Energy-based generator matching: A neural sampler for general state space
Authors: Dongyeop Woo
Minsu Kim
Minkyu Kim
Kiyoung Seong
Sungsoo Ahn
Venue: NeurIPS 2025 poster ()
Keywords: neural sampler
Boltzmann distributions
diffusion model
flow model
Primary area: probabilistic_methods
TL;DR: Energy-based training of neural continuous-time Markov processes in general state space.

Abstract: We propose Energy-based generator matching (EGM), a modality-agnostic approach to train generative models from energy functions in the absence of data. Extending the recently proposed generator matching, EGM enables training of arbitrary continuous-time Markov processes, e.g., diffusion, flow, and jump, and can generate data from continuous, discrete, and a mixture of two modalities. To this end, we propose estimating the generator matching loss using self-normalized importance sampling with an additional bootstrapping trick to reduce variance in the importance weight. We validate EGM on both discrete and multimodal tasks up to 100 and 20 dimensions, respectively.

Energy-based generator matching: A neural sampler for general state space
Dongyeop Woo!"
Minsu Kim 4.2
Minkyu Kim
Kiyoung Seong'
Sungsoo Ahn
KKorea Advanced Institute of Science and Technology (KAIST)
2 Mila - Quebec Al Institute
Abstract
We propose Energy-based generator matching (EGM), modality-agnosee ap- proach to train generative models from energy functions in the absence of data. Extending the recently proposed generator matching. EGM enables training of arbitrary continuous-ting Markov processes, c.g., diffusion, flow, and jump, and can generate data from continuous, discrete, and mixture of two modalities. To this end, we propose estimating the generator matching loss using noff-norml importance sampling with an additional bootstrapping trick to reduce variance the importance weight. We validate EGM on both discrete and multimodal tasks up to 100 and 20 dimensions, respectively.
Introduction
We tackle the problem of drawing samples from Boltzmann distribution (u) cxp(-E(x)) given only oracle access to the energy function E(x) and no pre-computed equilibrium samples. Such "energy-only" inference arises throughout machine learning, e.g., Bayesian inference [15] and statistical physics, e.g., computing thermodynamic averages [29], yet it remains intractable in high-dimensional or combinatorial state spaces. Classical methods based Markov processes, e.g.. Metropolis Haattool [28, 18]. Gibbs sampling [16], and Metropolis-adjuseee
Langevin algorithm [17, 33], provide asymptotically exact guarantees, but the cost of long mixing times. practice one must run chains for a vast number of steps to traverse metastable regions, and each transition incurs an expensive energy evaluation, limiting applicability to large-scale systems [32].
Deep generative models, particularly diffusion and flow models tied to continuous time processes [36, 25], offer compelling alternative: after training, cheap constant-coss network evaluation produce independent samples. Once trained, they do not require energy queries for inference. These models show great success in vision [34], language [24], and audio [21], and be transferred to unseen conditions. However, their success critically depends on data-driven training on a large number of true equilibrium samples, which are unavailable when only an energy function is known.
In response, researchers have designed new energy-driven algorithms, coined diffusion samplers, to train diffusion-based generative models to sample from the Boltzmann distribution. For example, path integral samplers (PIS) [42] and denoising diffusion samplers (DDS) [38] minimize the divergence between forward SDE and the backward SDE defined by Brownian bridge. Schound-Sadegee et al. [1] proposed iterated denoising energy matching (iDEM), a simulation -free approach to train diffusion models, highlighting the expensive simulation procedure for acquiring
samples from the diffusion-basee models. Generative flow networks (GFNs) [3, 22), originating from reinforcement continuous-time Markov process (CTMP) in the limit [6].
learning, can also be interpreted
However, there still exists a gap between the data- and energy-driven training schemes, particularly on the choice of state spaces and CTMP. Especially, Holderrieth et al. [19] recently proposed generator
*Correspondence to: dongyeop. www.greopwwokkaist..oo woo@kaist.ac.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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matching (GM), which allows unified data-driven training of continuous- time processes ranging from flow, diffusion, and jump processes for both continuous and discrete state spaces. This allows training generative model for hybrid data, e.g. amino acid sequences and 3D coordinates of protein [7], and greatly expands the available space of parameterization.
Contribution. We propose energy-based generator matching (EGM), an energy-driven training frame- work for continuous-time Markov processes parame- terized by neural network. EGM accommodates con- tinuous, discrete, and mixed state spaces and applies to various process types. including diffusion- flow-, and jump-based models. Our work greatly expands the scope of energy-driven training for continuous-time neural processes (See Table 1).
Table I: Comparison of sampling methods by state space type and underlying Markoy process. "D", "F", "J", "J (D)" denotes the diffusion, flow, continuous jump, and dis- crete jump process respectively.
Method
F
J
J (D)
PIS [42], iDEM [I], CMCD [39] iEFM [40], LFIS [37]. NFS [8]
V
X
X
X
V
X
LEAPS [20]
To this end, our EGM estimates the generator match- ing loss with self-normalized importance sampling, which requires approximating samples from the target distribution. However, this is challenging due the high variance in importance weights. To alleviate this issue, we introduce bootstrapping, which allows the generator to be estimated using easy-to-aphl -apppooxmma samples from the nearby future time step. It reduces the variance of importance weights, significantly boosting the sampler's performance.
EGM (ours)
We validate work through experiments on various target distributions: discrete Ising model and three joint discrete-continuood tasks, i.e., the Maussian-Bernoul restricted Boltzmann machine (GB- RBM) [9]. joint double-well potential (JointDW4), and joint mixture of Gaussians (JointMoG). Our findings demonstrate that the training algorithm enables parameterized CTMP to learn the desired distribution and is scalable to reasonably-size problems.
2 Preliminary: Generator matching on general state space
In this section, we provide preliminaries on generator matching (GM) [19], which allows generative modeling using arbitrary continuous-time Markov processes (CTMP).
Notation. Let S denote the state space with a reference measure v. We let Prarget denote the target distribution with samples 21 Puaget- We denote probability measure indicates integration with respect to p in the variable x. If p admits a density, and when no confusion arises, we use p both for the measure p(dr) and its density p(x) == de (x)) with respect to v.
S by p(dz), where "dx"
Overview of generator matching. GM relies on set of time-varying probability distributions (pen (dx|= (r) )ossss) depending on data point x S, coined conditional probability path. This induces corresponding marginal probability path (pr)oges1 via the hierarchical sampling procedure:
1 Ptargot Pajz (dx|x1)
x ~ (dr).
(I)
GM trains a Markov process (Xf)oses) parameterized by a neural network, to match the marginal probability path pt. After training, the induced probability path of the Markov process aligns with pt. enabling sampling from Prarget = Pi by simulating the trained process X².
Generators. A generator characterizes CTMP through the expected change of a test function f 3.S+PR in an infinitesimal time frame. It is a linear operator defined as:
E Ex (xx
[f [(x)
(xf(x) = dh 11=0
Ex (
[[(+++))
lim h=+0
h
where Pi+hit denotes the transition kernel of Markov process X,. If two Markov processes X and X have identical generators C and CJ, the processes equivalent.
Linear parametrization of generators. For commonly used Markov processes, e.g., diffusion, flow, and jump processes, the generator can be linearly parameterized as (xf = (Kf F(())) where K is an operator fixed for each type of Markov process, the parameterization. For example, F. u, for flow model with ODE dr = (z(r)d F, = of for diffusion with SDE dr = ar (a)ddur or F; = Q2 for discrete jump with transition rate matrix (),) described by the Kolmogorov equation (vp Σyes Qo )x))).
a vector space, and F, F.(x) E V is

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X1
Puje- (x)
9sle( (x)
Arlel (x)
X
X
(x)
xx(x)
Fx(x)
Prief 12)
(x)
P ef1(2)
Fir (x)
Po
Pr
Pt
Pr
P1
Pt
P1
Pr
P1
(a) Probability path
(b) Generator matching
(c) EGM
(d) EGM w/ bootstrapping
Figure I: Overview of energy-based generator matching (EGM). (a) The target probability path that
interpolates between the prior and the target distribution; aim estimate the (x) as (weighted) average of conditional generators. (b) GM draws I1 ~ (ij((-(z) with uniformly weighted Fiji (x).
(c) EGM draws 21 (1je(-(z) with importance weighted Fat (z). (d) EGM w/ bootstrapping draws 2, 9rje with importance weighted (2).
Marginalization trick. The key idea of GM is to express marginal generator Le that generates probability path Pt by conditional generator 441 for the conditional probability path Paja (-(21). This leads to the expression of the paramcterization F; for the generator C, using parameterization F of conditional generator L,1- To be specific, the marginalization trick is expressed as:
(xf(x) = -E (le)
(x = E., (- (47 (x)):
(3)
where (d)e(dii(a) is the posterior distribution, ie., the conditional distribution over data æ1 given observation time f. Intuitively, at any point (Ir t), the marginal generator L. steers in the average direction of endpoints xj sampled from the target distribution.
Conditional generator matching. GM trains neural network F. to approximate the parametrization Fr of marginal generator L, using a Bregman divergence D V R20:
(0) = Ez~Unif, s.mpp F
which is hard to minimize since F, intractable estimate. Instead, one can minimize the conditional generator matching (CGM) loss expressed as follows:
TCGM (0) R,~Uuii a. Parge 2n (pp()) [D(Ff (mx), (=:)))
(5)
One can derive that V LGM (0) = VeLCoM(0), and gradient-based minimization of the CGM loss equivalent to that of the GM loss.
3 Energy-based generator matching
In this section, we introduce energy- based generator matching (EGM), the first method for learning neural samplers on general state spaces via continuous-time Markov processes. EGM extends the GM framework energy-drives training by estimating the marginal generator with self-normalized importance sampling (SNIS). To this end, we define a variant of GM loss LaM in Equation (4):
LEGM (f) Bu- x. [D (F(()))) (x:))),
where F is the energy-driven estimator of the parametrization F; and pf is reference distribution whose support covers the probability path Note that, when F, F,, minimizing LEGM guarantees that the learned sampler E. recovers samples from the target distribution Ptarget-
We first present our scheme to compute the estimator Fr, then describe a bootstrapping technique for variance reduction of the importance weight, and finally provide the training algorithm. Figure 1 provides a visual overview of these estimators. We also include examples demonstrating how EGM accommodates diffusion, flow, and jump processes.
We include the derivation and example in Appendix for mixed state space.

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3.1 Marginal generator estimation via self-normalized importance sampling
Problem setup. We consider the sampling problem from density? (1(2) (,(x)/2 given only the unnormalized density Pi (x:) =
with intractable partition function Z = pi (dz).
Following GM, our goal is to train neural network F. to approximate the parametrization Fg of the marginal generator L, in Equation (3).
SNIS estimation of the generator. First, consider the following importance sampling estimator:
(x) = E EE, Merrauttla (b [wlz Zi 21) f(x)], (((zx11) == P1 Pii (x)
(m) (2r (7)
2ije ( (2)
Zp (x) (1) Vijp((r (1) (x))
where Pije is the target posterior and gi is a proposal kernel. We avoid computing intractable Zpr (z) for the importance weights by using SNIS scheme:
(0 x1) Pi (111))p (11))
ui (2, xxxx Zpe(z) En (et (2,
w(x
(8)
where w is the unnormalized importance weight that is tractable to compute. In this way, one can compute the self-normalized weight without knowing the normalization constant (x)) Further- more, when we use the same samples for estimating the generator and normalization constant, we obtain a low-variance estimator at the cost of inducing a bias.
based generator matching (EGM). The linear parametrization of the generator admits the importance sampling expression F((x) Exyogne(((l)) [2xx,
P.] (x)) Then the loss is:
(0) EE,,Uiir 2, upp D (1.) (xx)) f.(x)) E, ú(xx, Fij (x)
Σ, (x5
where x1 are sampled from the proposal (-zr) We implement F. using the LogSumExp trick, which provides numerically stable estimator even in the low-density region.
Example 1: Conditional optimal-transport (OT) path. EGM can be applied train neural flow sampler driven by the conditional OT path [25]. In the flow model the conditional probability path and its conditional velocity field are expressed by Puj1 (2r(2r) Mx (1 id) and
Meja (ar (x)) = 2i- for 21 E R.d. Choosing the proposal (1)((21(z)) x Peji one obtains,
x (1 (1)(<011)) = N F1;
t2 1))
(e(x) = Σ, (z, (8) Jup (3)2)))
(10)
tt
Σ, (x)))
Example 2: Discrete masked diffusion. EGM also supports masked diffusion path which has
recently gained popularity in language modeling. In the masked diffusion path?, the conditional probability path and the conditional transition
matrix are expressed P: (35/2)) kn (4) (1 - S (x) and ueja (y, x x((r)) = I-x; (5., (v) - 8y(()) where M E the mask token, Kr. [0 + R is a schedule, and 8, is Dirac delta distribution at æ. In this setting, the proposal and the estimator are,
ås
(x)((=)) = Unif(2r; M) (o M) ( M)
6,(y,x) E1 (= (u (V,
(1))
(x1)
3.2 Bootstrapping tricks for low-variance estimation
The quality of the estimator in Equation (9) depends critically the proposal distribution Zitt- In the ideal case, the proposal ZıIz that matches the posterior Pıjı ensures that EGM is unbiased and aligns with GM. However, the simple choice q1 (z) i (are) x Peja (2.(<r) leads to a mismatch with the true posterior Pi i, resulting in high variance in the importance weight.
TToough density may not be defined in general, it is given priori in the sampling problem. 4 'For a detailed description of flow model and masked diffusion path, see Appendix C.
DDscrrete masked diffusion is a discrete jump process which has transition rate matrix be (y, 2).

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For nearby future time step r > t, the posterior (Zr (2z) is similar to the backward transition kernel (T) because Pric (a, p-(==)
P [r.) pe(ms) ~ for continuous densities. Thus, a simple proposal (2, (rr) & Ptj- (z,,) ) can effectively match the posterior Prie (z,] reducing the variance of importance weights. To exploit this, derive generalized marginalization trick using the
intermediate state Ty..
Backward transition kernel with marginal consistency. To derive bootstrapping, we need to choose a backward transition kernel Pejr which satisfies the marginal consistency described by the Chapman-Kolmogoro equation:
= - f -ur--a------ par (dx,
(12)
Not every Peir with marginal consistency tractable. For instance, in the flow model, Pejr is deterministic, making density evaluation infeasible. We give examples of tractable backward transition kernels satisfying our condition for diffusion, flow, and masked-diffusion paths in Appendix C.
Bootstrapped marginalization trick. Now, we generalize Equation (3) and show that the marginal generator L, can be expressed as marginalization over generators Ctp for probability paths (Pej br, (oos)ss conditioned on time r, instead of the conditional generators Lij1- To this end, we define marginal consistency of conditional generators as follows:
Ex, (1m - [q)
(13)
With generators Lür conditioned on time r satisfying the consistency, we express the marginal generator Lt. We provide the corresponding proof in Appendix A.
Theorem 1. Let C denote the conditional generator for conditional probability path Per ( (xy) for 0 < t < S If the backward transition kernels Ptr satisfy the Equation (12) and the conditional generators satisfy Equation (13), then the marginal generator can be expressed as follows, regardless of::
(. = Eeprp. (fje
( (x:]]
(14)
where Prie Prie(dx,1a) is the posterior distribution (i.e., the conditional distribution over intermediate state F, given an observation x at time t).
Bootstrapped SNIS estimation of the generator. Using a proposal distribution (-)e(-1x), the marginal generator can be estimated as follows:
(xf(x) (o) x,, CTp (x)], w(I,x,) (e]e(=](z) Pr (x. (15)
(,]((x)()) pr (r))
Similar Section 3.1, we also consider the estimator based on the self normanormalized importance sampling scheme to reduce the variance of the IS estimator. With simple choice of (x,((=)(1))) x (1)l)))) unnormalized importance weight becomes:
pr
/ pr (=1(0,l11))(((d))))
( xry Ey oige uf (-1x)) 17 (x+)))
(x,)))
(16)
Then we obtain the following marginal estimators:
Lff(x) Ei C(ū x (3) ) f(z) E,
F((z) E, u(x (z, xt D (x)
(17)
w(r, 10.
where 27 sampled from the proposal gr (r (-(z:) and Pir is the parametrization of Sip
Intermediate energy estimator. Note that the unnormalized density P. is intractable, unlike 5 in Equation (9). Therfore, we train a surrogate p. (x) to approximate the unnormalized density of the intermediate state v,. This surrogate can be learned using the following estimator:
(r(xx) : (E) Me (x) ddr= (1, Er, you on (.l..) (f,(21)) (z1)),
(18)
5

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Algorithm 1 Iterated training with EGM loss with bootstrapping
Require:
Network Ft E replay buffer B 0. bootstrapping step size and batch size b. 1: while Outer-loop do
2: Sample (21 y from the simulation of the current sampler F. and set B + B f21)
while Inner-loop do
4:
if bootstrapping then Sample Unif(O, 1].r min(t + e, 21 x1 Update & to minimize LNEM as defined in Equation (20).
5:
and Pija (-(rr)))))
Compute bootstrapped estimator (2; Φb r) with the proposed sample af ~ grje(-
8:
Update 8 to minimize LEOM as defined in Equation (21).
else
10:
Compute SNIS estimator (x()) with the proposed sample x] ~ (-(r))
11:
Update 0 to minimize LEGM as defined in Equation (9).
12:
end if
13:
end while
14:
end while
where (z,) is the normalization constant of the proposal (1) (x,)) о (rr1()))))). The estimator is unbiased, but exhibits high variance. Therefore, we learn the energy of intermediate state, (z,) = log (z,, that can be expressed using the target energy function & (z)):
(x,) log Ex, ( (xx.))/xp((xpp ( (z1))) log Z1 (xrr,,
(19)
To learn the surrogate, we train on a general version of noised energy matching (NEM) [30] objective:
1 Σ exp(-E((11)) log (1rr) INEM (0) -E pp - E, (xrr llog K
(20)
where r1 ...., I1 (K) is sampled from qıju. Then our final loss from Equation (9) becomes:
LEOM-BS (0;;0) Ex ((P(((z;)) (x:))) P((zni))) exepp=EE )) F (ax)
Ex Ed ((()))
(21)
where 2/ is sampled from the proposal (-)x(-
Example 1: Conditional OT path. To apply bootstrapping with conditional OT path, we show that the backward transition kernel (e)r(=a)<rr) = N (rx; f.r.,, TT and conditional veloc- ity (xe , 20. (a+. fx.)) satisfy backward Kolmogorov equation with above pro-
posed conditional probability path Paja where o 1)2 fu2 (1 r)2. Choosing the proposal (2,,())) x (xe//2 (x,), one obtains,
grje (1, = =N Zei y 22
F p aid) (,11,)
Ee Pr (xt (x
(22)
E, (2
We check that the transition kernel Ptj, satisfies the assumption we proposed in Appendix C.
Example 2: Discrete masked diffusion. For masked-diffussin path, the backward transition kernel
and conditional transition rate matrix are given as Ptj (z,)) Mr 6,, (xx) + Kn- 8M (2xe and
Kt
& (y. 2. (x,) K..-Mi (8., (2) 8zz (u)). Then, the proposal and the estimator are
E.Or(a (8) (u.
(0re(=r/2 = M)
Fie Jas (Ir) +
1rr Kez Kr
(.(,,x)
(23)
Ky
Expr
3.3 Training details
We now describe the training algorithm for EGM, with the full procedure provided in Algorithm
NNte that goj (xr waa # M) (xr. the token flipped the data token does not change thereafter.

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Bi-level training scheme. To choose a reference distribution p that is close to Pt. we use hierarchical sampling procedure 2 B, 2, ~ (-1x)) with replay-buffee that approximates p1. This leads to the following form of EGM objective:
LEGM (0)
(PP(()))))))))
(24)
When the buffer remains close to pt. our loss approximates the generator matching loss in Equation (4). We achieve this by continuously updating the buffer, using the bi-level scheme introduced in Akhound- Sadegh et al. [1]. The bi-level scheme alternates between an outer loop and an inner loop, where in the outer loop the buffer B is improved by drawing samples from the current sampler FL. For the inner loop, the sampler F is trained to minimize the EGM loss with B held fixed. Because the sampler is updated in the inner loop, the samples collected the subsequent outer loop reflect its improved performance, thus progressively improving the buffer.
Forward-looking parametrization for masked diffusion. We also introduce a trick to further boost the training of masked diffusion samplers for graphical models via inductive bias on the estimator E, To this end, consider a graphical model defined on graph G = (V, E) with vertices V and edges E. It defines a distribution of vertex-wise variables x = {zi)iev using an energy function expressed as a product of edge-wise potentials (#t.) (E.))e i.c., pi (z) exp(-E(x))
[[(s)E ( ([",,>)) x²)
where we let æ² denote the variable associated with vertex V.
For these models, we can express the intermediate distribution (x(),) using fixed configurations, since once a token is converted from a mask to a data token, it remains fixed thereafter. To this end, let I, denote the set of edges with deterministic value time 4, and the intermediate distribution can be expressed as follows:
f pe(
(x() II
II
Vis(ai, dz1.
(25)
(A))EE
Hence, it suffices to estimate the contributions of the undetermined region, which leads parameterization E, as: & (xx) NNo( EEnyen log vy (
4 Experiments
In prior work, research has concentrated mainly on diffusion-based samplers, leaving purely discrete and multimodal settings underexplored. To fill this gap, we evaluate EGM on purely discrete and joint discrete-continuot tasks. Specifically, we employ jump-based neural samplers for purely discrete tasks and combine discrete jumps with continuous flow tackle multimodal tasks. These experiments demonstrate EGM's performance and versatility.
Our primary evaluation metric is the energy-W: 1 (E-W1) distance, consistently applied across tasks and complemented by qualitative visualizations. Since no jump-based or hybrid discrete- continuous neural sampler baselines exist, we include traditional Gibbs sampler [16]
competitive baseline. Although Gibbs sampling is guaranteed to converge given sufficiently many iterations, we use four parallel chains with 6000 steps each to reflect realistic computational budget for the per-sample cost and to yield reasonably competitive results. Detailed descriptions of the energy functions, evaluation metrics, and experimental protocols are provided Appendix D.
4.1 Discrete EGM on the 2D Ising Model
We assess the performance and scalability of EGM using the two-dimensional Ising model, varying both grid dimension and temperature. The Ising model is a canonical benchmark in probabilistic inference and sampling, with well-established ground-trune samples facilitating robust evaluation. The Ising model, whose complexity can be tuned via temperature and grid size, serves as ideal testbed for our jump-based sampler.
We report the quantitative results in Table 2, alongside comparisons to the traditional parallel-Gibbs sampler as a baseline. Given the absence of metric structure in the discrete Ising model, we adopt the Wi distance over energy and average magnetization M Σ; di metrics. EGM consistently matches or exceeds baseline performance across metries. Especially, bootstrapping improves the performance even in high-dimensional settings near critical temperatures B = 0.4.
7

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Table 2: Performance of EGM on Ising model in terms of energy W1 (E-WI) and magnetization Wi (M-W)). For reference, we report the metric of the Gibbs sampler. Each sampler is evaluated with three random seeds, and we report the mean + standard deviation for each metric. Results shown in bold denote the best result in each column.
Energy
5 5 Ising (d od=25) 25)
10 10 Ising (d = 100)
Parameters
8 = 0.2,
1.0
330.4,J
1.0
3= 0.2,
1.0
3 0.4,J 1.0
Algorithm 1
E-Wi 4
M-W
E-Wi 4
M-W; 4
8-Wi 4
M-W;
E-Wi 4
M-W1 L
Gibbs
0.10
0.06
0.7120.43
0.23
0,53g0.04
0.04
5.29 as
0.29
EGM (ours)
0 0.2010.05
0.02
3.7320.39
0.24
0.84
0.0210.00
19.94
0,36
+Bootstrapping (ours)
0.10
0.02 10.01
0.60
0.04
0.39
0.0610.07
2.51
0.24
Ising 5, 8=0.2)
Ising 5, 8=0.4)
Ising 8=0.2)
Ising 8=0.4)
0.15
0.2
0.15
0.04
0.10
0.1
0.0
-20
-20
-18
-80
-80
40
Enorgy
Enorgy
Enenyy
Enengy
Ising (5 x, 6=0.2)
Ising x5 (=0.4)
Ising (10x 10, 8=0.2)
Ising (10 x 10, 6=0.4)
2.0
1.5
Denety
1.0
0.5
-1.0
-0.5
0.0
0.5
1.0
1.0
-055
0.0
0.5
1.0
-0.50
-0.25
0.00
0.25
0.50
-1.0
5.
0.0
0.5
1.0
Magretization
Magustattion
Magnut ization
Maguetiaation
Ground Truth
Gibbs
EGM
Bootstrapping
Figure Comparison of energy (top) and magnetization (bottom) histograms for ground-truth samples and various sampling methods.
Qualitative evaluations presented in Figure 2 depict energy and average magnetization histograms for the Ising model with low temperature 3 = 0.4, showing EGM's ability to accurately capture the true energy distribution.
4.2 Multimodal EGM on GB-RBM, JointDW4 and JointMoG
We validate EGM on joint
sampling tasks using three synthetic benchmarks:
- restricted Boltzmann machine (GB-RBM) [9], double-well potential with type- dependent interactions (JoinDDW4), and We choose GB-RBM to show an application of sampling from the energy-based model. JointDW4 mimics simplified molecular sequence requence-structure structure co co-generation problem. JointMoG is a mixed-state extension of MoG that is common in ceffusion-sapllee adopt conditional-O0 OT or variance exploding paths as a conditional probability path for the continuous flow sampler, and a masked diffusion path for the discrete jump sampler.
joint mixture of Gaussians (JointMoG).
benchmarks. For all the experiments, we
Quantitative outcomes are summarized in Table 3. For GB- RBM, we also use x-W2 for the first two continuous dimen- sions. Qualitative results for GB-RBM and JointMoG in Fig- ure visually demonstrate EGM's capacity to capture all dis- tinct modes accurately. In Figure 4, we observe that EGM (with bootstrapping) correctly models the true energy distribution in JointMoG compared to the Gibbs sampler. Bootstrapping consistently improves performance in capturing multiple modes of the distributions.
0.20
Ground Truth
ECM
0.15
Bootstrapping
Density
Gibbs
0.10
0.05
10
20
60
Energy
Figure 4: Energy histograms on the samples from multiple samplers vs. ground truth of JointMOG.
5 Related Works
Diffusion samplers. Diffusion samplers cast sampling as stochastic control or denoising problem. Farly work, such as the path integral sampler (PIS) [42], required on-policy SDE simulations per update, leading to high computational costs. Subsequent variants explored different probability
8

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Table Performance of EGM on multimodal task. For reference, we report the metric of the Gibbs sampler. Each sampler is evaluated with three random seeds, and we report the mean + standard deviation for each metric. Results shown in bold denote the best result in each column.
Energy
GB-RBM (d 5) x-W2 +
JointDW4 (d 12)
JointMoG (d 20)
Algorithm 4
&-W1 +
E-W1 4
&-Wi +
Gibbs
0.77+0.03
5.12:t0.21
4.80#0.02
7.76=0.02
EGM (ours)
0.33
0.63-10.10
2.45
8.83
+ +Bootstrapping (ours)
0.20
0.61
1.65
1.20-0.32
(a) Gibbs
(b) EGM
(c) EGM + BS
(d) Gibbs
(c) EGM
(f) EGM + BS
Figure Sample plots of GB-RBM (a-c) and JointMoG (d-f). Samples are projected onto the
first two continuous dimensions. BS stands for bootstrapping. Contour lines represent the target distribution, and colored points indicate samples from each method.
paths [38, 5, 39] and divergences [39, 31], yet most still rely on solving or simulating SDEs to compute their objectives. To reduce this burden, off-policy methods like iDEM [1], iEFM [40], and BNEM [30] estimate the marginal score (or flow) directly and eliminate inner-loop simulation: PINN- based approaches such as NETS [2] similarly train samplers without rollouts. While these advances improve efficiency in continuous spaces, discrete or jump process samplers remain underexplored. Recently, LEAPS [20] introduced proactive importance sampling scheme for discrete state spaces using locally equivariant networks. However, it requires many energy evaluations at inference since escorted transport.
Generative flow networks (GFNs). GFNs [3, 22] formulate sampling as a sequential decision process: forward policy constructs an object step by step, and a backward policy enforces consistency so that trajectories terminate with probability proportional to target reward (or unnormalized density). Training objectives like trajectory balance [26] and detailed balance [4] enable off-policy updates. Furthermore, Falet et al. [12] propose the sampler for a sparse graphical model with a local objective similar to our forward-looking parametrization. Though diffusion sampler is special case of continuous-state continuous-time GFNs [41, 6], remains an interesting question that training a CTMP general state space, ours, can be unified under the GFNs framework.
Relation iDEM and BNEM. In EGM, choosing (1je(zz x Ptj (I (41) yields simple weights exp(-E(x1)). In the VE path on continuous space where target score identity [11] is available, the integrand can be replaced with the target score instead of the conditional score. This yields the same estimator used in iDEM [1] and NEM [30]. Yet, it remains an open question whether it's possible to derive the target score identity on general state space. Recently, Zhang et al. [43] introduce the target score identity on discrete states with neighborhood structure, which might improve the accuracy of our estimator on discrete space when combined.
(1 œ1) = P1 (z1)
Conclusion
We introduce energy-based generator matching (EGM), training method for neural samplers on general state spaces that directly estimates the marginal generator of continuous-timg Markov pro- cess (CTMP). EGM is the first neural-sampler framework to handle discrete, continuous, and hybrid multimodal distributions via principled snnerator-macchigg objective. We also propose boot- strapping scheme using intermediate energy estimates to reduce variance in importance weights. We empirically validate EGM on both high-dimensional discrete systems and hybrid discrete- continuous domains. Our results show that EGM, especially with bootstrapping, performs competitively when sampling from complex distributions with multiple modes. This work opens new avenues for training expressive neural samplers beyond diffusion-based models. Future work may explore unbiased and low- varvinriece estimation of the generator, Iearned proposal for better estimation, theoretical

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connections to physics-informed neural network (PINN) objectives, and unifying the training of general CTMP into the GFNs framework.
Acknowledgments and Disclosure of Funding
This work was partly supported by Institute for Information & communications Technology Planning (valuationnIITTP) grant funded by the Korea government(MSTT) (RS-2019-I1190075, Artificial In- telligence Graduate School Support Program ( (AST)), National Research Foundation of Korea(NRF) grant funded by the Ministry of Science and ICT(MSIT) (No. RSS2022-NR0721144 GRDCCGGlobal
Rescarch Development Center) Cooperative Hub Program through the National Research Foundation of Korca(NRF) grant funded by the Ministry of Science and IMT(MSIT) (No. 14-2024-00436166 and the Institute of Information & Communications Technology Planning & (valuation(IITP) grant funded by the Korea government(MSII) (RS-2025--0) 2025-02304967, Al Star Fellowship(KAlSTT), Minsu Kim acknowledges funding from KAIST Jang Yeong Sil Fellowship.
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A Proofs
A.1 Importance sampling for the generator estimation
This section provides a detailed derivation of the importance sampling estimator for the generator and its parametrization presented in Equation (9).
Existence of densities. Consider the measurable space (S, Σ) with E-measurable reference measure V (e.g., counting measure if S' is discrete and Lebesgue measure if S = R2), as introduced in the Section Since we focus a sampling problem, the target distribution P1 admits u-density du : S + Rzo. Assume that the joint probability measure Pt.1 is absolutely continuous with respect to the product measure y @ v. Then, all related probability measures Pt, Ptj1:P1) Pije admits corresponding v-densities, expressed as:
/ Pt.1 (dae, dr1)
Pe(xx))=
(26)
f (11 (1), @ (dx dr,)
(27)
( (1 v(dx)),
(28)
p((a))
(i,i dx1)
(1) @ v) (dx dr1)
(29)
Pt. (L. 211
Pe (T (I)))((
(30)
Pe(xt)
Pt, (11,
v (dr ((dr))p (dree)
(31)
Po(rz)
PPij( (ms)
Pr,1 dx1)
Pt,1 (11, @ v) (dz dr1)
(32)
pr, (We, 11)
Pr (Ir (dd (ddr))
(33)
(1(1)
Pt, (11, v ( P1(11)
(34)
where Pr,1( x1) denotes the density of the joint probability measure.
SNIS estimation of the generator. We introduce a proposal distribution Quee Σ x 5 - Rzo- satisfying absolute continuity conditions: P(j(-llx) < ()j(-+1), qua(-11r) < v and < q)pe(112) for all x E S. Using the marginalization trick Equation (3) and the Radon-Nikodym theorem [23. Chapter 7], we have:
Lxf(x) (4
(35)
dp (m (1]l)) (xx)
-Ber
(36)
dqu
20

---

Since the Bayes' rule holds for the v-density, i.e., P1)c(211222) Pon(er/21)p))))) obtain:
Pe(w))
dpilt (zi ( Citi)
(xf(x)) = (e
(37)
drip
dpilt (21 (2x) doipe
dv
= (errggie(llee))
(x1l
(38)
dv
(1 (x1)
Leli (axe) (110(11 (1t))
(39)
W We 11 (i ((x))
(40)
where 21) P)e(a1) ((x)))))())
(11((21(22) (xx) We estimate the normalization term Zp;(xx)
with tractable unnormalized density (i(x; v1) == (11((112z))
pi (((
E ( (((x,,)1)) = Ber
(41)
(1p(21)(2))
= - pr (100)))((((//2))((
(42)
z p((21)
Z Pipi (1n(z(/1)u(dd)
(43)
Z / (e
(44)
Zpe(ze).
(45)
Thus, we derive the self- nelf-nermalized importance sampling (SNIS) estimator for the generator:
(xf(xr) Besrgy(((lee)) [ an (
(46)
Exg~gy(((ar)) [1 21)]
SNIS estimation of the parametrization. Similarly, the SNIS estimator for the parameterization F is:
F((x,) = Ensrn
F (ax))
(47)
dpijt (x) (2x)) F (1re)
(
(48)
dqipe
(ije (Ire) par Fill (1re)
(49)
( (xn)
(eson)())))
w(1,, 21) Fil (xx))
(50)
Ezy m.) (x()
Specifically, the expression above suggests the Monte-Carlo (MC) estimator with K samples (1) o (K) ()))((12)) as follows:
ER (x x1 ) (xx) ΣK ww (t))
P((xx)
(51)
This is SNIS estimator of the parametrization F((x).
A.2 Proof of Theorem
For convenience, we repeat the theorem and its assumptions below.
21

---

Theorem 2 (Restatement of Theorem 1). Let Cir- denote the conditional generator for conditional
probability path P-,(- for S t s 1. If the backward transition kernels Ptly satisfy the Staapann-Kologgon equation,
Paj((re(21) = Pej- (dz, (rr) (dx,|
(52)
and the conditional generators für satisfy the marginal consistency follows,
Ex, (e (mn, (F)r)
(zx).
(53)
Then the marginal generator can be expressed as follows, regardless of
(+s(xx) = Peiclr)
(54)
where Prie (r)e(dzy(z) is the posterior distribution (i.e., the conditional distribution over intermediate state Er given an observalion x at time 1).
Proof. Define the marginal generator conditioned at time as fir == (x.- [4pp (a)).
Although this definition depends explicitly on r, we aim to demonstrate its independence from r. would result in conflicting objectives at times t < Ti, 12 for distinct T1: 72. The proof proceeds in two main steps:
This invariance is crucial since any dependence
1. Verify that the marginal generator Ltr generates the probability path (pt)oges
2. Show the marginal generator's independence from (i.e., Lty = Lt).
To establish the first step, it suffices to verify that the Kolmogorov Forward Equation (KFE) holds for the probability path (prlosesr and the generator Leyi
d Ezcpp [/ (xx) = Ezcopi (er (xt) for 0sigrá
(55)
di
The KFE is satisfied for the conditional probability path ptir by definition:
dt Earepy (x. ,.)[f(=))] = (xy-pe)-((1.))) (l
0 O
(56)
Thus, we have:
Eze Ler (2:a) Expop E mmpee((ma))
(
(57)
Exy BaINpa)r((a))) ((a)
(58)
d Eupopo dl Barrpan- [f (2t))
(59)
dt Barwpe BeterpeBaaaeee Baupee (xx))
(60)
dt Enrop [f(]]]
(61)
Hence, Ler indeed generates the probability path (pr)osesr-
Next, we demonstrate the independence of Ley from the choice of r:
fxrl(xt) (p. [c))(()))]
(62)
E Ee. f(r)
(63)
Eeromp(lee)) (())(
(64)
C.f((x))
(65)
where the marginal consistency assumption Equation (13) applied the third equality. This concludes the proof.
22

---

A.3 Derivation of bootstrapping estimator for generator estimation
We derive a bootstrapping estimator for the marginal generator and its parametrization proposed in Equation (21), based on the marginalization trick Equation (54).
Bootstrapped SNIS estimation of the generator. Assume that the backward kernel Pri- admits v-density. Then, the posterior Pri also admits v-density. Let the proposal distribution grle E Rgo satisfy Pr (-lr(-l)),, (r/e(-]z) u, and < grje(-lz) for all S. Applying the same change-of-measure trick as before:
(xf(x))= = Exr (p)
(66)
dprit (3r (20 Lipp (x(e) = me
(67)
dgrlt du (1r (er (1te) dprit (12) dgrle
Begrgg. (-12a)
(68)
du
Grle (2,)(1)))))) fir (2x1)
(69)
pr (Ir, Par (1e ((r)))))))))
B2, ((nalel)))))))
Pe (r)o)) fip (
(70)
wo x. ) Stp [1x)]
(71)
where the importance weight wxre is given by
w(x,, (r)o((r,)31e)) (x)((1))
Dr (1,))p r (1re)2r)) Pe (2x, (roe) (xr))))))))))
(72)
To estimate the unnormalized density (e(x))) we define the unnormalized importance weight:
Fr (3r))(rr(3)((z)))
w(I,, 24)
(73)
(x, (xc))
and compute:
(x())) / Pey (1r()))))))))) ( (xr) (((d,,)
(74)
Par (as (r (2r)
Gele (r1
(75)
(pp (x))
(76)
dr ((xn(ax))
E Ee (+10))
[1(( xr)]
(77)
Thus, the marginal generator can be expressed in SNIS form as:
E (a (10,,
(xf(x;)
(78)
(e [w x+]]
Bootstrapped SNIS estimation of the parametrization. Now derive similar expression for the parametrization of the generator. Suppose the conditional generator Lif admits a parametrization Fe. such that.
(xx) = (CC (xx))
(79)
23

---

where K. is an operator fixed for each type of Markov processes. From the marginalization trick again:
L.f(x)) (epepp)))) ((rs(())
(80)
(eprenitlle)) (K(((1), Pelp (1)))
(81)
((Kf(Z)) E Benepni((ae) (ze)
(82)
F(()))
by linearity of the inner product. Thus, the marginal generator parametrized by
F (z)) (, (zx)).
(83)
By applying the same importance sampling trick, we obtain the SNIS estimator:
F((2a)= = Ez, Pere(p
P (1x)
(84)
dprit (2r (x) (1te)
Ez, (el
(85)
dgrlt
Pr (2) (12)
-Eg ((12))
Filt (xxz)
(86)
9r (I, (2x)
En
( 2r F (21)
(87)
Specifically, this yields the following Monte Carlo estimator using K samples (1) gop(-1z))
(K)
...., Er
ER (1 27 (5) F (x1)
(x))
(88)
Ei 10( (3))
A.4 Analysis on bias and variance the IS and bootstrapping estimator
In this section, we investigate the bias and variance of the IS and bootstrapping estimator. The goal of the analysis is to support the bootstrapping estimation with a theoretical argument and demonstrate that it provides a lower variance estimator compared to the IS method. To do so, we first derive asymptotic bounds on bias and variance for estimation. Then, we derive the bounds for bootstrapping estimation, assuming that the intermediate energy is well-trained. Under this assumption, we show that bootstrapping reduces estimation variance compared to IS, demonstrating that bootstrapping trades off the estimation variance with the model learning bias.
Bias and variance of IS estimation. We show that the SNIS estimator's s error decays as O(L/VK) and its bais and variance decays as O(1/K), where K is the sample size. Proposition 1. Consider an unnormalized density (x() and a conditional generator Fil1 (I:) evaluated on a sample #1 ~ qujt- Suppose p1 (I(x)) and (2(()))) are sub-Gaussian. Then there exists constant c(z;) such that with probability least 8,
( (((a)))F((zz))) S sc(z))) log(2/0)
where F((xx) denotes the SNIS estimator from Equation (9) using K samples æ1 1..., 11
(K)
Σx
-= ĐI (z (8) F (Iz)
F((x;)
ΣK in (x)
Moreover, its bias and variance are expressed as:
1
1
Bias [i] Kp((z;}² (-Cov (or Farx:)) +
K2
24

---

4Varlpi (21)
Varl
( (1 + [1=)())))
Proof. Let and B denote the numerator and the denominator of F,, respectively, i..
K
=
Pi
D (ot),
B= B (1
(89)
K
We also let [[A] = A = Fe and B[ = B = (r()). By Hoeffding's inequality for sub-Gaussian random variables, there exists constant C' such that
log (2/))
1og(2/6)
A A S V
β - BB s CV
(90)
K
K
with probability. (Here, = v Varlpn (x1) a possible choice.)
Since we have bounds for both numerator and denominator Fi, we can also bound the error of F to Fú. The result is as follows:
|| (((z)) F((1))
Å
A
ÅB AB
=
(91)
B
B
BB
|||||
B
B
+
BBBAA
B
A
(92)
BB
1
log(2/0)
(93)
BB
K
20 Pe(x.)
log(2/0)
log(2/8)
(1 (IF((E)))
c(zu)y
(94)
K
K
where we assume sufficiently large K such that B 2 B.
Now, for sufficiently large K, the Taylor expansion of F. is expressed as:
A 1 (A A) B
Ba B (B- B)
B² (A- A) B)+
B3 B3 (B - B)² + +0
(B-
B
K2
To derive the final equation for the bias term, one can express the bias term as follows:
A 1/2 Cov[A, B]
A
Bias Fi] E[LA]
Varl
B3
Since Cov(Å, B] Cov fúr Fal1 p1) /K and Var Varlt1) K, one obtains the conclusion for [i.s
To derive the final equation for the variance term, one can combine the sub- Sabb-aussiantty F, with the error bound on [[(((a.) - F((x))) -
Bias and variance of bootstrapping estimation. Now, derive analogous asymptotic bounds for the bias and variance of the bootstrapping estimator. We assume a fully trained intermediate energy model, which allows us to isolate the impact of estimation bias the intermediate energy while disregarding any neural network learning bias.
Proposition 2. Consider fully trained surrogate energy model (x = EJE, (2,)), where (x is a biased energy estimator from Equation (20). Let F((xE; E,) be the bootstrapping estimator using E, as the intermediate energy function:
E, exp(-E F.r (xt)
F(
(95)
Ex,((EEE,,))))
25

---

Then, the bias and the variance bound of the bootstrapping estimator with surrogate model Fo(mx; EX) are given by:
Fias{Fi(Eri (iaa( + O
K2
Varlpr (2rr) Varl (ar(Fi(())];
(xx; (
Varlin (x1))
where Pe(2s) is the estimator without bootstrupping. Since Varlār (x+)] < Varlpi (21)), the variance of bootstrapping estimator is smaller than the F((
Proof. The bias of the energy estimator &, is given as follows (refer to Corollary 3.2 of [30]):
Bias ( = Varlir +0
(96)
(x,
K2
Thus, we can approximate the surrogate energy as,
= [(,,,,= = (x,) + Varlp1 (x1)
(97)
2x²
:=b(x))
Plugging the above equation into the Fr E.)). we get,
E)); = E. exp( (app-EE(11) Ef () E (1) (x)
(xx)) - 6(xz²) (2 )) F e/r (x)
(98)
exp(( 87 (zf
E, expl E, (xt)) - )x)
Here, b(x()) is close to 0 and concentrated to:
my = b(x,) Var pp (x1))
(99)
(2,)
when is close to and K is sufficiently large. To keep notation concise, let w, be the self-normalized importance weight with true energy (z,,) and fn()) be conditional generator with the sample (
exp( Exp(-E,(11))) W44- Σ, exp((E
pa(()) t/r (1)).
(100)
Then, by applying first-order Taylor expansion to the (1 8 and approximation b(z(() ma. we obtain:
E, w. expep (b((,- )) f/r
(101)
Σ, ws exp(exppppp (bb(),"))) ((,)"))) ))
Σ, wi(1 b(xz²) (1) 1
(102)
Σ, E,w;(1 b(xx))) ))
E, wr 1/r - E, w;b(xx 10
(103)
1- E, w.W""""")))
F()) Wi t/r
Σ (rb((a))) FF(1))
++ (msllott)
(104)
Σ wi (())
mb Σ wi wii
(1 + ms)
(105)
E, expl E, (2f )) (x)
(1 - m 11m)
(106)
Ex (at)))
(1 - m F((
(107)
26

---

Therefore, the bias of the bootstrapping estimator Fi Eg9 is:
Bias F. ET)] = EE (x;; - F((2))
(108)
mf) Fz(
(109)
= (1 m + Bias [x); - ((()))
(110)
(1 m Bias) (zri (x)) - m2 (x)))
(111)
[ias([((11; (x1)
(F((ze) + Bias[F((zri&))))
(112)
4pf(x;)K²
[ras[[[(Fr;;,,]]] K2
(113)
Similarly, the variance of the bootstrapping estimator is:
Ex,; (1 m (+) (ar(F((():
(114)
since m, < 1 for sufficiently large K.
The variance of the IS estimator (xxe without bootstrapping is given by:
Varlf = Varrli (xx,) (1 + (+:))))
(115)
p((x,)
Also, the variance of the bootstrapping estimator Fr(x,;; E,) with true energy is given by:
Var(F((In::.)1 Aar(p,,(2,)) (1 [(=(()
(116)
Consequently,
Varlp, Var (rx)) Varlńi (x1))
Var(f((x:;E))) v Var(F((ar:
(117)
Because Varlp, (x+)] (arl) the variance of the bootstrapping estimator is smaller than the P.(),). Consequently, with an fully trained energy model, we reduce the variance of SNIS generator estimation.
27

---

B Generator estimation in the multimodal spaces
Our estimator can also be applied the mixed state spaces S X x Y within the generator matching framework. Let ( (+1a1)))oees and (Pej(-lyn bogeg1 denote the conditional probability paths on the X and Y. respectively, and let Lij and 2tp2 denote the corresponding conditional generators
for X and y1 E Y. Assume these generators are parameterized by Fil [0, 1] + Vi and
Fiji [0, x S + V2, respectively. For the joint space S = X x Y, we consider the factorized
conditional path:
Pen (dæ,, dy [1, v1) = (1)((de((z1) Paja (dyelvi),
where It, 21 X and ye.91 91 Y.
According to Proposition 5 in Holderrieth et al. [19), the conditional generator associated with Piji admits the following parameterization:
F (I), vc) (Fi] Fill (vv) 8/1
where the sum, scalar product, and inner product are naturally defined over the tuple (:-)) Vi V2. Thus, the importance sampling estimator for the parameterized generator can be written as:
Fe Mt. Ex, Pap (..)..))))
Fql (Ire, Mt]]
(118)
[dije (I11, yn (re, M6) (f (are), Pl)(
(119)
dqip
As in the uni-modality case, this leads self-normalized importance sampling estimator, which can be directly extended to the bootstrapping setting. This demonstrates the generality and flexibility of our framework in handling molli-modal spaces.
28

---

C Example of EGM with application to flow and masked diffusion
C.1 Generator of flow and jump model
In this section, we provide the definition of flow and discrete jump models, their generators and parametrizations. For the case diffusion processes or more rigorous derivations, refer the reader to Holderrieth et al. [19]. The discrete jump model is often referred to as a continuous Markov chain (CTMC).
Flow model. Let the state space be = Rd, and let ut R x [0.1] 1] + R be a time-dependent vector field. The flow X₃ is defined by the following ordinary differential equation:
dX.
(X.), Xo ~ po-
(120)
dt
By definition of the generator, the generator of the flow model is given by
[E] |X,, =x] f(x)
(xf())
lim
(121)
h-+0
h
[X + (X,) + o(h))) = x) f(x)
lim
(122)
h-+0
[[f(X,) + AVF [ o(h))X = =x] =x))
lim
(123)
h-40
h
= =Vf())"u, u,
(124)
where first-order Taylor expansion. Hence, the generator of the flow model admits the following linear parametrization:
(.f(x) (KI(()), (x))), Kf(x)) = Vf(x),
(125)
i.e., the generator parameterized by the ODE vector field uz, and EGM aims to learn Be via its conditional counterpart Mr|1-
Discrete jump model. Let the state space be discrete with |S| < co, and define the time-
dependent transition rate matrix Qi S x S x [0, 1] + R such that Qx( - (y,
and (y, 2 0 for all y F The CTMC is defined by the transition rule:
Xith ~ (t = 6x, + (2
(126)
We derive the generator informally; see Davis [10] for formal treatment:
[[ Xr z] f()
Lxf(x)= lim
(127)
[X (h)) f(Xo))Xe x, Jump in [t, (Jump in [t, h))
(128)
h=00
h
(Xf(h)+)) f(X)))x, = x. No jump in [t, + (No in [t, t + h))
lim
h-+0
h
(129)
Eg f(x)) Qi(gaah
(z,
lim
(130)
((() - f())) Σ f()) DeS
(131)
ya
Therefore, the generator of the CTMC can be linearly parameterized as:
C,f(x) (KS(x), Kff()) (f()) f(x))ves, (a, Σ by.
(132)
VE
i.e. the generator parameterized by the transition rate matrix Qels x), and EGM aims learn Q.
via its conditional form Quj--
29

---

Remark on linear parametrization. Under mild regularity conditions (e.g.. Feller processes), Holderrieth et al. [19] shows that Markov processes on both discrete and continuous state spaces can be universally expressed via linear parameterizations:
1. Discrete state space (|S) < oc): The generator parameterized by the transition rate matrix Qt. corresponding to CTMC.
2. Euclidean space (S = IR:); The generator is parameterized as a combination of flow,
diffusion, and jump components.
This implies that. like GM, EGM is capable of modeling wide range of Markov processes on both discrete and Euclidean spaces.
C.2 Application to the conditional OT flow model
This section details the application of the EGM framework to flow models defined via the conditional optimal transport (CondOT) path.
Definition of the CondOT path. The conditional OT probability path is defined as:
X, LX2 + (1 (1-).xo
(133)
where X1 Pt. Xo ~ Po WO /). and Xo: Xi independent. It linearly interpolates between Gaussian prior and the target distribution. By construction, the conditional distribution is given by:
Paj (23g 1x) = N2x,; tz 11)
(134)
EGM on the CondOT path. First, consider naive implementation of EGM with simple proposal distribution defined as:
(nje (21 (a)) x Pet1 (2)
|1 ) = N (cre) 1x1, (1 t)²r)
(135)
|| 2(1 t)²
tæ 11
x exp
(136)
|| マ 11
= exp
عرما x2
(137)
which implies that
(1)((11);)=)) N
(1
Tt
(1 (1-1)² 1)2
1
(138)
t t²
This choice yields simple importance weight of the form w(x,, 21) (x11) (zz))Z1/((z))
Using the identity from Equation (51), the estimated vector field ux(rr) becomes:
E. B,(a) 21(9)
(x(x)) = Zin(ze) "oji (xx)
(139)
E. (x(2))
Zzje(())
(1) E, (x )uyji (Ir)
(1)
(140)
E.
where 21
(1)x)-)))). This is precisely the same estimator used in Woo and Ahn [40] for the flow-based sampler.
Assumption check for bootstrapping. Next, derive the bootstrapping estimator. We construct the backward transition kernel Palr satisfying the marginal consistency Equation (52):
t²
o11,, (x)e)=) = N(xei Cr. o.l), or (1 -1-
72 7)
(141)
We verify the consistency via:
Paj (2s (2rla1) )dx, = Peji (2)l2a))
(142)
30

---

Using reparameterization tricks X, = f,2, + x,
= +
Et 1 Er, we have:
L (rXi + (1-r) Vo
X,
(143)
= 1X + 1 (11 - -r), + vora
(144)
tx, + (1 -)dc, 4 ~0,,0,
(145)
where d denotes that two random variables have same distribution. Thus, marginal consistency holds. The conditional vector field Mzlr defined as:
Tt (xt Ir 20:
Uk ( 2r +
(146)
This vector field arises naturally from differentiation of the reparameterization
X = 1 X, + Vo,No
= X, X + Vo.Xo
(147)
X, + Vot
X, - t _x,)
(148)
Vor
X, + di
_ X,,
Xt
(149)
20,
r
Now, verify that the conditional vector field War satisfies the marginal consistency Equation (53):
Ex, Peii, (-lor (xel ueju (1)
(150)
With (1)1 =
(epplll being Gaussian, its mean is explicitly: Pr (m))
t(1 r)
ro+
(ri1 21)
r(1 - t)2
tt +
(1 - t)2 21
(151)
Direct calculation confirms consistency:
E (101 (
[
{
2r +
ot
Ir
(152)
20,
r
1 Ey (e
+
2t
Le (o) [x.r]
Ex
20,
(153)
(01, + di
t ((n1)(01102))
(154)
20:
The first term 1 Hej,r reduces to,
(111) =
12 1)2 Ut
{(1 - r)2
(1 1)2
T2
(155)
t(1 r)222, + +200011
(156)
r²(1 -t)2
t(1 r)2
, t)2 (It txri) I1.
(157)
where we used o; = (1 - 1)2 -1 - r) in the third equality.
The part of second term 24 | (211 .F) reduces to,
{2 - r)2
to:
St - (111, = xt 72 - 1)2 2h
(1 1)22 41
(158)
r² t)2 (1 -r)
to:
r²(1 - t)²
Dt.
(1 - 1)22 11
(159)
to
(1 1)22 :t
(1 1)2
11
(160)
31

---

(1 where we used (1 - r)2 in the third equality. Put it all together, we conclude that,
1 1 (2)) +
de (x+
t
Ex Pp ( n.) (1)p((()(())))
- [rriLe((01,5e))
(161)
20t
t(1 r)2
de
( 01
to
r²(1 - 1)2
(Te tx1) + x1 +
20, {1 -1-1) -11=)
(1
1)2 I1
(162)
t(1 - r)
61
(Iz Lm1) + +
(xz tar))
(163)
r²( 1)2
2(1 1)2
((1 r)2
x2 + di 3t tm
+ 21
(164)
(1 - 1)2
(1 t) Tt tx
+1
(165)
(1 t)²
(166)
Uzj1 (Iee (21)))))
(167)
which implies the proposed transition kernel (1)r(1))),,))) and conditional vector field utjr (x,,)
satisfies the assumption of our Theorem 1.
Bootstrapped estimator for the CondOT. Lastly, we define the bootstrapping estimator for the
CondOT flow model using proposal as follows:
gr (r (rn) x (2)/(2)/()))) = =N(01 Ir, oeT)
(168)
|, - 1
exp
(169)
2
||,r, - 1
exp
2 r2 pz az
(170)
which implics that
,2
(.je(1r(111)) = N
Tri It, 22 (001))
(171)
The bootstrapping estimator is then given by:
(x1)) = Σx K
(x axth Juee E w(x )
((I,, 2x) = (x,,)) exp(-E;(xx)),
(172)
where samples dr 24 (K) dr (-12:)) and & (xr,) is learned energy estimator.
(1)
C,3
Application to the masked diffusson model
This section describes how the EGM framework can be applied to discrete jump models using the masked diffusion path.
Definition of masked diffusion path. We define the masked diffusion path as follows:
Ki 85 (xxe + (1 (1- Kit Keg
(173)
Ky
where Ki 10. + Ryn is an increasing function satisfying Ko = 0, 1, M is the mask token, and 6, is the Dirac-delta distribution centered æ. Next, we derive the conditional transition rate matrix generating the conditional probability path Ptrr (-(a,)) Starting from the Kolmogorov forward
32

---

equation, we have:
d
Kit
dt Polr Porr (pr((elar))
(Ōzz (yee) )xr(
(174)
Kr
1
((15r 1it (et) (ye) - (Nir me))as((e))
(175)
Hi Hrr hit
R2
Rr
(6x, - ((1)((e)227))
(176)
te
Σ Ky Ke
(8z, (2/1 82. Pe
(177)
Σ Helr (Ye (1)
(178)
thus obtaining Uzjr(1es (e
(6z, (We) 6z, (w+)).
EGM on the masked diffusion path. We first introduce naive implementation of EGM using simple proposal distribution defined as:
(21(2212))) (2)(21/=1)) k.dx; + k1)6
(179)
(180)
which implies:
S M) (x M) (2, (x1)
(1)(21))))
(181)
(x + M)
This yields the simple importance weight (i 21) (11) (/)e) Following Equation (51), the estimator for the transition matrix ue (Vt, 24) becomes:
E/ (1(21) (y), 21 (x
(ye, Fx)
(182)
where samples (1)
(K) (1)r())
In practice, the state space [V] factorizes along dimensions, where D is sequence length and [N] NJ. We thus factorize the masked diffusion path as follows:
D II ph (x)la))
(x]lat = ked (2t) + (1 - kx)
(183)
where E [N] denotes the i-th token of the sequence x E S. The proposal Qıje and the transition rate matrix (z æ) factorize accordingly. The proposal factorizes as:
(11((11 = II (=fl=)) x (1tla))) i=1
(184)
where li a proposal defined over each dimensions. The transition matrix factorizes as:
Σ 1=1
(185)
where - denotes the z without i-th token and uf is transition rate for each dimension. Hence, our neural network is trained to predict the D x N matrix NN. (I2, t) + (14(vi, 1 vie[M]
Assumption check for bootstrapping. The backward transition kernel Pajr of masked diffusion satisfies the marginal consistency since it defines the Markov process (noising process of masked diffusion). Thus, it is suffice to show that the conditional transition rate matrix uej- 24 (r)) satisfies the marginal consistency Equation (53):
Eg 2 [uejr (Vr, Iz [2|22)]] dan (v(1))
(186)
33

---

This condition can be confirmed via explicit calculations. (M 1 (2s((a))
Note that P=|1a(Er(21.22) =
(L.H...) Σ
(ōz, (v:) - 5r, (1+)) (r
(187)
Rir
Kit
hit
(6x, (vk) - 8e (v:)) Per (a Iir (1
(188)
Kt
Pej1 (2)((2))
rip
Σ
(6xx (Ht) - 8z (Be)) (x (2,
(189)
(Kr
Ke Pelp (ot (21)
((Sss (Br) dr,
(1)
(190)
(Kr Kz) Ke) (2te (x
+ (6z, (ve) dz, (we ( (ore /2r (2
(191)
Aie
(Nr ke)pep (2)) (des (vr) da, (x)0) me))
(192)
+ (6e, (wr) da, (we) (a)
(193)
(x; = x
di (Mr me){ ] (ōx, (2e) - Ez (1+)) (1- #x Kr
(194)
(x; = M)
Ai (dxs (ye) Ōzr, (v:))
(195)
lit
= Uol1 (ye, (Wee22e(00)) (2:1
(196)
where we used the fact that pr 1 (zr (z,((r)) > for only I, M or x,, = 21 in the fourth equality. Hence, the proposed transition kernel Pejr and conditional transition rate matrix Ugj satisfies the assumption of our Theorem 1.
Bootstrapped estimator for masked diffuusion. Lastly, we define the bootstrapping estimator for the transition rate matrix of masked diffusion model. We use the following proposal:
Kt 82, (xx) (1 (1-
(x)()1)))))))) x (2elxx)))))==
(197)
Por
Nrr
K1
(xt + M, vir = xz)
0
(I: * M, Ir, # x2)
(198)
1
(x, M, x, # M)
Kr
(xtz = M, xi, = M)
which implies,
82 (Ir)
(n+ M)
(2, (xc)
(199)
CCa(11
No. 1) K,
(m==M)
HR ....,
where the mask token is the last token M = N and Cat is the categorical distribution with unnormal- ized weight.
The bootstrapping estimator is given by: Σx =i=1 (( Jugjr ) (Ve, (1)
ú (11,
T (.x,, 2.) (x(xx) (2 (200)
EK u6 ((
where samples xf
(K) arje (12z)) and Ex is learned energy estimator.
Learning energy with generalized NEM objective. We train the E with the following estimator for the intermediate energy:
(x,, log Ex, (-I.) /exp( (ex(((E()) log Z1 (z.).
(201)
34

---

For masked diffusion, the partition function Z1 (2,) explicitly depends on Ir, is given by:
(1,) = Σ Pri1 Pr/1 (2r
(202)
Σ Ky (2+r + (1 - (x)))((()))
(203)
(N- - 1)(1-k3) Ks)
(I, = M)
(204)
Kr
(I, # M)
where N is the number of token in the state space S [N]
35

---

D
Additional details on the experiments
In this section, provide detailed descriptions of the experimental tasks, evaluation metrics, and experimental setups used throughout this work. The code is available at here.
D.1 Task details
Discrete Ising model. We consider the Ising model defined on a 2D grid {-1, 1] with size L. The energy function & [-1, + R is given by:
E(x) = B
I,Z; + Σ (1.1)
(205)
where {i,j) denotes pairs of neighboring spins, is the interaction strength, μ is the magnetic moment, and | is the inverse temperature. We employ periodic boundary conditions and specifically focus on the ferromagnetic setting (J > 0) without external fields (μ = 0), reducing the energy function to:
{(x) = -3J Σ T.TT (F.)
(206)
We fix the interaction strength at J = 1.0 and examine various temperatures through B.
For evaluation, approximate ground truth samples generated using extended Gibbs sampling run consisting of 10k burn-in steps, thinning every 10 steps, and 4 parallel chains, collecting 300k samples in total.
GB-RBM. The Gaussian- Bernoulli Restricted Boltzmann Machine (GB-RBM) task involves two continuous visible units following Gaussian distributions and three binary hidden units following Bernoulii distributions, with the energy function:
Caussian-Bernoulli
E(11, = Σ [2] al2 22) - E- Wa2,
(207)
where 3 a R2, Z2 b E {0. 13 Σ E. and multiple modes, specifically six modes in continuous dimensions (see Figure 5). We set:
R2x3 Parameters are selected induce
10
10
[0,0], b b= -- -5,, = W
(208)
0
10
0
Approximately ground truth samples are generated using Gibbs sampling with 10k burn-in steps, 100-step thinning intervals, and 100 parallel chains, collecting 100k samples.
JointDW4. JoinDDW4 exemplifies the molecular sequence-structure co-generation task, extending the classical four four-particle double-well (DW4) benchmark with particle-tpe--ependdee interactions. This setup includes 4 particles in 2D space, each assigned discrete types, yielding 12-dimensional (8 continuous, discrete) energy function:
EJoinDDW4 (x, t)
] alti (d), do) + ((t,, - do) + c(t,, (d) - do) (209)
1 2T
1
Σ
2.j
where dej = ||, - xz ll2 is Euclidean distance between the particle i, and I, (1,2) 2) is the type
of particle i. The parameters are set follows:
3.0
-2.5
0.8
0.4
al.,-) 0, -= 1. do = .b=
-2.
-2.8
c
0.4 0.6
(210)
where b(tz, ty) = bi.s, and c(t,, t,) Cest,"
Ground truth samples similarly obtained from Gibbs sampling, running 10k burn-in steps, thinning every 50 steps, across 100 parallel chains, collecting 100k samples in total.
JointMoG. The JointMoG extends Gaussian mixture benchmark commonly used for evaluating diffusion samplers. includes one continuous dimension x R and one binary dimension {-1
1 202
Ezn.ainnMoc b)
|| bl(..
(211)
36

---

Figure 5: Ground truth sample plot of GB-RBM (left) and JointMoG (right). Samples are projected onto the first two continuous dimensions.
with standard deviation a. We scale this model to 20 dimensions (10 continuous, 10 discrete) for benchmarking:
EjoimMon (2, b)
Σ
20 - - b:112.
(212)
with 0.3 to create clearly separated modes. Exact sampling is possible by first sampling discrete variables uniformly and subsequently sampling continuous variables from corresponding Gaussians, providing exact evaluation samples.
D.2 Metries
Evaluation metrics our experiments primarily utilize Wasserstein distances, computed via the Python Optimal Transport (POT) library [13] using exact linear programming. Specifically, measure the distances between 2000 empirical samples generated by our samplers and 2000 ground truth samples uniformly selected from extensive Gibbs sampling exact sampling processes.
The Wasserstein distance of order p between two probability measures μ and v is defined as:
Wolu,)) v) ( inf where (1) (- E P(X X) T(Ax x X) μ((A), x(x x B) v(B) the of all couplings between and v, and d(x, y) denotes the metric on the space.
TTTTim)) / d(z, ()
1/p
(213)
Energy 1-Wasserstein (E-W)). We use the Energy 1-Wasserstein distance our primary evaluation metric. measures the I-Wasserstein distance between the empirical distributions of energy values computed from generated and ground truth samples. This metric is universally applicable across all sampling tasks and effectively captures discrepancies in the energy distributions regardless of the underlying state space and Markov processes involved.
Magnetization I-Wasserstein (M-W1). For the discrete Ising model, we additionally employ the magnetization 1- I-Wasserstein distance. Magnetization for given spin configuration {-1,1]4x defined as the average spin:
M(x) = 1
Σ Z4.
(214)
This metric assesses the discrepancy in magnetization distributions between generated and ground truth samples. Particularly in low-temperature scenarios (e.g.. = 0.4). the system exhibits distinct modes around extreme magnetization values, making this metric especially sensitive to capturing difficulties in multimodal sampling.
37

---

Table The best for the learning rate for the learned intermediate estimator.
combination for EGM and Bootstrapping (BS). Flow LR stands
Tasks
Method
Hidden dim.
of layers
Flow LR
EGM
256
3
Ising 5, 0.2
BS
256
3
2 x 10-3
0.05
EGM
256
3
Ising 5 5, 0.4
BS
256
3
10
0.05
EGM
1024
Ising 10 x 10, g = 0.2
BS
512
3
10
0.05
EGM
256
3
Ising 10 x 10, 8 = 0.4
BS
2048
10
0.05
EGM
128
6
GB-RBM
BS
128
6
10
0.01
EGM
128
6
JointDW4
BS
128
6
10-3
0.01
EGM
128
JoinMMo
BS
128
10 3
0.01
Sample 2- Wasserstein (x-W2). Specifically used for the GB-RBM task. the sample 2- Wasserstein distance evaluates discrepancies between the empirical distributions of generated and ground truth samples projected onto the first two continuous dimensions. A high sample 2- Wasseesttein distance coupled with low energy 1- I-Wasserstein may indicate mode collapse within certain low-energy modes. Due to interpretability concerns (e.g., a poor sampler generating trivial solutions might misleadingly score well), we do not employ this metric for tasks beyond GB-RBM.
D.3 Experimental setup
We performed grid search determine the optimal hyperparameters for each experimental task and method, evaluating each configuration using three random seeds.
As baseline, report the performance of traditional Gibbs sampler [16]. Specifically, Gibbs sampling with four parallel chains, each performing 6000 steps, collecting a total of 24,000 samples. For evaluation purposes, we uniformly subsampled 2000 samples from this set.
Across experiments, we employed 2000 Monte Carlo samples for estimations and a training batch size of 300. Both EGM and bootstrapping utilized 100 outer-loop iterations, with each iteration collecting 2000 samples into buffer with a maximum size of 10k. The inner-loop iterations were set to 100 for EGM and 1000 for bootstrapping. We adopted linear masking schedule (k; I), a linear conditional OT schedule (a, = t), and an exponential variance exploding (VF) schedule )²). All samplers were trained using the AdamW optimizer with an initial learning
(a,- max Gras
rate 10 applying cosine learning rate schedule with 7min - Training was conducted on an NVIDIA 3090 GPU (24GB VRAM).
-WVDIAA3090
For bootstrapping, the intermediate energy model E was trained with separately tuned learning rate. Bootstrapping step sizes of c E {0.01, 0.05) were evaluated, and an exponential moving average (EMA) was applied to stabilize estimates from 8
In multi-moda tasks, we applied weighted loss combining discrete transition rate matrix prediction errors and continuous drift prediction errors: LEGM L + Aenttinea with fixed coefficients Atise = 5.0 and Aconi = 1.0.
Additional task-specific details provided below, and optimal hyperparameters are summarized in Table 4.
Discrete Ising model. We employed 3-layer MLP with sinusoidal time embeddings for both the intermediate energy function and the transition rate matrix. Each discrete token representing spin values -1 was embedded in 4 dimensions. Following Gat et al. [14], the transition rate matrix u4(y. (y, æ) was parametrized using a probability denoiser (1)())))) analogous to the Ii #1-prediction in the flow models. Hidden dimensions were explored within (256, 512), with additional trials at [1024, 2048} for the 10 x 10 Ising grid.
38

---

GB-RBM. We utilized a 6-layer residual MLP with 128 hidden units, 4-dimensional discrete embedding, and 64-dimensional continuous embedding. Discrete and continuous embeddings were concatenated and fed into the shared 6-layer MLP. Separate predictor networks subsequently estimated the continuous drift and discrete transition rate matrix. The conditional OT path performed best for both EGM and bootstrapping. We clipped the regression target F; at a maximum norm of 20 and the energy estimator & at 100 to stabilize training.
JointDW4. The network architecture matched that used in GB-RBM. The conditional OT path again yielded optimal performance for both methods. Regression targets F, and energy estimates & were clipped at maximum norms of 100 and 1000, respectively.
JointMoG. We maintained the same 6-layer residual MLP structure as GB-RBM. The VE path achieved superior performance for both methods, configured with maax = 2.0 and min 0.01. The regression targets and energy estimates were clipped maximum norms of 100 and 1000, respectively.
39

---

E Limitations and Discussion
We have presented an energy-driven training framework for continuous-time Markov processes (CTMPs). Our method introduces energy-based importance sampling estimator for the marginal generator and proposes an additional bootstrapping scheme to reduce the variance of importance weights. By lowering this variance, we demonstrate that the bootstrapping approach significantly enhances the sampler's 's performance.
Limitations of our work. Despite the strengths of our approach, several limitations remain. First, we have not extensively evaluated the method high-dimensional tasks due to limited computational resources. While our framework performs well on benchmarks of moderate scale, its scalability complex nigh-dimensional domains-such protein conformer generation- question.
an open
Second, observe that the training process be unstable. We hypothesize that this instability stems from the simultaneous optimization of the CTMP and the energy model. This joint training often leads to degraded sampling performance. We apply exponential moving average updates the energy model, which empirically stabilizes training. Nonetheless, further investigation is required to improve robustness.
Third, our estimator incurs bias due to self-normalized importance sampling and the potential mismatch between the proposal and the true posterior distributions. This bias may compromise the accuracy generator estimation, particularly when the proposal diverges significantly from the posterior. Although the bootstrapping scheme helps reduce this mismatch, its effectiveness depends on the intermediate energy estimator's quality, which may introduce additional bias.
Comparison to LEAPS. We compare our method to LEAPS [20], a neural sampler designed for discrete spaces. Our framework more general in that applies to arbitrary state spaces and Markov processes, including both continuous and discrete cases, whereas LEAPS is limited to discrete domains. Even when instantiated with a discrete sampler, EGM and LEAPS differ fundamentally. EGM relies the prescribed conditional probability paths that mix the target distribution, while LEAPS is built on the escorted transport with temperature annealing. In continuous domains, it has been shown that geometric annealing paths can lead to optimal drifts with high Lipschitz constants [27], which limits sampler performance; whether a similar issue arises in discrete spaces remains an open question.
Additionally, EGM does not utilize an MCMC kernel (analogous Langevin preconditioning in continuous settings), whereas LEAPS explicitly relies on this mechanism. We believe exploring both directions-leveraaee and omitting Langevin preconditioning or MCMC comnels-offed promising avenues for future research.
40

---

Additional results
F.1
Additional qualitative results
We provide additional qualitative results for the experiments in Section 4. In Figure 6, we plot the energy histogram of the GB-RBM and JointDW4 compared to the ground truth sample. The Gibbs sampler baseline, EGM, and Bootstrapping match the ground truth energy histogram. However, the Gibbs sampler on GB-RBM suffers from mode collapse as demonstrated in Figure 3.
0.8
Ground Truth
Ground Truth
ECM
0.5
EGM
0.6
Bootstrapping
Bootstrapping
Gibbs
0.4
Gibbs
Density
Density
0.4
0.3
0.2
0.2
0.1
0.0
0.0
10
15
-10
Energy
Energy
Figure 6: Energy histograms of various samplers the GB-RBM (left) and JointDW4 (right).
F.2 Effective sample size of our MC estimator
We present quantitative evidence of the bootstrapped estimator's superiority over the naive EGM. Since the true marginal generator is intractable, we assess estimator quality via the effective sample size (ESS):
w (E -i=1 72
ESS
(215)
where v; denotes the unnormalized importance weight with the i-th proposed sample and is the total number of MC samples. We report the normalized ESS to indicate the fraction of effectively used samples. Figure shows the average normalized ESS over the course of training. The bootstrapped estimator maintains a significantly higher ESS during training, confirming its improved utilization of proposed samples compared to the naive EGM.
1.0
Bootstrapping
EGM
0.8
0.6
ESS
0.4
0.2
0.0
20
40
60
80
100
Outer iteration
Figure 7: Effective sample size (ESS) of the Monte Carlo estimator during training on the Ising model (10 x 10, B = 0.4). ESS is evaluated each regression point and then averaged across all points. In the early training phase (shaded region). the energy model is insufficiently trained, so ESS estimates are unreliable.
41

---

F.3 Additional quantitative results
Comparison to additional baselines. Because our proposed method applicable any state space and any Markov process, it can also be directly applied to purely continuous settings, where most neural samplers have been developed. Although many of these existing methods are specialized for continuous domains, we include them as references to contextualize our results. For discrete settings, we additionally include specialized baselines to further demonstrate the performance of our approach. As emphasized earlier, prior neural samplers are typically designed for either continuous or discrete domains and often rely on domain- speciffc architectures to achieve efficiency, whereas our method imposes no such constraints.
For continuous tasks, we compare against iDEM and PIS on the Many benchmark [35]. For discrete tasks, we compare against LEAPS on a 15 x 15 Ising model.
Table 5: Performance of EGM compared to PIS and iDEM on ManyWei Well32. BS denotes boot- strapping.
Table 6:
Performance of EGM compared to LEAPS on the Ising model (15 x 15, 0.28). BS denotes bootstrapping.
Algorithm
8-W1 4
x-W2 4
of modes
Algorithm
E-W; &
M-W1 +
EGM + BS
3.30
7.07
4
EGM
0.89
0.06
PIS
3.39
6.14
EGM BS
0,65
0.04
iDEM
5.53
7.74
4
LEAPS
0.68
0.01
LEAPS + MCMC
0.49
0.01
The results show that EGM matches slightly outperforms existing diffusion-based samplers (iDEM, PIS) in continuous settings. In the discrete case, LEAPS performs comparably to EGM+ BS (bootstrapping). Notably, LEAPS combined with MCMC achieves slightly better performance than EGM+BS, albeit at the cost of significantly more energy evaluations (45B vs. 85B).
Computational complexity analysis. For reference, Table summarizes the computational plexity of EGM compared to a classical Gibbs sampler. We report the number of energy evaluations, wall-clock time per training epoch, and GPU memory usage 10 10 Ising model with B 0.4. All experiments are conducted on an NVIDIA RTX 3090 GPU.
Table 7: Computational complexity of EGM compared to the Gibbs sampler.
Method
Energy evals
Wall-clock time (1 epoch)
Memory footprint
EGM
200M
0.16
12 GB
EGM BS
200M
0.038
12 GB
Parallel Gibbs
280K
1 GB
Effect of bootstrapping time step. We also study how performance varies with the time gap = r - t > 0 between two time steps r and t. Table reports the results on a 5 x 5 Ising model with
3=00 0.4.
Table Effect of the bootstrapping time gap c on performance.
E-W1
M-W 4
0.01
2.84
0.32
0.02
0.96
0.09
0.05
0.84
0.08
0.10
0.58
0.08
0.20
1,90
0.09
0.50
3.73
0.20
As decreases, bootstrapping generally improves performance. However, if becomes too small, the scale of the conditional generator Fij can grow rapidly, causing large fluctuations in loss magnitude
42

---

across time steps. This leads to unstable neural network optimization due to inconsistent gradient scales. Consequently, choosing moderately small is critical for stable training.
EMA and training stability. Finally, we examine the effect of applying an exponential moving average (EMA) the parameters of the energy model. EMA a standard stabilization technique in reinforcement learning. used to mitigate the moving-target problem when regressing on a learned value function. A similar effect observed here: applying EMA significantly stabilizes training when the generator is conditioned on learned energy model. Table 9 shows results with and without EMA on a x Ising model with 0.4. All other hyperparameters are fixed.
Table 9; Effect of EMA on training stability.
E-W1 +
M-W1
With EMA
0.24 10.08
2.5110.16
Without EMA
0.45+0.13
Using EMA substantially reduces the variance of both energy-based and magnetization metrics, confirming its effectiveness in improving training stability.
43


Reviews:
[{"review_id": "YpFXxXl3Mc", "reviewer": "Reviewer_Vwnf", "summary": "This paper introduces Energy-based Generator Matching (EGM), which extends the generator matching framework to scenarios where only energy functions are available rather than pre-computed equilibrium samples. The authors propose using self-normalized importance sampling to estimate the marginal generator in the absence of training data, along with a bootstrapping technique that leverages intermediate time steps with learned energy estimators to reduce variance in importance weights. EGM aims to provide a unified framework for training neural samplers across arbitrary continuous-time Markov processes (diffusion, flow, and jump processes) on general state spaces including continuous, discrete, and mixed modalities. The method is validated on several tasks including the 2D Ising model, Gaussian-Bernoulli RBM, and joint discrete-continuous sampling problems.", "questions": "- Missing Baseline Comparisons: Why weren't key energy-based neural samplers like iDEM, DDS, or PIS compared? These methods also work in energy-only settings and would provide much more meaningful baselines than Gibbs sampling alone. Can you provide experimental comparisons showing where and why EGM outperforms these existing approaches?\n\n- The paper acknowledges training instability and bias from self-normalized importance sampling but provides limited analysis. Can you provide theoretical or empirical analysis of: (a) convergence guarantees for the bi-level training scheme, (b) bias bounds for the SNIS estimator, and (c) conditions under which the method is expected to be stable?\n\n- Scalability Demonstration: The experiments are limited to moderate dimensions (≤100). Can you demonstrate the method's effectiveness on higher-dimensional problems where the advantages over traditional MCMC would be more compelling? What are the computational scaling properties compared to existing energy-based neural samplers?\n\n- Relationship to Existing Methods: When examining your estimator in Equation (9) and comparing to iDEM's approach, the techniques appear conceptually similar (both use importance sampling with energy-based weights). Can you clearly articulate the technical differences and when EGM would be preferred over existing methods?\n- While Theorem 1 justifies the bootstrapping theoretically, can you provide more rigorous analysis of when and why it reduces variance? Under what conditions might bootstrapping fail or introduce additional bias through the learned energy estimator?", "limitations": "No limitations are discussed in the main text.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "# Strengths\nThe paper tackles an important problem since many real-world applications naturally provide energy functions rather than equilibrium samples. I also appreciate the theoretical contribution of extending generator matching to energy-only settings, which addresses a practically important limitation. The unified framework supporting different Markov process types across general state spaces is conceptually valuable, and the mathematical development appears sound. The bootstrapping technique with Theorem 1 providing theoretical justification for using intermediate time steps is a clever contribution.\n\n# Weaknesses\nThat being said, I have serious concerns mostly about the experimental evaluation of this paper. \n\n- The authors of this paper fail to compare against key existing energy-based neural samplers. For example, with discrete tasks, LEAPS should have been compared, and for energy-based sampling, several relevant baselines exist.\n\n- Moreover, the experimental scope is insufficient for the claims made. The largest experiments are on 100-dimensional Ising models and 20-dimensional mixed problems, which doesn't demonstrate scalability to the high-dimensional problems where such methods would be most valuable. The authors acknowledge training instability issues but provide only ad-hoc solutions (exponential moving averages) without thorough analysis.\n- The technical quality is solid but not exceptional. The bias introduced by self-normalized importance sampling and potential proposal-posterior mismatch is mentioned but not rigorously analyzed. I think the paper would benefit from theoretical guarantees about convergence properties and bias bounds. The relationship between EGM and existing energy-based samplers like iDEM is unclear. For example, Equation (9) appears conceptually similar to existing importance sampling approaches in the energy-based sampling literature.", "was_revised": true, "final_justification": "I have initial doubts on some points of the work, which has been answered by the authors in their rebuttal. Therefore I recommend weak acceptance to this work.", "rebuttal": "Thank you for your thoughtful and encouraging review. We appreciate your recognition of our theoretical contributions. Please find our detailed responses below.\n\n### **W1, Q1: Missing baseline (e.g., iDEM, PIS, DDS, and LEAPS)**\nWe agree on the importance of more baselines; however, we would also like to emphasize that EGM is not directly comparable with the baselines. EGM is a general framework applicable to discrete, continuous, or even discrete-continuous hybrid tasks, while the baselines focus on one of the discrete or continuous tasks.\n\nNevertheless, we provide a comparison with the baselines. For continuous tasks, we compare with iDEM and PIS on the many-well 32 [6]. We omit DDS from the comparison since it performs similarly to PIS. For discrete tasks, we compare with LEAPS on the 15 x 15 Ising model.\n\n**Continuous task (manywell 32):**\n|Method|Energy w1|Sample w2| # of modes found (in projection plot) |\n|-|-|-|-|\n| EGM + BS | 3.30| 7.07|4|\n| iDEM |5.53|7.74|4|\n| PIS | 3.39 | 6.14 |1 (mode collapsed)|\n\nAs the results show, EGM matches or slightly outperforms existing diffusion-based samplers (iDEM, PIS) in the continuous setting. \n\n**Discrete task (Ising 15 x 15, $\\beta = 0.28$):**\n|Method|Energy w1|Magnet w1|\n|-|-|-|\n|EGM| 0.89 | 0.06 |\n| EGM + BS | 0.65 | 0.04 |\n| LEAPS| 0.68 | 0.01 |\n| LEAPS + MCMC | 0.49 | 0.01 |\n\nFor this case, LEAPS is on par with EGM+BS (bootstrapping). However, one can also observe that LEAPS + MCMC is better than EGM+BS at the cost of using more energy calls (45B vs. 85B).\n\n---\n### **W2-1, Q3: Scalability to high-dimensional problems?**\nPlease refer to our response to W1, Q1, where we present results on a higher-dimensional (d = 225) Ising model to support the scalability of EGM.\n\nAdditionally, our primary goal is to expand the design space (e.g., probability path, Markov transition type) and the application of energy-driven training (e.g., multi-modal case), rather than scaling a method to higher dimensions than previous research. Scaling to higher-dimensional tasks remains a challenge for all energy-driven training algorithms.\n\n---\n### **W2-2, Q2-3: Analysis on training instability is missing.**\n\nWhile the theoretical analyses on stability are outside of the scope of our paper, to alleviate your concern, we provide an empirical analysis of the training stability. \n\nApplying EMA to the energy model can stabilize the training, and we provide a result with and without EMA (on Ising 10 x 10, $\\beta=0.4$). All the other hyperparameters are fixed.\n\n| Metric         | Energy w1  | Magnet w1 |\n| --------       | ---------  | --------  |\n| With EMA       | 2.51±0.16  | 0.24±0.01 |\n| Without EMA    | 10.08±8.65 | 0.45±0.13 |\n\nUsing EMA on the energy-model parameters reduces variance in energy w1 and magnetization score. EMA addresses the moving‑target issue when the generator is regressed on a learned energy model. It is a well‑established technique in reinforcement learning for stabilizing value‑function updates.\n\n---\n### **W3-1, Q2-2: Lack of bias bounds introduced by self-normalized importance sampling.**\nWe show that the SNIS estimator's error decays as $O(1/\\sqrt{K})$, and its bias and variance decay as $O(1/K)$, where $K$ is the sample size. Below, we sketch the proof, and the full rigorous derivation will appear in the revised version. \n\n**Proposition.** Consider an unnormalized density $\\tilde p_1(x_{1})$ and a conditional generator $F_{t|1}^{x_1}(x_{t})$ evaluated on a sample $x_1 \\sim q_{1|t}$. Suppose $\\tilde p_1(x_1)$ and $\\Big\\lVert \\tilde p_1(x_1) F_{t|1}^{x_1}(x_t) \\Big\\rVert$ are sub-Gaussian. Then there exists a constant $c(x_t)$ such that with probability at least $1-\\delta$,\n$$\n\\left\\lVert \\hat F_t(x_t) - F_t(x_t) \\right\\rVert \\leq c(x_t) \\sqrt{\\frac{\\log (2/\\delta)}{K}},\n$$\nwhere $\\hat F_t(x_t)$ denotes the SNIS estimator from eq. 9 using $K$ samples $x_1^{(1)}, \\\\ldots, x_1^{(K)}\\sim q_{1|t}$: \n$$\n\\hat F_t(x_t) = \\frac{\\sum_{i=1}^K \\tilde p_1(x_1^{(i)})F_{t|1}^{x_1^{(i)}}(x_t)}{\\sum_{i=1}^K \\tilde p_1(x_1^{(i)})}.\n$$\nMoreover, its bias and variance are expressed as:\n$$\n\\text{Bias}[\\hat F_t] = \\frac{1}{Kp_t(x_t)^2} \\left(- \\text{Cov}\\left[\\tilde p_1 F_{t|1}, \\tilde p_1\\right] + F_t(x_t)\\text{Var}[\\tilde p_1] \\right) + O\\left(\\frac{1}{K^2}\\right),\n$$\n$$\n\\text{Var}[\\hat F_t] = \\frac{4\\text{Var}[\\tilde p_1(x_1)]}{p_t^2(x_t)K}(1 + \\lVert F_t(x_t)\\rVert)^2,\n$$\n\n**proof sketch)** By Hoeffding's inequality for sub-Gaussian random variables, there exists a constant $C$ such that \n$$\n\\left| \\frac{1}{K} \\sum_{i=1}^{K} \\tilde p_1(x_1^{(i)}) - \\tilde p_t(x_t) \\right| \\leq C \\sqrt{\\frac{\\log (2/\\delta) }{K}},\n$$\n$$\n\\left\\lVert \\frac{1}{K} \\sum_{i=1}^{K} \\tilde p_1(x_1^{(i)}) F_{t|1}^{x_1^{(i)}}(x_t) - \\tilde p_t(x_t)F_t(x_t) \\right\\rVert \\leq C \\sqrt{\\frac{\\log (2/\\delta) }{K}},\n$$\nwith $1 - \\delta$ probability. (Here, $C = \\sqrt{\\text{Var}[\\tilde p_1(x_1)]}$ is a possible choice.)\n\nSince we have bounds for both numerator and denominator of $\\hat F_t$, we can also bound the error of $\\hat F_t$ to $F_t$. The result is as follows:\n\n$$\n\\left\\lVert \\hat F_t(x_t) - F_t(x_t) \\right\\rVert \\leq \\frac{2C}{p_t(x_t)}(1 + \\lVert F_t(x_t)\\rVert)\\sqrt{\\frac{\\log(2/\\delta)}{K}}=c(x_t)\\sqrt{\\frac{\\log(2/\\delta)}{K}}.\n$$\nNow, let $\\hat{A}$ and $\\hat{B}$ denote the numerator and the denominator of $\\hat F_t$, respectively. We also let $\\mathbb{E}[\\hat{A}] = A$ and  $\\mathbb{E}[\\hat{B}] = B$. For sufficiently large $K$, the Taylor expansion is expressed as: \n$$\n\\hat F_t =\\frac{A}{B}+\\frac{1}{B}(\\hat{A}-A) - \\frac{A}{B^2}(\\hat{B}-B)-\\frac{1}{B^2}(\\hat{A} - A)(\\hat{B} - B) + \\frac{A}{B^3} (\\hat{B} - B)^2 + O\\left(\\frac{1}{K^2}\\right).\n$$\nTo derive the final equation for the bias term, one can express the bias term as follows:\n$$\n\\text{Bias}[\\hat F_t] = \\mathbb{E}[\\hat F_t] - \\frac{A}{B}=-\\frac{1}{B^2}\\text{Cov}[\\hat{A}, \\hat{B}]+ \\frac{A}{B^3}\\text{Var}[\\hat{B}].\n$$\nSince $\\text{Cov}[\\hat{A}, \\hat{B}] = \\text{Cov}\\left[\\tilde p_1 F_{t|1}, \\tilde p_1\\right] / K$ and $\\text{Var}[\\hat{B}]=\\text{Var}[\\tilde p_1]/K$, one obtains the conclusion for $\\text{Bias}[\\hat F_t]$.\n\nTo derive the final equation for the variance term, one can combine the sub-Gaussianity of $\\hat F_t$ with the error bound on $\\left\\lVert \\hat F_t(x_t) - F_t(x_t) \\right\\rVert$.\n\n---\n### **W3-2, Q4: Relationship to iDEM is unclear.**\n\nWhile both iDEM and EGM use importance sampling with energy-based weights, they rely on different score identities with distinct applicability.\n\niDEM is based on the target score identity (eq. 5 in [37]), which requires access to the target energy gradient and is restricted to the VE probability path. In contrast, EGM adopts the denoising score identity (eq. 2 in [37]), which only requires the conditional generator and applies broadly to general Markov models and state spaces.\n\nThe two approaches also differ in variance behavior across time. As shown in [37], combining both estimators can reduce variance in data-driven training. Extending this idea to energy-driven training is a promising direction for future work.\n\nIn short, EGM complements and extends iDEM by supporting more general settings.\n\n[37] De Bortoli, Valentin et al. Target Score Matching, arXiv:2402.08667, 2024.\n\n---\n### **Q2-1: Can you provide a theoretical or empirical analysis of convergence guarantees?**\n\nWhile a formal convergence analysis is beyond the scope of this work, to address your concern, we instead provide an empirical study on the convergence behavior of both EGM and the proposed bootstrapping method. \n\nWe report the number of training epochs required for convergence. We define convergence as the point at which the evaluation metric (e.g., energy Wasserstein-1) shows no further significant decrease. (on Ising 10 x 10, $\\beta = 0.4$)\n\n| Method | # of training steps|\n|-|-|\n|EGM| 10K | \n|EGM + BS| 60K |\n\n---\n### **Q5: Can you provide a more rigorous analysis of when and why bootstrapping reduces variance?**\n\nWe prove that bootstrapping reduces variance compared to the non-bootstrapping estimator. We first show that the bias and variance of the bootstrapping estimator with a surrogate energy are close to those of the estimator using the true intermediate energy (i.e., the effect of energy estimation bias is negligible). We then show that bootstrapping with the true energy achieves lower variance than the non-bootstrapping estimator.\n\nThis result suggests that, when the surrogate energy is well-trained, bootstrapping effectively reduces the variance of generator estimation.\n\nWe state the proposition below. A full proof is provided in our response to Reviewer meoi's comment (W1) and will be included in the revised manuscript.\n\n**Proposition.** Consider an fully trained surrogate energy model $\\mathcal{E}_r^{\\phi}(x_r) = \\mathbb{E}[\\hat{\\mathcal{E}}_r(x_r)]$, where $\\hat{\\mathcal{E}}_r(x_r)$ is a biased energy estimator (from eq. 20). Let $\\hat F_t(x_t; \\mathcal{E}_r)$ be the bootstrapping estimator using $\\mathcal{E}_r$ as the intermediate energy function:\n$$\n\\hat F_t(x_t;\\mathcal E_r) = \\frac{\\sum\\_i \\exp (-\\mathcal{E}\\_r(x_r^{(i)})) F\\_{t|r}\\^{x_r^{(i)}}(x)}{\\sum\\_i \\exp (-\\mathcal{E}\\_r(x_r^{(i)}))}.\n$$\n\nThen, the bias and the variance bound of the bootstrapping estimator with surrogate model $\\hat F_t(x_t; \\mathcal{E}_r^\\phi)$ are given by:\n$$\n\\text{Bias}[\\hat F_t(x_t;\\mathcal{E}_r^\\phi)] = \\text{Bias}[\\hat F_t(x_t;\\mathcal{E}_r)] +  O\\left(\\frac{1}{K^2}\\right),\n$$\n$$\n\\text{Var}[\\hat F_t(x_t;\\mathcal{E}_r^\\phi)] = \\text{Var}[\\hat F_t(x_t;\\mathcal{E}_r)] = \\frac{\\text{Var}[\\tilde p_r(x_r)]}{\\text{Var}[\\tilde p_1(x_1)]}\\text{Var}[\\hat F_t(x_t)].\n$$\nwhere $\\hat F_t(x_t)$ is the estimator without bootstrapping. Since $\\text{Var}[\\tilde p_r(x_r)] < \\text{Var}[\\tilde p_1(x_1)]$, the variance of bootstrapping estimator is smaller than the $\\hat F_t(x_t)$.\n\n---\nWe hope our responses have addressed your questions and concerns. Please let us know if there is anything we may have missed or if any points require further clarification."}, {"review_id": "Ba5SjvAUpQ", "reviewer": "Reviewer_meoi", "summary": "Setting\n--\n\nThis is a method to train a continuous time Markov process in order to draw samples from the Boltzmann model:\n\np_{target} \\propto \\exp ( - E(x) )\nwith known energy E(x).\n\nThe goal is to have a single approach to generate  from continuous, discrete, mixed modalities. The method is demonstrated concretely to have as special cases diffusion, flow, and masked diffusion models. \n\nThe work builds on the recent paper of Holderrieth et al, ``Generator Matching'' [16], which\nalready unifies  diffusion models, flow matching, and discrete diffusion models;\nexplores jump processes, and allows for multimodal generation (continuous, discrete, mixed); but is data driven (vs the Boltzmann model in this submission).\n\nAs noted by [16], diffusion, flow, and jump processes are instances of continuous time Markov processes with a linear parametrization. The authors use F_t(x) to define the linear parametrization of a given infinitesimal generator which goes from noisy to clean.    The reparametrization trick (3) relates F_t to the conditional generator with parametrization F_{t|1}^{x_1}. The loss (4) gives a loss from the true F_t to the model F_t^{\\theta}. \n\nApproaches\n--\n\nA first approach is Energy-based Generator matching, where an approximation to F_t (x_t) is obtained with a self normalized importance weighted estimator with respect to distribution q_{1|t}, with the weights being a function of the unnormalized \\tilde p, and the normalization constant being computed as in (8). Eq. (4) then becomes eq. (9). \n\nThe challenge is in the design of q_{1|t}. A naive approach with high variance is to set it to p_{t|1}.  A second ``bootstrapping'' approch with lower variance is to use a nearby time r to t, under the weaker assumption  p_{r|t} \\approx p_{t|r}.  Eq. (14) gives the equation for L_t as a function of L_{t|r}^{x_r} and p_{r|t}.\n\nThen p_{r|t} is estimated using an importance sampler wrt q_{r|t}, which is chosen to be approximated by p_{t|r}, with lower variance.\n\nThis leads to another challenge: that the weights require \\tilde p_r (x_r). The estimator in eq. 18 has a similar high variance problem to the first approach, in that it approximates q_{1|r} \\propto p_{r|1}.  The model for \\tilde p_{r} is trained by noised energy matching.", "questions": "Is there a reason for eq. 5 being included? Is it used?", "limitations": "yes", "rating": 3, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "Strengths:\n--\n\nthe idea is original (notwithstanding the dependence on [16], which is clearly and appropriately cited), the method is clearly explained, and performance on discrete and mixed continuous-discrete examples outperforms a baseline classical Gibbs sampler.\n\n\n\nMain theory limitation:\n--\n\nIt is not obvious to me that eq. 18 *provably* gets around the high variance problem, since it again makes the approximation  q_{1|r} \\propto p_{r|1} for a large jump in time from r to 1 (assuming t is close to r, to allow for small variance in that step).\n\nSo it seems you have just shifted the same problem to another part of the reasoning.\n\nIt is also not at all clear why, or if, the fact that the surrogate loss is used (eq. 20) to get a parametric model for \\tilde p_r solves the variance problem in a meaningful way since the loss is with respect to (19), which has high variance.  This would limit the accuracy with which one could accurately compute \\tilde p_r .\n\nCan you give a more formal arguemnt for why ``bootstrapping'' should be expected to guarantee lower variance? I understand that empirically it appears to do better, but I wonder if there is a theoretical argument to be made?\n\n\nMain experiment limitation:\n--\n\nThe evaluation was performed only on discrete (Ising) and joint discrete-continuous tasks, however the method also applies for the continuous setting. In order to evaluate the efficacy of the method, and in particular the issues that arise in eq. 18 and following, it is reasonable and necessary to benchmark the method against established methods (including those cited by the authors) for sampling from purely *continuous* EBMs (parallel tempering/SMC samplers/SOTA diffusion samplers).", "was_revised": true, "final_justification": "I am maintaining my score due to continued concerns regarding the theoretical contribution.", "rebuttal": "Thank you for your thoughtful and constructive review. We sincerely appreciate your recognition of the originality of our method. Please find our detailed responses to your comments below.\n\n### **W1: Energy estimator still has high variance, so why is bootstrapping better?**\n\nThank you for raising this important concern. Our bootstrapping helps training through a trade-off between approximation bias and estimation variance. That is, our algorithm replaces the high variance Monte Carlo approximation of intermediate energy $\\mathcal{E}_r$ with a biased surrogate model $\\mathcal{E}_r^\\phi$. Similar to actor-critic training in reinforcement learning, this trade-off helps stabilize the training of the marginal generator by replacing the high-variance (unbiased) target with a low-variance (biased) target.\n\nTo further alleviate your concern, we formally prove that the bootstrapping estimator achieves a lower variance compared to the original estimator.\n\n**Proposition.** Consider a fully trained surrogate energy model $\\mathcal{E}_r^{\\phi}(x_r) = \\mathbb{E}[\\hat{\\mathcal{E}}_r(x_r)]$, where $\\hat{\\mathcal{E}}_r(x_r)$ is a biased energy estimator (from eq. 20). Let $\\hat F_t(x_t; \\mathcal{E}_r)$ be the bootstrapping estimator using $\\mathcal{E}_r$ as the intermediate energy function:\n$$\n\\hat F_t(x_t;\\mathcal E_r) = \\frac{\\sum\\_i \\exp (-\\mathcal{E}\\_r(x_r^{(i)})) F\\_{t|r}\\^{x_r^{(i)}}(x)}{\\sum\\_i \\exp (-\\mathcal{E}\\_r(x_r^{(i)}))}.\n$$\n\nThen, the bias and the variance bound of the bootstrapping estimator with surrogate model $\\hat F_t(x_t; \\mathcal{E}_r^\\phi)$ are given by:\n$$\n\\text{Bias}[\\hat F_t(x_t;\\mathcal{E}_r^\\phi)] = \\text{Bias}[\\hat F_t(x_t;\\mathcal{E}_r)] +  O\\left(\\frac{1}{K^2}\\right),\n$$\n$$\n\\text{Var}[\\hat F_t(x_t;\\mathcal{E}_r^\\phi)] = \\text{Var}[\\hat F_t(x_t;\\mathcal{E}_r)] = \\frac{\\text{Var}[\\tilde p_r(x_r)]}{\\text{Var}[\\tilde p_1(x_1)]}\\text{Var}[\\hat F_t(x_t)].\n$$\nwhere $\\hat F_t(x_t)$ is the estimator without bootstrapping. Since $\\text{Var}[\\tilde p_r(x_r)] < \\text{Var}[\\tilde p_1(x_1)]$, the variance of bootstrapping estimator is smaller than the $\\hat F_t(x_t)$.\n\n**proof sketch)** The bias of the energy estimator $\\hat{\\mathcal{E}}_r$ is given as follows (refer to Corollary 3.2 of BNEM [25]):\n$$\n\\text{Bias}[\\hat{\\mathcal{E}_r}(x_r)] = \\frac{\\text{Var}[\\tilde p_1(x_1)]}{2p_r^2(x_r)K} + O\\left(\\frac{1}{K^2}\\right)\n$$\nThus, we can approximate the surrogate energy as,\n$$\n\\mathcal{E}_r^\\phi(x_r) = \\mathbb{E}[\\hat{\\mathcal{E}}_r(x_r)]=\\mathcal{E}_r(x_r)+\\underbrace{\\frac{\\text{Var}[\\tilde p_1(x_1)]}{2p_r^2(x_r)K}}\\_{:=b(x_r)}.\n$$\n\nPlugging the equation into the $\\hat{F}_t(x_t;\\mathcal{E}_r^\\phi)$, we get, \n\n$$\n\\hat F_t(x_t;\\mathcal{E}_r^\\phi) = \\frac{\\sum_i \\exp (-\\mathcal{E}\\_r^\\phi(x_r\\^{(i)})) F\\_{t|r}^{x_r\\^{(i)}}(x_t)}{\\sum_i \\exp (-\\mathcal{E}_r^\\phi(x_r\\^{(i)}))} = \\frac{\\sum_i \\exp(-\\mathcal{E}_r(x_r\\^{(i)}) - b(x_r\\^{(i)}))F\\_{t|r}^{x_r\\^{(i)}}(x_t)}{\\sum_i \\exp(-\\mathcal{E}_r(x_r\\^{(i)}) - b(x_r\\^{(i)}))}\n$$\n\nHere, $b(x_r^{(i)})$ is close to 0 and concentrated to:\n\n$$\nm_b= b(x_t)=\\frac{\\text{Var}[\\tilde p_1(x_1)]}{2p_r^2(x_t)K},\n$$ when we choose $t$ close to $r$ and large $K$. Then, by applying first-order Taylor expansion to the $\\hat F_t(x_t;\\mathcal{E}_r^\\phi)$ and approximation $b(x_r\\^{(i)}) \\approx m_b$, we obtain:\n\n$$\n\\hat F_t(x_t;\\mathcal{E}_r^\\phi) \\approx (1 - m_b^2)\\frac{\\sum_i \\exp(-\\mathcal{E}_r(x_r\\^{(i)}))F\\_{t|r}\\^{x_r\\^{(i)}}(x_t)}{\\sum_i \\exp(-\\mathcal{E}_r(x_r\\^{(i)})))} = (1 - m_b^2) \\hat F_t(x_t;\\mathcal{E}_r)\n$$\n\nTherefore, the bias of the bootstrapping estimator $\\hat F_t(x_t;\\mathcal{E}_r^\\phi)$ is:\n\n$$\n\\text{Bias}[\\hat F_t(x_t;\\mathcal{E}_r^\\phi)] = \\mathbb{E}[\\hat F_t(x_t;\\mathcal{E}_r^\\phi)] - F_t(x_t) = \\text{Bias}[\\hat F_t(x_t;\\mathcal{E}_r)] - m_b^2 \\left(F_t(x_t) + \\text{Bias}[\\hat F_t(x_t;\\mathcal{E}_r)]\\right) = \\text{Bias}[\\hat F_t(x_t;\\mathcal{E}_r)] +  O\\left(\\frac{1}{K^2}\\right)\n$$\nSimilarly, the variance of the bootstrapping estimator is:\n$$\n\\text{Var}[\\hat F_t(x_t;\\mathcal{E}_r^\\phi)] = (1-m_b^2)^2 \\text{Var}[\\hat F_t(x_t;\\mathcal{E}_r)] \\approx \\text{Var}[\\hat F_t(x_t;\\mathcal{E}_r)]\n$$\nsince $m_b < 1$ for sufficiently large $K$.\n\n\nAlso, the variance of the bootstrapping estimator $\\hat F_t(x_t;\\mathcal{E}_r)$ with true energy is given by: \n$$\n\\text{Var}[\\hat F_t(x_t;\\mathcal{E}_r)]=\\frac{4\\text{Var}[\\tilde p_r(x_r)]}{p_t^2(x_t)K}(1 + \\lVert F_t(x_t)\\rVert)^2,\n$$\n\nConsequently,\n$$\n\\text{Var}[\\hat F_t(x_t;\\mathcal{E}_r^\\phi)] \\approx \\text{Var}[\\hat F_t(x_t;\\mathcal{E}_r)] = \\frac{\\text{Var}[\\tilde p_r(x_r)]}{\\text{Var}[\\tilde p_1(x_1)]}\\text{Var}[\\hat F_t(x_t)].\n$$\nBecause $\\text{Var}[\\tilde p_r(x_r)] \\ll \\text{Var}[\\tilde p_1(x_1)]$ is very small, the variance of the bootstrapping estimator is smaller than the $\\hat{F}_t(x_t)$. Consequently, with an fully trained energy model, we **reduce the variance of the SNIS generator estimation**. \n\n[25] OuYang et al. BNEM: A boltzmann sampler based on bootstrapped noised energy matching, ICLR 2025.\n\n---\n### **W2: Benchmark on the continuous task is missing.**\n\nThank you for the suggestion. To evaluate EGM in a continuous setting, we added results on the many-well 32 benchmark, comparing against iDEM and PIS, a diffusion-based baseline.\n\n\n**Continuous task (manywell 32):**\n|Method|Energy w1|Sample w2| # of modes found (in projection plot) |\n|-|-|-|-|\n| EGM + BS | 3.30 | 7.07 |4|\n| iDEM | 5.53 | 7.74 | 4|\n| PIS | 3.39 | 6.14 | 1 (mode collapsed) |\n\nAs the results show, EGM matches or slightly outperforms existing diffusion-based samplers (iDEM, PIS) in the continuous setting. \n\n---\n### **Q1: Is there a reason for Eq. 5 being included?**\nEq. 5 was included to illustrate how to solve the intractability of marginal generator matching for data-driven training. However, since it is not used later in the paper, we will remove it in the revision.\n\n---\nWe hope our responses have addressed your questions and concerns. Please let us know if there is anything we may have missed or if any points require further clarification."}, {"review_id": "a771SOxvxo", "reviewer": "Reviewer_29F6", "summary": "This paper tries to solve the problem of sampling from distribution with given energy function (unnormalized log probability). Given a target distribution, the authors introduce a series of intermediate distributions similar to diffusion/flow matching model. They then introduce a generator function $F_{t}^{\\theta}$, which is trained with the generator matching loss. To reduce the variance in the importance weight, the authors further introduce an additional bootstrapping trick. The authors evaluate their method on several toy examples and compared with the basic Gibbs Sampler.", "questions": "No extra questions", "limitations": "Yes, the authors have adequately addressed the limitations and potential negative societal impact", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 2, "originality": 2, "strengths_and_weaknesses": "***Strength:***\n\nThis paper proposes a valid sampler and provides sufficient derivations and theoretical justifications to facilitate reader understanding.\n\n***Weakness***\n\nThe proposed method is only evaluated on relatively simple distributions and is compared solely with the basic Gibbs sampler. In fact, the task of sampling from a given energy-function remind me of the research of Energy-Based Model (EBM), where the objective is to learn the underlying unnormalized data distribution.\n\nA classical approach to EBM training is based on Maximum Likelihood Estimation, which involves alternating between updating the energy function and sampling from the given energy function. To accelerate the sampling process, Cooperative Training [1] introduces an auxiliary generator trained via MCMC-based teaching. Subsequent works [2, 3] extend this framework to multi-scale or multi-noise-level variants, which bear strong resemblance to the generator design adopted in this paper.\n\nImportantly, EBM training is typically conducted on highly complex, multimodal distributions—such as natural images or 3D shapes—where the energy functions are implemented as deep neural networks. In contrast, the distributions studied in this paper are relatively simple, fixed, and analytically tractable. Given appropriate hyperparameter settings, it is plausible that existing methods from [1, 2, 3] would already be capable of addressing the tasks considered in this paper.\n\nTherefore, I have concerns regarding the motivation and significance of the proposed approach.\n\n[1] Cooperative Training of Descriptor and Generator Networks\n\n[2] Learning energy-based models by cooperative diffusion recovery likelihood\n\n[3] Improving Adversarial Energy-Based Model via Diffusion Process", "was_revised": true, "final_justification": "My previous concern lies in the relation between the proposed method and the previous EBM model works. The authors raise an interesting point on possible mode collapse during rebuttal, which resolved my concern. Thus, I increase my rating to 4.", "rebuttal": "We appreciate your insightful comment and the connection to existing EBM literature. While these works are relevant, we would like to highlight the key difference in problem setting, as described below.\n\n### **W1: It is plausible that existing methods from [1, 2, 3] would already be capable of addressing the tasks considered in this paper.**\n\nOur task is fundamentally different from the suggested works [1, 2, 3]. The suggested papers focus on energy-based \"models\", while our work focuses on energy-based \"training\". The suggested papers aim to train an energy function \"from data\", while our work aims to train a model \"from energy functions\".\n\nWe hope our responses have addressed your questions and concerns. Please let us know if there is anything we may have missed or if any points require further clarification."}, {"review_id": "7d0n1nCHKu", "reviewer": "Reviewer_UQ9j", "summary": "The paper addresses the problem of drawing independent samples from an unnormalised target distribution $p_{\\rm target}(x)\\propto \\exp(-E(x))$ when only an energy function E(x) is available. Building on the recent generator matching (GM) framework, which trains continuous-time Markov processes (CTMPs) to follow a prescribed probability path via minimisation of a Bregman divergence between marginal and conditional generators, the authors introduce and energy-driven training of CTMPs by estimating the marginal generator via self-normalised importance sampling (SNIS).  Under this approach, a proposal distribution $q_{1|t}$ is used to draw candidate endpoints, and weights are computed from the energy function to approximate the intractable posterior over data endpoints. One of the challenges with the naive implementation of this scheme is that the variance in the importance weights can be high. To alleviate this issue, the authors introduce a bootstrapping approach to reduce the high variance of importance weights by introducing an intermediate at time $r>t$ proposal. The paper includes an empirical study on synthetic benchmarks up to 100 dimensions (Ising grids at various temperatures) and joint discrete–continuous tasks (GB-RBM, double-well sequence–structure co-generation, mixture-of-Gaussians). In all cases, the proposed EGM scheme appears to outperform a parallel Gibbs baseline in terms of energy-Wasserstein distance.", "questions": "1) Choice of $r$: It would seem that, based on Section 3.2, the method works well when $r$ and $t$ are close, and therefore their densities are similar. However, for the experiments I do not believe $r$ (or rather $\\epsilon$) was stated in the main paper. It would be interesting to test the method's efficacy as a function of $\\epsilon$. \n\n2) Comparisons to generator matching: I was quite surprised that the paper does not compare against generator matching, flow matching, or any other method (with the exception of Gibbs sampling). Can the authors explain why they haven't compared against other than Gibbs sampling? \n\n3) Real-world examples: Can the authors explain why they haven't used a real-world datasets within their empirical study?", "limitations": "The authors do not really address the limitations of their method. Had they explored more comparisons to alternative approaches, this might have led them to better understand the settings under which this method may not perform as well as competing approaches. \n\nThe authors do not discuss the negative societal impacts of their work. However, for this type of paper, I do not believe that this a significant issue.", "rating": 4, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 4, "significance": 3, "originality": 3, "strengths_and_weaknesses": "$\\underline{\\text{Quality}}$\n\nThe paper is methodologically and theoretically sound. The authors present very clear derivations of all of the important concepts in the paper, e.g. importance sampling, generator matching, etc. without any obvious omissions. The paper is very well-structure and easy to read. The method itself is clearly described in the algorithm box, making it easy to recreate the methodology. The empirical study is convincing, in terms of how the method compares to the Gibbs sampler approach. \n\nHowever, there are a few weaknesses that might be possible to address. The numerics are a bit simple. The paper omits any systematic analysis of computational complexity. There is no clear guidance on the number of energy evaluations required, the wall-clock time per training epoch, or the memory footprint relative to baselines like Gibbs sampling or diffusion-based methods. There is also a lack of comparisons to other methods, e.g. generator matching (with the exception of Figure 1). Additionally, while variance reduction via self-normalised importance sampling does seem to offer an improvement, the bias introduced by this scheme, and also the surrogate energy approximation, is only discussed informally. Currently, there are no theoretical bias bounds, nor does the empirical study show how bias scales with the number of importance samples, or how surrogate accuracy impacts the results. Finally, all empirical validation is confined to synthetic tasks in at most 100 dimensions, leaving the method’s performance on large-scale, high-dimensional, or real-world problems unknown. \n\n$\\underline{\\text{Clarity}}$\n\nThroughout the manuscript, notation is clear and terms such as: variables, generators, and kernels are introduced in a systematic fashion. The notation also follows a similar style to other papers in this area, which helps readers to compare this paper to other similar works. The figures, especially Figure 1, are well-presented and help to understand the methodology and its relationship with other approaches. The pseudocode in Algorithm 1 offers a clear, step-by-step recipe that helps with comprehension and implementation. \n\nOverall, the paper is very clear, and I think readers who have read the related papers, e.g. flow matching, will find it easy to follow this paper. However, some of the notation occasionally is a bit complicated (but probably hard to avoid) e.g.  symbols such as $\\tilde p$, w, and multiple kernels appear in rapid succession, forcing readers to frequently backtrack to distinguish unnormalised densities from importance weights. Furthermore, some critical results, especially those concerning the intractable backward kernels and the precise role of the surrogate energy, are relegated almost entirely to the appendices.\n\n$\\underline{\\text{Significance}}$\n\nA key contribution of this work is its unified treatment of diverse data modalities. By casting EGM in a measure-theoretic framework, the authors demonstrate how continuous, discrete, and hybrid sampling problems can all be addressed under a single continuous-time Markov process paradigm. This is a nice contribution which addresses a gap in energy-based generative modelling. The bootstrapping trick, which introduces a learned surrogate energy at an intermediate time for variance reduction, represents an interesting approach that has potential utility beyond the specific context of EGM, suggesting broad applicability to other importance-sampling and sequential Monte Carlo methods. Moreover, the method directly addresses scenarios in which only an unnormalised energy is available and no pre-sampled data exist, offering practical value in scientific and engineering settings where normalisation constants cannot always be computed.\n\nHowever, the true significance remains to be demonstrated in large-scale, applied domains. The current empirical evidence is restricted to synthetic benchmarks, leaving open the question of whether the theoretical benefits translate to real-world gains in fields such as molecule generation or image generation. The empirical comparisons are also narrowly focused on parallel Gibbs sampling; without head-to-head evaluations against cutting-edge alternatives. For example, it's a bit surprising that there are no comparisons with any diffusion-based schemes or flow matching, or even generator matching (especially given that this paper builds on generator matching). Therefore, the claim of broad applicability is weakened. Finally, while the unification of modalities is appealing, some may view the contribution as an incremental extension of existing generator-matching, rather than a unique and novel contribution.\n\n$\\underline{\\text{Originality}}$\n\nEmbedding importance sampling directly into generator matching is a new idea and an interesting methodological contribution. Although generator matching and diffusion-based samplers have been explored previously, to the best of my knowledge, this paper is the first to integrate self-normalised importance sampling in the marginal-generator estimation step, supported by a clear theoretical justification. The introduction of a bootstrapping mechanism via a learned surrogate energy at an intermediate time is quite creative, offering a new approach to control the variance, which I haven't seen before in the energy-based sampling literature. \n\nThat said, the work does draw heavily on existing frameworks. The mathematical foundation closely parallels that of iDEM and other diffusion samplers, and the bootstrapping idea, while inventive, relies on surrogate modelling techniques. This is potentially an issue because this now adds an additional layer of approximation which wouldn't be needed if the importance proposal used the simpler $q \\propto p$ form. The discussion of connections to GFlowNets and discrete flow models, though present, is somewhat limited (e.g. no numerical comparisons). It would have been interesting to see a deeper exploration of these relationships which could have further highlighted the paper’s distinctive contributions. Ultimately, the originality hinges on the effective combination of the importance sampling, with bootstrapping, within generator matching. This is interesting in itself, but could be seen by some people as a specialised rather than a broadly transformative advance.", "was_revised": true, "final_justification": "I appreciate the authors' response to my queries and for the extended results. However, as I noted to the authors, I still believe this paper sits on the border of acceptance and I maintain that position.", "rebuttal": "### **W1-1: No analysis on computational complexity.**\nTo answer your question, we present a table of computational complexity, which includes the number of energy evaluations required, the wall-clock time per training epoch, and the memory footprint on the discrete Ising model 10 x 10, $\\beta =0.4$.\n\n| Method   | # of energy evals | Wall-clock time (1 epoch) | Memory footprint |\n| -------- | --------   | -------- | -------- |\n| EGM      | 200M       | 0.16     | 12GB     |\n| EGM + BS | 200M       | 0.038    | 12GB     |\n| Parallel Gibbs | 280K | -        | 1GB      |\n\n---\n### **W1-2: The bias and the variance of self-normalized importance sampling are not formally discussed.**\n\nWe provide a formal analysis addressing both the bias from self-normalized importance sampling (SNIS) and the surrogate energy approximation.\n\nSpecifically, we show that (1) the bias and variance of the SNIS estimator decay as $O(1/K)$, and (2) with a well-trained surrogate energy, the bootstrapping estimator closely matches the bias and variance of the bootstrapping estimator using the true energy. Moreover, bootstrapping reduces the variance compared to the non-bootstrapped estimator.\n\nWe state the propositions with proof sketches; for the second proof, see our response to Reviewer meoi (W1). A full rigorous proof will be included in the revised version.\n\n**Proposition.** Consider an unnormalized density $\\tilde p_1(x_{1})$ and a conditional generator $F_{t|1}^{x_1}(x_{t})$ evaluated on a sample $x_1 \\sim q_{1|t}$. Suppose $\\tilde p_1(x_1)$ and $\\Big\\lVert \\tilde p_1(x_1) F_{t|1}^{x_1}(x_t) \\Big\\rVert$ are sub-Gaussian. Then there exists a constant $c(x_t)$ such that with probability at least $1-\\delta$,\n$$\n\\left\\lVert \\hat F_t(x_t) - F_t(x_t) \\right\\rVert \\leq c(x_t) \\sqrt{\\frac{\\log (2/\\delta)}{K}},\n$$\nwhere $\\hat F_t(x_t)$ denotes the SNIS estimator from eq. 9 using $K$ samples $x_1^{(1)}, \\\\ldots, x_1^{(K)}\\sim q_{1|t}$: \n$$\n\\hat F_t(x_t) = \\frac{\\sum_{i=1}^K \\tilde p_1(x_1^{(i)})F_{t|1}^{x_1^{(i)}}(x_t)}{\\sum_{i=1}^K \\tilde p_1(x_1^{(i)})}.\n$$\nMoreover, its bias and variance are expressed as:\n$$\n\\text{Bias}[\\hat F_t] = \\frac{1}{Kp_t(x_t)^2} \\left(- \\text{Cov}\\left[\\tilde p_1 F_{t|1}, \\tilde p_1\\right] + F_t(x_t)\\text{Var}[\\tilde p_1] \\right) + O\\left(\\frac{1}{K^2}\\right),\n$$\n$$\n\\text{Var}[\\hat F_t] = \\frac{4\\text{Var}[\\tilde p_1(x_1)]}{p_t^2(x_t)K}(1 + \\lVert F_t(x_t)\\rVert)^2,\n$$\n**proof sketch)** By Hoeffding's inequality for sub-Gaussian random variables, there exists a constant $C$ such that \n$$\n\\left| \\frac{1}{K} \\sum_{i=1}^{K} \\tilde p_1(x_1^{(i)}) - \\tilde p_t(x_t) \\right| \\leq C \\sqrt{\\frac{\\log (2/\\delta) }{K}},\n$$\n$$\n\\left\\lVert \\frac{1}{K} \\sum_{i=1}^{K} \\tilde p_1(x_1^{(i)}) F_{t|1}^{x_1^{(i)}}(x_t) - \\tilde p_t(x_t)F_t(x_t) \\right\\rVert \\leq C \\sqrt{\\frac{\\log (2/\\delta) }{K}},\n$$\nwith $1 - \\delta$ probability. (Here, $C = \\sqrt{\\text{Var}[\\tilde p_1(x_1)]}$ is a possible choice.)\n\nSince we have bounds for both numerator and denominator of $\\hat F_t$, we can also bound the error of $\\hat F_t$ to $F_t$. The result is as follows:\n$$\n\\left\\lVert \\hat F_t(x_t) - F_t(x_t) \\right\\rVert \\leq \\frac{2C}{p_t(x_t)}(1 + \\lVert F_t(x_t)\\rVert)\\sqrt{\\frac{\\log(2/\\delta)}{K}}=c(x_t)\\sqrt{\\frac{\\log(2/\\delta)}{K}}.\n$$\nNow, let $\\hat{A}$ and $\\hat{B}$ denote the numerator and the denominator of $\\hat F_t$, respectively. We also let $\\mathbb{E}[\\hat{A}] = A$ and  $\\mathbb{E}[\\hat{B}] = B$. For sufficiently large $K$, the Taylor expansion is expressed as: \n$$\n\\hat F_t =\\frac{A}{B}+\\frac{1}{B}(\\hat{A}-A) - \\frac{A}{B^2}(\\hat{B}-B)-\\frac{1}{B^2}(\\hat{A} - A)(\\hat{B} - B) + \\frac{A}{B^3} (\\hat{B} - B)^2 + O\\left(\\frac{1}{K^2}\\right).\n$$\nTo derive the final equation for the bias term, one can express the bias term as follows:\n$$\n\\text{Bias}[\\hat F_t] = \\mathbb{E}[\\hat F_t] - \\frac{A}{B}=-\\frac{1}{B^2}\\text{Cov}[\\hat{A}, \\hat{B}]+ \\frac{A}{B^3}\\text{Var}[\\hat{B}].\n$$\nSince $\\text{Cov}[\\hat{A}, \\hat{B}] = \\text{Cov}\\left[\\tilde p_1 F_{t|1}, \\tilde p_1\\right] / K$ and $\\text{Var}[\\hat{B}]=\\text{Var}[\\tilde p_1]/K$, one obtains the conclusion for $\\text{Bias}[\\hat F_t]$.\n\nTo derive the final equation for the variance term, one can combine the sub-Gaussianity of $\\hat F_t$ with the error bound on $\\left\\lVert \\hat F_t(x_t) - F_t(x_t) \\right\\rVert$.\n\n**Proposition.** Consider an fully trained surrogate energy model $\\mathcal{E}_r^{\\phi}(x_r) = \\mathbb{E}[\\hat{\\mathcal{E}}_r(x_r)]$, where $\\hat{\\mathcal{E}}_r(x_r)$ is a biased energy estimator (from eq. 20). Let $\\hat F_t(x_t; \\mathcal{E}_r)$ be the bootstrapping estimator using $\\mathcal{E}_r$ as the intermediate energy function:\n$$\n\\hat F_t(x_t;\\mathcal E_r) = \\frac{\\sum\\_i \\exp (-\\mathcal{E}\\_r(x_r^{(i)})) F\\_{t|r}\\^{x_r^{(i)}}(x)}{\\sum\\_i \\exp (-\\mathcal{E}\\_r(x_r^{(i)}))}.\n$$\nThen, the bias and the variance bound of the bootstrapping estimator with surrogate model $\\hat F_t(x_t; \\mathcal{E}_r^\\phi)$ are given by:\n$$\n\\text{Bias}[\\hat F_t(x_t;\\mathcal{E}_r^\\phi)] = \\text{Bias}[\\hat F_t(x_t;\\mathcal{E}_r)] +  O\\left(\\frac{1}{K^2}\\right),\n$$\n$$\n\\text{Var}[\\hat F_t(x_t;\\mathcal{E}_r^\\phi)] = \\text{Var}[\\hat F_t(x_t;\\mathcal{E}_r)] = \\frac{\\text{Var}[\\tilde p_r(x_r)]}{\\text{Var}[\\tilde p_1(x_1)]}\\text{Var}[\\hat F_t(x_t)].\n$$\nwhere $\\hat F_t(x_t)$ is the estimator without bootstrapping. Since $\\text{Var}[\\tilde p_r(x_r)] < \\text{Var}[\\tilde p_1(x_1)]$, the variance of bootstrapping estimator is smaller than the $\\hat F_t(x_t)$.\n\n---\n### **W1-4, Q3: Performance on high-dimensional real-world problems is unknown.**\n\nWhile the current experiments are in relatively low-dimensional settings, the Ising model is a widely used benchmark with relevance to real-world physics problems. More broadly, energy-driven training is still at an early stage, and much of the existing literature focuses on validating methods on synthetic or controlled tasks. Scaling to high-dimensional real-world problems is a promising direction for future work in this field.\n\n---\n### **W2-1: A bit complicated notation $\\tilde{p}, \\tilde{w}$ and multiple kernels appear in rapid succession, which may force readers to backtrack.**\n\nThank you for the helpful comment. We understand that the use of notations such as $\\tilde{p}$ and $\\tilde{w}$, along with multiple kernels introduced in close succession, can hinder readability. In the revised version, we will make an effort to briefly clarify these notations whenever they reappear, so that the exposition remains accessible without requiring readers to backtrack.\n\n---\n### **W2-2: Some important results (intractability of the backward kernels) are primarily deferred to the appendices.**\n\nThank you for the feedback. We agree that the intractability of the backward kernels is important, and we will move the key parts of this analysis from the appendix to the main text in the revised version to improve clarity.\n\n---\n### **W3-1, Q2: No comparisons with other cutting-edge baselines (e.g., any diffusion-based schemes or generator matching)**\n\nWe agree on the importance of more baselines; in response, we include results on the continuous many-well 32 benchmark comparing our method with diffusion-based samplers iDEM and PIS:\n\n**Continuous task (manywell 32):**\n|Method|Energy w1|Sample w2| # of modes found (in projection plot) |\n|-|-|-|-|\n| EGM + BS | 3.30 | 7.07 |4|\n|iDEM | 5.53 | 7.74 |4|\n| PIS | 3.39 | 6.14 |1 (mode collapsed)|\n\nRegarding Generator Matching (GM) and Flow Matching (FM), they typically assume access to samples from the target distribution. As such, they are not directly applicable to our problem formulation. We believe that the included baselines (iDEM and PIS), which also operate in energy-only settings, offer a more relevant and fair comparison and help address your concern.\n\n---\n### **W3-2: Some may view EGM as an incremental extension of GM.**\nWhile it is true that our method builds on known components such as importance sampling and generator matching, we believe that their combination in the energy-only setting with bootstrapped estimation represents a significant and broadly applicable contribution. \n\nUnlike generator matching that requires access to data, EGM operates entirely from energy evaluations, making it applicable to a wide range of domains where sampling is difficult or impossible — including discrete, continuous, and hybrid state spaces. In that sense, we view EGM not as a narrow specialization, but as a general framework for neural sampler training across arbitrary state spaces and stochastic processes (diffusions, flows, and jumps).\n\n---\n### **W4-1: The discussion of connection to other methods (GFlowNet) is presented, but somewhat limited (e.g., no numerical comparison).**\nThank you for the suggestion. While our main focus is on proposing a continuous-time neural sampler for general state spaces, we agree that a deeper investigation into the connection with GFlowNets would enrich the broader context of our work. We consider this an interesting and promising direction, and we would like to explore both theoretical and numerical comparisons with GFlowNet-based methods in future work.\n\n---\n### **Q1: Method's efficacy depending on the choice of bootstrapping time step $\\epsilon$?**\nWe present a table showing how the performance of bootstrapping varies with the time gap $\\epsilon = r - t > 0$ between two time steps $r, t$ (on Ising model 5 x 5, $\\beta = 0.4$).\n| $\\epsilon$   | Energy W_1 | Magnet W_1 | \n| --------| --------| --------|\n| 0.01 | 2.84|  0.32 |\n| 0.02 | 0.96|  0.09|\n| 0.05 | 0.84|  0.08|\n| 0.1  | 0.58|  0.08|\n| 0.2  | 1.90|  0.09|\n| 0.5  | 3.73|  0.20|\n\nAs $\\epsilon$ decreases, bootstrapping improves performance. However, if $\\epsilon$ becomes too small, the scale of the conditional generator $F_{t|r}$ increases rapidly, leading to large fluctuations in the loss magnitude across time. This makes neural network optimization unstable due to inconsistent gradient scales. Hence, selecting a moderately small $\\epsilon$ is crucial for stable training."}]

Author rebuttal:

Meta-review: This paper adapts generator matching (GM) to the case where the target distribution is given as a normalized positive function, using importance sampling and bootstrapping techniques to reduce variance of the important sampling estimator.

Most reviewers found the problem tackled by the paper important and believe this paper introduces a valuable theoretical advancement in generalization of GM to energy functions. 
The main criticism in the reviews was about: 1) missing comparisons against relevant baselines; 2) theory questions regarding high variance, missing convergence proof and bias bounds, and connection to other energy based sampler; 3) method not tested in scale, experiments include relatively simple distributions; and 4) method somewhat incremental over GM. Points 1 and 2 were addressed to reasonable extent in rebuttal and two reviewers accordingly revised their scores. 3 and 4 are valid points but do not seem to hinder the significance of the suggested approach (e.g., scale can happen in follow up paper). Overall this paper seems to be in the weak accept regime.
Decision: Accept (poster)