- is the stationary distribution of the energy model and qu- is the optimal CP-Flow distribu- tion.
For the first term, following Lemma 1 in (21]. we analyze the differential form in primal space.
Lemma 1 in [1 tells that the differential form of Algorithm 2 primal space is
dx,
di
2 (X,) (x.) dW,
[-[vG((a. - ((()))) -/aaa
(s(x))
(25)
[V' T ( (x) (VGG dt v222))=++x2))x (x. dW.
= (V - (X. H V/ +1) di + -
where we denote f [X'
(V (Vf Vf (Xx)) and H - -v
This is weighted Langevin dynamics with shifted drift ja (the reason of the convergence to a biased limit).
Now consider the fokker-Planck equation for the conditional density Pejo (xz xo). For the drift b = H H ' + f. applying Lemma in [5T]. we have
(p)(x)
/ Oppi (x mo) pa (xO) dag
--v (eeo (V Go(z) (G(()) + (13.,y))G(()))
V (P)ooo(())] po (x0) dxo (ove) / - (pr (V-(Go(o) Go(z)V/(x))) (p.Go(z)) -
V (or pol(ko())dzo)
Pt (otee f P.GoV log
pr [ poniio(())daa))
dao
po- (r)
(ole / PiGo log
Pt dxo
(or (olcRo(())do))
Dg- (x)
(26)
where the last equality is because Now consider the KL-ddiveggecce
log P
V(log + fo-).
di DKa(pillpo--=
=
dp: di pa
log Pt dx +
pa. po- dl di
1 dpt dx /
dpt
log
Pt. dx
(27)
pg*
25

---

According to Equation 26 we have
d dt
DxL (pllpopo))
_ dpt dt po
log
Pt dæ
/ (oat
P.Guv
log
Pt
dxg
log
Pt. dx /v. (pr poloRodzoo)
Pt dr log Pt po
ps-
pa-
po droda
--
/ pole /
V log
Pt
Go,
V log
Pt
dxoda + I pole (i, V log
po
po-
Pt
Bpoo K
(p. log
Pt.
E..
log
Po-
[[20]_1
po-
$ PEp.
log
Pt
+ Epo.
[11i||
1 + 4 Bpt
V
log
Pt
Po-
p
[
38 13 (pe
(po.) + Epo. [|
2
(28)
The third equality refers to the integration by parts formula (VV((z), u(x)))z = =
-f
v(z)dd (z.) The first inequality is because x y S | + + llyll2 and the last inequality is from Mirror LSI (Assumption 4.1.
Under Assumption 4.2
4.4
let M == exp (2CD/ We have
p ||[
I 72 Boo.
VVG - VG (zo) (l (1
-tVf (zo) v2 / V (v'G(z)) (x+)) 1/2 dW,
=²² E
[[
(29)
< 2 2 | (zo) [ / (z.) [[
< 2 2
where the second inequality we use Itô isometry and (a bb2 < 2(a² + 1))) Then if & S h we have
d dt
3,3 -2 D (xcllllppo)) 2 [ 2
D kt (p.llo)
(30)
which is
d
(e 2 (KI Se (2 2 Md)
(31)
di
Integrate it for 0 < h.
38
D.D. (po.) DrI(pollpo))
e
38h
- (2
(32)
3
Then
(rI(pollpo.) se "
2
38h x
(33)
3,9
Iterating the recursion.
(xI(lxlp Se 3/ -rr.llll 33
(2-21212 + 4
(34)
Using Lemma 6 in 21 for initialization, picking the assumed stepsize, after k 2 (M-2dd/3 we have D DxL ([po.) < 8.
26

---

Using Pinsker's inequality, we establish:
6 dr $ V 2 DKi(pelpo-) < V2
(35)
The second term, drv (pe=, ((-(=----)), qa-), represents the approximation error from running the LMC for a finite number steps T instead of running it to convergence. This term can be bounded by summing the incremental changes over the 7 steps. Let po Po- and pr qa- be the distributions at the start and end of the sampling chain. By the triangle inequality for TV distance, we have
drv (Pe-: 90-) - Ein drv (Pt.Pt-1). We can analyze the single-step change using the Fokker- Planck equation:
Opr(x) V Øt
n² + Vepa(x)
(36)
2
From this, estimate the incremental change drv (pt, Pt-1) (pt,)ptt-)))) Summing over T steps gives total error of: dpv :0-) ~ O(T)).
v DKL O
The third term leverages the universality property of CP-Flow (Theorem 3 in 20 Given that the initial noise distribution absolutely continuous with respect to the Lebesgue measure, there exists a sequence go such that drv(go, Pdara) < 63 as n N.The optimality of CP-Flow (Theorem 4 in (20)) further guarantees almost sure convergence in distribution of qo, to the optimal Brenier map go-, ensuring that drv (ga- Pdata) < 63.
Combining these results, we conclude that drv (pe, Pdam) <6 / 82 + 83.
I Experimental Details
Parameter
Value
Dataset Size
50,000 samples
Dynamic Mirror Map VG
CP-Flow block
Depth
20
Dimh
32
VG Optimizer
Adam
VG Activation
Gaussian Softplus
VG Initial Learning Rate
0.005
EBM V
4 linear layers Adam
Vf Optimizer
f Activation
Swish 0.005
f Initial Learning Rate
Batch Size
128
Reported Results
After and 10 epochs
Table 4: Experimental setup for toy dataset
Table 4 outlines the experimental setup for Eight Gaussian toy dataset experiment. This setup includes a dataset size of 50,000 samples, using single CP-Flow block with the Gaussian Softplus activation function for the Dynamic Mirror Map VG. The optimizer for VG is Adam, with and an initial learning rate of 0.005. The EBM Vf comprises four linear layers with Swish activation, also utilizing the Adam optimizer, and initial learning rate of 0.005. The batch size for this setup is 128, with reported results after and 10 epochs.
Table 5 presents the experimental setup for the CIFAR-10, 10, SVHN, and CelebA datasets. We use multi-scale structure, involving two CP-Flow blocks, followed by invertible downsampling, and then another two CP-Flow blocks. All ICNN architectures had two hidden layers. The strong convexity
27

---

Parameter
Value
Datasets
CIFAR- 10, SVHN, CelebA
Dynamic Mirror Map VG
2 CP-Flow blocks
ICNN Architecture
2 hidden layers
GG Optimizer
Adam
VG Activation
Gaussian Softplus
VG Initial Learning Rate
5e-4
VG Weight Decay
5e-5
Mirror Steps
10
Mirror Step Size
le-2
EBM V
blocks with convolutional layers
ff Optimizer
Adam Swish
f Activation
f Initial Learning Rate
5e-3
Batch Size
128
Reported Results
After 200 epochs
Table 5: Experimental setup for CIFAR-10, SVHN, and CelebA datasets
CIFAR-10
SVHN
CelebA
Training time (hours)
10.5
12.6
40.2
Table 6: Training time each dataset eight 3090 GPUs
parameter (Section 2.2 design the negative energy function. The following Table 6 presents training time of our model on each dataset on eight 3090 GPUs.
lc-4. For the EBM in CPMLA, 3 blocks network is used
In the CPMLAprt setting, we first pretrain a CP-Flow on training examples, and then train a 30-step mirror Langevin sampling, whose parameters are initialized randomly, together with the pretrained CP-Flow by following Algorithm 2
J More Image Generation Results
In Section 5.2
we have shown generated examples from CPMLA. In this section, first show the examples generated by LMC on CIFAR-10, 10. Then we compare the examples generated by the CP-Flow only and CPMLAprt Figure and and our generated examples are meaningful.
9
We can see huge difference between two algorithms
Models
#Para
FID.
NT-EBM
23.8M
78.12
EBM-FCE
44.9M
37.30
GLOWW
44.2M
45.99
Flow++ only
28.8M
92.10
CoopFlow
45.9M
21.16
CPMLA
17.39M
20.85
Table 7: Model size performance comparison (lower is better). CPMLA lightweight yet highly effective, showcasing superior efficiency.
28

---

Figure 6: Image reconstruction on the CIFAR-10. The right column showcases the original images. The left and middle columns feature fow-generated images and the reconstructed images, respectively. We can see that the reconstruction is almost the same as the original one, which solidifies the stance that CPMLA functions effectively sampling algorithm.
Figure 7: LMC on CIFAR-10
(a) CIFAR-10
(b) SVHN
(c) CelebA
Figure 8: CP-Flow results
29

---

CPMLA CIFAR-10
(b) CPMLAprt CIFAR-10
(c) CPMLAprt CelebA
(d) CPMLAprt SVHN SVHN
Figure CPMLAprt results
30


Reviews:
[{"review_id": "oNMsiHkqYN", "reviewer": "Reviewer_EWFX", "summary": "The paper proposes the Convex Potential Mirror Langevin Algorithm (CPMLA), an algorithm that uses mirror langevin dynamics with a convex potential flow as dynamic mirror map for sampling energy-based models.", "questions": "1. I think more background on mirror Langevin dynamics may help\n2. can you comment on the requirement of strong convexity on page 3\n3. line 121-125 is a bit hard to follow\n4. can you discuss the computational cost compared to previous works\n5. The proofs are well-rewritten but can you make it clearer which part is novel and which part is standard technique\n6. how is it compared with other types of generated models like diffusion or flow-matching models? what are the advantages and disadvantages", "limitations": "yes", "rating": 5, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 4, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths\n1. very clear writing\n2. interesting method that combines the merits from several lines of work\n3. valid theoretical analysis\n4. reasonable and convincing experimental results\n\nno significant weakness. see question section", "was_revised": false, "final_justification": "", "rebuttal": "We are grateful for your positive assessment and your constructive questions.\n\n**Q1: I think more background on mirror Langevin dynamics may help.**\n\n**A1:** We agree that a more detailed introduction to mirror Langevin dynamics (MLD) will benefit readers. In the revised version, we will expand the Related Work section to provide a self-contained explanation of MLD, including its geometric intuition and how it generalizes standard Langevin dynamics. To clarify, MLD extends standard Langevin dynamics by operating in a 'curved' Riemannian geometry defined by a convex potential, rather than the 'flat' Euclidean space. When the potential is quadratic, MLD reduces exactly to standard Langevin dynamics.\n\n**Q2: Can you comment on the requirement of strong convexity on page 3?**\n\n**A2:** As stated in Line 115, this strong convexity of $G$ is essential: to ensure that $\\nabla G$ defines a valid mirror map, $G$ must be strongly convex. \n\n**Q3: Line 121-125 is a bit hard to follow.**\n\n**A3:** This paragraph explains how CP-Flow is trained and the tricks for avoiding computing the full Hessian matrix. We first explain the motivation behind these tricks, then describe the detailed method and implementation. To improve readability, we will present the formulas in a more step-by-step and less abrupt manner, making the derivation easier to follow.\n\n**Q4: Can you discuss the computational cost compared to previous works?**\n\n**A4:** We have conducted new experiments on CIFAR-10 to provide a direct comparison with our main baseline, CoopFlow.\n\nWe compared them from two perspectives:\n\n1.  **Total Training Time under Memory Constraints:** We configured both methods to use the maximum possible batch size that would fit into 24GB of VRAM. Under this practical, resource-constrained setting, the estimated total training time for CoopFlow is approximately **38 hours**, which is significantly longer than CPMLA's **10.5 hours**.\n\n2.  **Per-Iteration Cost:** To provide a more direct comparison, we also estimated the per-iteration time for a fixed batch size. We calculated that CoopFlow would take approximately **12.0s/iter**, while CPMLA takes **15.7s/iter**.\n\nWhile CPMLA is marginally slower on a per-iteration basis, this is expected, as the CP-Flow is inherently more computationally complex than the standard Normalizing Flow used in CoopFlow.\n\nIn summary, the slightly higher per-iteration cost is a well-justified trade-off. The significantly shorter total training time under realistic hardware limitations highlights CPMLA's superior efficiency for both generation and sampling, particularly when computational resources are constrained.\n\n**Q5: The proofs are well-rewritten but can you make it clearer which part is novel and which part is standard technique?**\n\n**A5:** In Line 53-60, we have stated the theoretical contributions and improvements. In the appendix, we will restructure the proofs and add commentary to explicitly distinguish our novel contributions from standard proof techniques. \n\n**Q6: How is it compared with other types of generated models like diffusion or flow-matching models? What are the advantages and disadvantages?**\n\n**A6:** This is an important question for positioning our work. \n*   **Advantages of CPMLA/EBMs:** EBMs offer more modeling flexibility as they only need to specify an unnormalized energy function, unlike normalizing flows or diffusion models which require specific architectures or tractable noise processes. Our method, by improving EBM sampling, makes this flexibility more practical.\n*   **Disadvantages:** Training EBMs can be more challenging due to the intractable partition function. While models like flow-matching currently achieve state-of-the-art sample quality (FID scores), our work closes the gap for EBMs while offering the aforementioned flexibility. Our primary contribution is to the EBM literature, making them more competitive."}, {"review_id": "0Ftv1j7Nvo", "reviewer": "Reviewer_GaGj", "summary": "This paper combines Alternative Forward Discretization Scheme (MLA_AFD) of Mirror Langevin Monte Carlo and CP-Flow dynamic, in order to achieve an efficient sampler. This sampler is then used to training models with intractable normalizing constant. Theoretical analysis is conducted, along with intensive simulation studies. Its simulation results are better than CoopFlow.", "questions": "Can you discuss the choice of alpha?\n\nCan you provide more explanation of $G_\\vartheta$? Does $\\nabla G_\\vartheta$ approximate an OT before two measure $\\mu$ and $\\nu$? If so, what are $\\nu$ and $\\mu$?\n\nIn Algorithm 2, both $G_\\vartheta$ and $G^*_\\vartheta$ appear. Is it typo? Algorithm 2 also writes \"Number of steps in dual space $\\eta$.\" I suppose it is a typo.\n\nAccording to (8), the variance in dual space depends on $y_t$ or $x_t$, which is the samplers. But in algorithm 2, the variance of $\\xi$ depends on the data point $x_i$. I am confused.\n\nAlgorithm 2 has two loops: inner loop is about sampling, and the outer one is about training of $\\vartheta$ and $\\theta$. I don't see any discussion about the number of iterations or step size for the outer loop. \n\nAssumption 4.1: what is \\pi? Do you mean $p_{theta}$ or $p_{data}$.\n\nFor all assumptions, the indices $\\theta$ and $\\vartheta$ are omitted. Do you mean the assumptions hold for any $theta$ and $\\vartheta$? If so, how reasonable are these assumptions?\n\nWhat is the index $t$ for $p_t$ in Thm 4.5?\n\nI cannot understand why the convergence result (Thm 4.5) doesn't depend on the number of iterations and step size for the outer loop. I also don't understand why it doesn't depend on the choice $f_\\theta$. Eventually, the algorithm samples from $p_\\theta$ for some $\\theta$ (say, the last iteration of $\\theta$ in the outer loop). If the family of $p_\\theta$ has a huge gap to $p_{data}$, how can one retrieve $p_{data}$?\n\nIn the proof, line 551: I can't understand the relationship between $d_{TV}(p_t,p_{t-1})$ and  $d_{TV}(p_{\\theta^*},p_{\\vartheta^*})$", "limitations": "Yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 1, "significance": 2, "originality": 3, "strengths_and_weaknesses": "Strength: The presented simulation results are impressive.\n\nWeakness: \nThe intuition of using CP-Flow is not clear. What is its theoretical or intuitive advantage over other choices, say normalizing flow?\nMany notations in the paper are not clear to me. The readability of this paper needs to improve.\nI am not 100% convinced by the theory.\nThe efficiency presented by this paper is mainly about the inference stage. There is no discussion aboutthe  training cost.", "was_revised": true, "final_justification": "The author has carefully addressed my question. I had some misunderstanding of the paper. With all typos corrected and better representations, I think the paper is close to the standard of acceptance.", "rebuttal": "We thank you for your detailed feedback.\n\n**W1: The intuition of using CP-Flow is not clear. What is its theoretical or intuitive advantage over other choices, say normalizing flow?**\n\n**A1:** CP-Flow is a method with a well-established theoretical foundation. Its universality and optimality are guaranteed by Brenier's Theorem [1]. The key advantage of using a CP-Flow, implemented via an ICNN, is that the learned potential function $G_\\vartheta$ is **guaranteed to be convex** by its architecture. This convexity is crucial for the mirror map in MLD to be well-defined and for the theoretical analysis to hold. A standard normalizing flow does not inherently provide a convex potential, which would make it unsuitable as a mirror map in our framework without significant modifications and loss of guarantees.\n\n[1] Chin-Wei Huang, Ricky T. Q. Chen, Christos Tsirigotis, and Aaron Courville. Convex potential flows: Universal probability distributions with optimal transport and convex optimization. Learning, 2020.\n\n**W2: Training Cost.**\n\n**A2:** Thank you for raising this point about training cost. We have conducted new experiments on CIFAR-10 to provide a direct comparison with our main baseline, CoopFlow.\n\nWe compared them from two perspectives:\n\n1.  **Total Training Time under Memory Constraints:** We configured both methods to use the maximum possible batch size that would fit into 24GB of VRAM. Under this practical, resource-constrained setting, the estimated total training time for CoopFlow is approximately **38 hours**, which is significantly longer than CPMLA's **10.5 hours**.\n\n2.  **Per-Iteration Cost:** To provide a more direct comparison, we also estimated the per-iteration time for a fixed batch size. We calculated that CoopFlow would take approximately **12.0s/iter**, while CPMLA takes **15.7s/iter**.\n\nWhile CPMLA is marginally slower on a per-iteration basis, this is expected, as the CP-Flow is inherently more computationally complex than the standard Normalizing Flow used in CoopFlow. In summary, the significantly shorter total training time under realistic hardware limitations highlights CPMLA's superior efficiency.\n\n**Q1: Can you discuss the choice of alpha?**\n\n**A1:** $\\alpha$ is a regularization parameter (introduced in Line 116) that ensures the Hessian of the potential $\\nabla^2 G_\\vartheta$ is strictly positive definite, i.e., $\\nabla^2 G_\\vartheta(x) \\succeq \\alpha I$. In practice, $\\alpha$ is a small constant (1e-4) that we treat as a hyperparameter, which is consistent with previous literature [1]. We will clarify its role and how it's set in the experimental section.\n\n[1] Chin-Wei Huang, Ricky T. Q. Chen, Christos Tsirigotis, and Aaron Courville. Convex potential flows: Universal probability distributions with optimal transport and convex optimization. Learning, 2020.\n\n**Q2: Can you provide more explanation of $G_\\vartheta$? Does $\\nabla G_\\vartheta$ approximate an OT before two measure $\\mu$ and $\\nu$? If so, what are $\\nu$ and $\\mu$?**\n\n**A2:** Yes, $\\nabla G_\\vartheta$ indeed approximates the optimal transport (OT) map, where $\\mu$ is the noise distribution and $\\nu$ is the data distribution [1]. In our framework, it also serves as the mirror map, mapping from the primal to the dual space.\n\n[1] Chin-Wei Huang, Ricky T. Q. Chen, Christos Tsirigotis, and Aaron Courville. Convex potential flows: Universal probability distributions with optimal transport and convex optimization. Learning, 2020.\n\n**Q3: In Algorithm 2, both $G_\\vartheta$ and $G_\\vartheta^*$ appear. Is it typo? Algorithm 2 also writes \"Number of steps in dual space $\\eta$.\" I suppose it is a typo.**\n\n**A3:** The use of $G_\\vartheta$ and $G_\\vartheta^*$ is not a typo. $\\eta$ is indeed a typo; it should refer to the step size, not the number of steps. We will correct this in the revision.\n\n**Q4: According to (8), the variance in dual space depends on $y_t$ or $x_t$, which is the samplers. But in algorithm 2, the variance of $\\xi$ depends on the data point $x_i$.**\n\n**A4:** The variance in the dual space should indeed depend on the current sample $x_t$, not the data point $x_i$. We will correct this in the revised version.\n\n**Q5: Algorithm 2 has two loops... I don't see any discussion about the number of iterations or step size for the outer loop.**\n\n**A5:** As stated in the beginning of Algorithm 2, the inner loop corresponds to the number of mirror Langevin sampling steps $T$. The outer loop iterates over each batch; after all batches are processed, the next epoch begins. Therefore, there is no step size associated with the outer loop.\n\n**Q6: Assumption 4.1: what is $\\pi$? Do you mean $p_\\theta$ or $p_{\\text{data}}$. For all assumptions, the indices $\\theta$ and $\\vartheta$ are omitted. Do you mean the assumptions hold for any theta and $\\vartheta$?**\n\n**A6:** As stated in Line 171, $\\pi$ in Assumption 4.1 refers to the target distribution, which is $p_{\\text{data}}$. The assumptions are stated without indices because they are satisfied by the natural properties of neural networks and the training process. Please see the response to W1 of Reviewer JSTt on assumptions for more details on their validity.\n\n**Q7: I cannot understand why the convergence result (Thm 4.5) doesn't depend on the number of iterations and step size for the outer loop...**\n\n**A7:** Theorem 4.5 provides a non-asymptotic bound on the distance to the target distribution $p_{\\text{data}}$ after a sufficient number of iterations. The number of iterations and step sizes are implicitly embedded in the conditions required to reach the error bounds presented. The theorem essentially states that *if* the EBM and the CP-Flow are trained to a certain approximation accuracy (the epsilon terms), *then* the resulting sampler will be within a certain TV distance of the true data distribution. We will rephrase the discussion around Theorem 4.5 to make it clear that it's a characterization of the *best achievable error* of our framework.\n\n**Q8: In the proof, line 551: I can't understand the relationship between $d_{\\text{TV}}(p_t, p_{t-1})$ and $d_{TV}(p_{\\theta^\\*},q_{\\vartheta^\\*})$**\n\n**A8:** $d_{TV}(p_{\\theta^\\*},q_{\\vartheta^\\*})$ is the approximation error of the EBM, which measures the gap between the model with optimal parameters and the target distribution. This can be quantified using the finite-step Fokker-Planck equation. Let $p_0 = p_{\\theta^\\*}$ and $p_T = q_{\\vartheta^\\*}$, then $d_{TV}(p_{\\theta^\\*}, q_{\\vartheta^\\*}) \\leq \\sum_{t=1}^T d_{TV}(p_t, p_{t-1}) \\sim O(T \\eta)$. This result follows from the triangle inequality for the TV distance. Since $T$ is a finite constant, the order of magnitude does not change. We will clarify this point in the revision."}, {"review_id": "omUtoPBunB", "reviewer": "Reviewer_SLf1", "summary": "This paper proposes CPMLA, a mirror Langevin method with a data-driven mirror map to improve EBM sampling. It achieves exponential convergence in theory and shows superior sampling quality and inference efficiency in experiments.", "questions": "1. I suspect training time is an issue for CPMLA. The number of Mirror Langevin steps \nT needs to be large enough to obtain accurate samples, and the number of parameter update iterations must also be sufficient to learn an effective mirror map. How do the authors balance them? This point needs further discussion.\n\n2. Related to the previous question, beyond sampling time, it is important to compare the training time of CPMLA with other baseline methods. If CPMLA’s training time is significantly longer, then the advantages in sampling quality need to be assessed more carefully.\n\n3. Regarding Algorithm 1, does it require storing the entire Hessian H? For high-dimensional sampling problems, this could lead to substantial memory overhead.\n\n4. Please clarify why a direct comparison between NCSN++ and CPMLA is inappropriate.\n\n5. In Algorithm 2, why is the noise \\$\\xi\\$ in each subroutine sampled from \\$\\nabla ^2 G(x_i)\\$ rather than from the current \n \\$\\nabla ^2 G(x)\\$ at the immediate \\$x\\$? Although this choice may improve the algorithm’s efficiency, is there any theoretical guarantee or intuitive explanation for doing so? Can this be verified?\n\n\n6. Typos: In the input of Algorithm 2, I guess eta should refer to the step size, not the number of steps; in line 121, “model density” may be incorrect; in Algorithm 2, after updating y, the subscript k should be incremented.", "limitations": "yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "Strengths:\n\n1. The data-driven mirror map makes the mirror map better suited to the data characteristics, enabling faster sampling.\n\n2. Both the theoretical analysis and experimental results demonstrate the efficiency of CPMLA in sampling.\n\nWeaknesses:\n\n1. The algorithm appears to simply combine mirror map updates with sampling. It requires repeatedly sampling from the initial point, without fully leveraging previously obtained samples $x_{out}$", "was_revised": true, "final_justification": "Recommended score:4. My primary concerns regarding training time and memory overhead have been addressed. However, I maintain some reservations about the experimental validation. The current evaluation appears limited in scope. This method involves JVP and Hessian approximations, and thus requires more rigorous validation through: (1) evaluation on larger-scale datasets to properly assess computational efficiency, and (2) comprehensive ablation studies (particularly on JVP computation strategies and Hessian approximation approaches) to demonstrate its effectiveness.", "rebuttal": "Thank you for your feedback and focus on the practical aspects of our algorithm.\n\n**W1: The algorithm appears to simply combine mirror map updates with sampling. It requires repeatedly sampling from the initial point, without fully leveraging previously obtained samples $x_{\\text{out}}$.**\n\n**A1:** Our algorithm does re-initialize the MCMC chain for each parameter update. This is a standard practice in settings like Persistent Contrastive Divergence (PCD) to avoid chain collapse and ensure that the gradients for the model parameters are estimated based on samples from the current model distribution, preventing feedback loops from stale samples. Relying on previous samples may also lead to insufficient diversity in generated samples. Moreover, our use of CP-Flow accelerates the sampling process. We will add a sentence to clarify the motivation for this design choice.\n\n**Q1 & Q2: Training Time.**\n\n**A2:** Thank you for raising this point about training time. We have conducted new experiments on CIFAR-10 to provide a direct comparison with our main baseline, CoopFlow.\n\nWe compared them from two perspectives:\n\n1.  **Total Training Time under Memory Constraints:** We configured both methods to use the maximum possible batch size that would fit into 24GB of VRAM. Under this practical, resource-constrained setting, the estimated total training time for CoopFlow is approximately **38 hours**, which is significantly longer than CPMLA's **10.5 hours**.\n\n2.  **Per-Iteration Cost:** To provide a more direct comparison, we also estimated the per-iteration time for a fixed batch size. We calculated that CoopFlow would take approximately **12.0s/iter**, while CPMLA takes **15.7s/iter**.\n\nWhile CPMLA is marginally slower on a per-iteration basis, this is expected, as the CP-Flow is inherently more computationally complex than the standard Normalizing Flow used in CoopFlow.\n\nIn summary, the slightly higher per-iteration cost is a well-justified trade-off. The significantly shorter total training time under realistic hardware limitations highlights CPMLA's superior efficiency for both generation and sampling, particularly when computational resources are constrained.\n\n**Q3: Regarding Algorithm 1, does it require storing the entire Hessian $H$?**\n\n**A3:** No, we do not store the full Hessian matrix. For high-dimensional data, this would be infeasible. In fact, Algorithm 1 is specifically designed to avoid storing the full Hessian [1]. We use Hessian-vector products, which can be computed efficiently using standard automatic differentiation libraries (e.g., a \"double backward\" pass) without ever instantiating the full Hessian. We will add this motivation in the revision.\n\n[1] Chin-Wei Huang, Ricky T. Q. Chen, Christos Tsirigotis, and Aaron Courville. Convex potential flows: Universal probability distributions with optimal transport and convex optimization. Learning, 2020.\n\n**Q4: Please clarify why a direct comparison between NCSN++ and CPMLA is inappropriate.**\n\n**A4:** NCSN++ belongs to a different family of generative models than EBMs. Our work's main goal is to improve the sampling efficiency and quality for the EBM family, making them more competitive. While both can generate data, their underlying principles and trade-offs differ. Score-based models often excel in raw generation quality (FID scores) on benchmark datasets, but EBMs offer greater modeling flexibility since the energy function is not constrained to be the gradient of a density (a score).\n\n**Q5: In Algorithm 2, why is the noise $\\xi$ in each subroutine sampled from $\\nabla^2 G(x_i)$ rather than from the current $\\nabla^2 G(x)$ at the immediate $x$?**\n\n**A5:** While using the Hessian at $x_i$ could in principle improve the algorithm's efficiency, I must admit this is a typo. The variance in the dual space should indeed depend on the current sample $x_t$, not the data point $x_i$. Otherwise, the geometry of the target distribution would be leaked via the Hessian at the very beginning, which violates the basic principle of Langevin sampling. We will correct this and other typos in the revised version."}, {"review_id": "gHeGOx2Ih9", "reviewer": "Reviewer_JSTt", "summary": "This paper introduces a novel framework for energy-based models using Mirror Langevin Descent with a convex mirror map learned via an input-convex neural network. By preconditioning Langevin dynamics through learned mirror mappings, the method accelerates sampling and improves efficiency. It provides the first theoretical guarantees for mirror Langevin algorithms with neural network-based mirror maps. Empirically, the approach achieves high-quality generation across benchmarks such as CIFAR-10, SVHN, and CelebA.", "questions": "See weakness.\n\nFurther questions:\n\nIn Lines 160-165, could the authors elaborate on how this trick affects sample quality or convergence in practice?", "limitations": "yes", "rating": 4, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 2, "originality": 2, "strengths_and_weaknesses": "Strengths:\n\nThis paper presents a novel framework that combines Mirror Langevin Descent with a learned convex mirror map, enabling efficient sampling for energy-based models. It provides the first convergence guarantees for mirror Langevin algorithms with neural network-based mirror maps and demonstrates empirical performance on standard benchmarks.\n\nWeakness:\n\n- The authors provide some explanation of Assumptions 4.1–4.4 in Appendix G, but do not offer justification for why these assumptions are satisfied. Could you give some examples? \n\n- I am somewhat confused about the sources of approximation error for the EBM and the CP-Flow in Theorem 4.5. It would be helpful if the authors could summarize these errors clearly, perhaps in the form of a lemma, and provide sufficient justification.\n\nThe proposed method is not competitive with score-based models such as NCSN++, which achieve significantly better FID scores.\n\n\nMinors:\n\n- Line 54: whats the relaxed log-concavity assumptions?\n\n- whats H in Equation 6?", "was_revised": true, "final_justification": "The authors have addressed my question and promised to improve the clarity in the final version.", "rebuttal": "Thank you for recognizing the novelty of our framework and theoretical guarantees. Your questions are very helpful for strengthening the paper.\n\n**W1: The justification for the assumptions is weak.**\n\n**A1:** Our assumptions are standard and common in the analysis of Langevin-type algorithms [1, 2]. Below we provide a concise justification for each one.\n\n*   **Assumption 4.1 ($\\beta$-Mirror Log-Sobolev Inequality):** This is a foundational assumption about the target data distribution, widely adopted in the theoretical analysis of mirror Langevin methods. It is difficult to verify theoretically or empirically for complex, high-dimensional, unknown target data distributions in our image generation experiments, so making this assumption is reasonable and necessary.\n\n*   **Assumption 4.2 ($\\zeta$-Self-Concordance):** We empirically validated this assumption on CIFAR-10. We estimate the Frobenius norms of the Hessian $\\nabla^2 G_\\vartheta(x)$ and five random directional third derivatives $\\nabla^3 G_\\vartheta(x)[\\vec{v}]$ (for five random directions $\\vec{v}$) using Hutchinson's estimator. This technique avoids instantiating large matrices by relying on efficient Hessian-vector products. We compute the proxy metric $\\hat{\\zeta}\\_{\\text{proxy}} = \\frac{\\|\\nabla^3 G\\_\\vartheta(x)[\\vec{v}]\\|\\_F}{\\|\\nabla^2 G\\_\\vartheta(x)\\|\\_F^{1.5} + \\epsilon}$ We tracked this metric across all training checkpoints. The value of $\\hat{\\zeta}_{\\text{proxy}}$ consistently remains small and stable (in the range $[10^{-4}, 10^{-2}]$), providing strong empirical support that this assumption holds in practice.\n\n*   **Assumption 4.3 (L-Relative Lipschitz):** The potential function $G$ is designed to be strongly convex, meaning its Hessian $\\nabla^2 G(x) \\succeq \\alpha I$ for some $\\alpha > 0$. In practice, we use **gradient clipping** on the EBM, which ensures that $||\\nabla f(x)||$ is bounded by a constant $C$. This directly leads to $||\\nabla f(x)||_{[\\nabla^2 G(x)]^{-1}} \\le (1/\\sqrt{\\alpha}) ||\\nabla f(x)|| \\le C/\\sqrt{\\alpha}$. Thus, the assumption holds by setting $L = C/\\sqrt{\\alpha}$.\n\n*   **Assumption 4.4 (Weaker $\\gamma$-Relative Smooth):** Similarly, the gradient of EBM, $\\nabla f$, is Lipschitz with some constant $L_f$ (determined by the network architecture and enforced by weight decay and gradient clipping). The potential $G$ is also smooth. This allows us to bound the relative smoothness, and the assumption holds by setting $\\gamma = L_f / \\alpha$.\n\nWe will integrate these justifications into the main body of the paper. \n\n**W2: I am somewhat confused about the sources of approximation error for the EBM and the CP-Flow in Theorem 4.5.**\n\n**A2:** As stated in Line 56, the approximation error comes from modeling both the CP-Flow and the EBM with deep neural networks. It is related to the expressive power of the EBM's and CP-Flow's neural network architecture. We will add a clearer paragraph to summarize this.\n\n**W3: The proposed method is not competitive with score-based models such as NCSN++.**\n\n**A3:** As discussed in Table 1, a direct comparison is not appropriate. NCSN++ is a score-based diffusion model, which belongs to a different family of generative models than EBMs. While both can generate data, their underlying principles and trade-offs differ. Score-based models often excel in raw generation quality (FID scores) on benchmark datasets, but EBMs offer greater modeling flexibility since the energy function is not constrained to be the gradient of a density (a score). Our work's main goal is to improve the sampling efficiency and quality for the EBM family, making them more competitive.\n\n**W4: what's the relaxed log-concavity assumptions? what's $H$ in Equation 6?**\n\n**A4:** As stated in Line 491, the \"relaxed log-concavity\" refers to the target distribution not needing to be log-concave (relative $\\mu$-strongly convex of $f$ to $G$). Assumption 4.1 is a form of relaxed log-concavity. As stated in Line 121, $H$ in Equation 6 is the Hessian of the potential $G$.\n\n**Q2: In Lines 160-165, could the authors elaborate on how this trick affects sample quality or convergence in practice?**\n\n**A2:** This trick has no effect on convergence, but it simplifies computation and improves efficiency in our experiments. Since the diffusion term already contains the coefficient $\\sqrt{2\\eta \\nabla^2 G_{\\vartheta}(x_i)}$, we are simply moving the Hessian part into the Gaussian noise generation, which is essentially equivalent. Similar approaches have been adopted in related literature [3]. In practice, this trick avoids repeated and complex matrix square root computations, thus improving sampling efficiency.\n\n[1] Jiang, Q. (2021). Mirror Langevin Monte Carlo: the case under isoperimetry. Advances in Neural Information Processing Systems, 34, 715-725.\n\n[2] Ahn, K., & Chewi, S. (2021). Efficient constrained sampling via the mirror-Langevin algorithm. Advances in Neural Information Processing Systems, 34, 28405-28418.\n\n[3] Srinivasan, V., Wibisono, A., & Wilson, A. (2024, June). Fast sampling from constrained spaces using the Metropolis-adjusted Mirror Langevin algorithm. In The Thirty Seventh Annual Conference on Learning Theory (pp. 4593-4635). PMLR."}, {"review_id": "mrDgtWy3EN", "reviewer": "Reviewer_T4a4", "summary": "This paper proposes a new sampling algorithm for energy-based models. The improved efficiency, according to the authors, mainly comes from the improved sampling geometry by a dynamic mirror map modeled by a neural network model. The authors provide a theoretical analysis proving convergence under a set of assumptions. In addition, the paper presents extensive experiments demonstrating strong empirical performance, particularly in terms of convergence speed and sample quality.", "questions": "I often found the writing and presentation of the paper quite confusing. Below are some representative issues, though similar concerns appear throughout. I strongly recommend a comprehensive rewrite to more clearly highlight the main ideas, motivation, and contributions.\n1. Lines 91-98: It looks like this paragraph is talking about how to estimate the parameter $\\theta$. But what is the ``log-likelihood objective''? What is $p_{data}$? And using $\\mathcal{L}$ to denote a gradient is very confusing. The $L$ notation is used later in equation (5) to denote a linear layer. \n2. Equation (4): What is $X$? How is it related to $x$?\n3. Section 2.2 looks like it is introducing the ``Convex Potential Flow'' yet it ends by discussing implementation details for computing the derivative of the log-determinant. The logical structure of this section feels disorganized and difficult to follow.\n4. Line 116: The $G_{\\theta}$ is used with an addition of a regularization term, which could very significant in some cases, especially if $G_\\theta$ has a negative singular value. This modification may substantially affect accuracy. In addition, I disagree with the statement that $\\nabla^2 G_{\\alpha} \\succeq \\alpha I$. \n5. Figure 5: The \"dotted line\" should be \"dashed line\"?\n6. Equation (7): Again, notations $X_t$ and $Y_t$ are not defined. \n7. I appreciate the convergence result, but it is based on four distinct assumptions. What do these assumptions mean in practice? Are they reasonable in the context of neural networks and real-world distributions? Are there known examples where these assumptions hold? These are critical questions and should not be relegated to the appendix—they deserve a clear discussion in the main text.\n8. Many details are missing. For example, what does ``one iteration'' of CPMLA mean in practice? According to Algorithm 2, there are two loops, and depending on which one is being referred to, the interpretation of “one iteration” could differ significantly.\n9. The sentence just below Line 224 references Table 7, which appears in the appendix. Referencing new results presented only in the appendix in the main body is not a good idea.", "limitations": "The authors discuss some limitations of their work in Section 6. I appreciate this inclusion and think that the paper does not pose any potential negative societal impact.\n\nFor open questions that the authors are not yet able to answer, I encourage the authors to explicitly include these points in the limitations section.", "rating": 4, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 1, "significance": 2, "originality": 3, "strengths_and_weaknesses": "This paper is targeted at addressing an looking-important problem in machine learning, improving the sample efficiency for energy-based models. \n\nIt has both clear strengths and notable weaknesses.\n\nFirst and foremost, the writing is not very accessible, particularly for readers who are not experts in simulation. The paper lacks a comprehensive introduction to the overall setup, notations, and necessary preliminaries. Many symbols are introduced without proper definitions, and the main ideas are often buried under layers of technical details. Additionally, many technical components are presented without sufficient motivation, making the paper difficult to follow. I strongly recommend reorganizing the structure to better highlight the motivation and main contributions early on.\n\nIn terms of originality, as someone not an expert in simulation, the idea of using a mirror map (modeled via a neural network) to improve sampling efficiency looks interesting. However, I am not well-positioned to assess the novelty of this idea within the broader simulation literature.\n\nRegarding theoretical guarantees, the paper establishes convergence under four assumptions (Assumptions 4.1–4.4). Assumptions 4.2–4.4 are relatively common in the optimization literature, but Assumption 4.1 is less common. However, I would not expect Assumptions 4.2-4.4 to hold in general when using deep neural networks. It would also be helpful if the authors provided more intuitive explanations for Assumption 4.1: why it is reasonable in their setting, and what it implies about the behavior of their algorithm. The convergence result itself is not particularly surprising under these assumptions, but it is a valuable contribution.\n\nThe experimental results are quite impressive. The proposed method appears to work well in practice. However, I would have appreciated more details on the experimental setup, particularly regarding how different baseline methods were configured and compared.", "was_revised": true, "final_justification": "The authors' response solved part of my questions and concerns about this paper. For this reason, I have adjusted my assessment and raised 1 point.", "rebuttal": "Thank you for your feedback.\n\n**Q1. Lines 91-98: What is the \"log-likelihood objective\"? What is $p_{\\text{data}}$? Using $\\mathcal{L}$ to denote a gradient is very confusing.**\n\n**A1:** $p_{\\text{data}}$ is the empirical data distribution defined by the training dataset. The \"log-likelihood objective\" refers to maximizing the log-likelihood of the data under the model $p_\\theta$, which is the standard objective for training an EBM. The gradient of this objective is what's shown. We have already used $\\mathcal{L}$ and $L$ to distinguish between the gradient of the log-likelihood and the linear layer. We will change this notation to $\\nabla_\\theta \\log p_\\theta(x)$ to be unambiguous. \n\n**Q2 & Q5. Equation (4): What is $X$? How is it related to $x$? Equation (7): Again, notations $X_t$ and $Y_t$ are not defined.**\n\n**A2 & A5:** $X$ is a random variable, and $x$ is a specific realization. $X_t$ and $Y_t$ are the stochastic processes in the primal and dual (mirror) spaces, respectively.\n\n**Q3. Section 2.2 ... feels disorganized.**\n\n**A3:** Section 2.2 is structured as follows: first, we present the theoretical guarantees of CP-Flow; next, we describe the practical implementation of CP-Flow; and finally, we explain how to optimize CP-Flow. We will reorganize and clarify this section in the revision to make it more accessible.\n\n**Q4. Line 116: ...I disagree with the statement that $\\nabla^2 G_\\alpha \\succeq \\alpha I$.**\n\n**A4:** The statement is correct. The modified potential $G_\\alpha(x) = G_\\vartheta(x) + (\\alpha/2) \\|x\\|^2$ has a Hessian $\\nabla^2 G_\\alpha(x) = \\nabla^2 G_\\vartheta(x) + \\alpha I$. Since $G_\\vartheta$ is convex ($\\nabla^2 G_\\vartheta(x) \\succeq 0$), its Hessian is positive semi-definite, so $\\nabla^2 G_\\alpha(x) \\succeq \\alpha I$. This makes the modified potential strongly convex.\n\n**Q6. What do these assumptions mean in practice? ... they deserve a clear discussion in the main text.**\n\n**A6:** These assumptions are stated in the main text as being adopted from prior work [1], and they are standard in the study of Langevin convergence. We will move the discussion of these assumptions to the main paper and provide much more intuition and justification. Please see the validation of assumptions at the response to W1 of Reviewer JSTt.\n\n[1] Qijia Jiang. Mirror langevin monte carlo: the case under isoperimetry. Neural Information Processing Systems, 2021.\n\n**Q7. Many details are missing. For example, what does \"one iteration\" of CPMLA mean in practice?**\n\n**A7:** \"One iteration\" in our experiments refers to one update of the parameters $\\theta$ and $\\vartheta$ (the outer loop in Algorithm 2). We will clarify this terminology in the revision.\n\n**Q8. The sentence just below Line 224 references Table 7, which appears in the appendix...**\n\n**A8:** We will integrate the key results from the appendix that are referenced in the main text into the main body of the paper.\n\n**Q9. On Novelty and Originality.**\n\n**A9:** Our key and novel contribution is being the first to propose and successfully train a data-driven, neural network-based mirror map for the mirror Langevin algorithm in the context of generative modeling. We also provide the first theoretical convergence guarantees for this specific setting. Previous works have typically relied on handcrafted or fixed mirror maps, and were limited to simulations on simple toy models. By learning the dynamic mirror map with a CP-Flow, we provide a new perspective for accelerating sampling for complex, high-dimensional distributions in EBMs."}]

Author rebuttal:

Meta-review: This paper proposed a a new method for sampling from energy-based models which combines the mirror-langevin algorithm and a neural network-based mirror map learned from data. All reviewers vote for acceptance of this work, noting its empirical results are impressive (GaGj, T4a4). While initial reviews shared a significant concern regarding the paper's clarity and presentation (GaGj, JSTt, T4a4), the authors provided a thorough rebuttal that successfully addressed them. Therefore, I recommend acceptance and encourage the authors to revise the final version to reflect these clarifications.
Decision: Accept (poster)