Discrete Guidance Matching: Exact Guidance for Discrete Flow Matching
Authors: Zhengyan Wan
Yidong Ouyang
Liyan Xie
Fang Fang
Hongyuan Zha
Guang Cheng
Venue: ICLR 2026 Poster (2026)
Keywords: Discrete Flow Matching
Guidance
Posterior Sampling
Primary area: transfer learning, meta learning, and lifelong learning

Abstract: Guidance provides a simple and effective framework for posterior sampling by steering the generation process towards the desired distribution. When modeling discrete data, existing approaches mostly focus on guidance with the first-order approximation to improve the sampling efficiency. However, such an approximation is inappropriate in discrete state spaces since the approximation error could be large. A novel guidance framework for discrete data is proposed to address this problem: we derive the exact transition rate for the desired distribution given a learned discrete flow matching model, leading to guidance that only requires a single forward pass in each sampling step, significantly improving efficiency. This unified novel framework is general enough, encompassing existing guidance methods as special cases, and it can also be seamlessly applied to the masked diffusion model. We demonstrate the effectiveness of our proposed guidance on energy-guided simulations and preference alignment on text-to-image generation and multimodal understanding tasks.

Published as a conference paper at ICLR 2026
DISCRETE GUIDANCE MATCHING: EXACT GUIDANCE FOR DISCRETE FLOW MATCHING
Zhengyan Wan"
Yidong Ouyang" Liyan Xie
Fang Fang"
Hongyuan Zha
Guang Cheng²
Equal contribution
*Corresponding author
School of Statistics, East China Normal University
2 Department of Statistics, University of California, Los Angeles
3 Department of Industrial and Systems Engineering, University of Minnesota
4 School of Data Science, The Chinese University of Hong Kong, Shenzhen
ABSTRACT
Guidance provides simple and effective framework for posterior sampling by steering the generation process towards the desired distribution. When modeling discrete data, existing approaches mostly focus guidance with the first-order approximation to improve the sampling efficiency. However, such an approxima- could
tion is inappropriate in discrete state spaces since the approximation be large. novel guidance framework for discrete data is proposed to address this problem: we derive the exact transition rate for the desired distribution given a learned discrete flow matching model, leading to guidance that only requires a single forward pass each sampling step, significantly improving efficiency. This unified novel framework is general enough, encompassing existing guidance methods as special cases, and it can also seamlessly applied to the masked diffusion model. We demonstrate the effectiveness of our proposed guidance on energy-guided simulations and preference alignment on sik-too-mmae gen- eration and multimodal understanding tasks. The code is available at [https: Maaiicee-Matcooo Matching
com h/t.thaaa..cc///aaaaaaaaaaaaaaaa
INTRODUCTION
Discrete diffusion models (Austin et al., 2021: Campbell et al. 2022 Sun al. 2023 Lou et al. 2024) and discrete flow-based models (Campbell et al.) 2024] Gat et al. 2024; Shaul et al. 2025 Qin et al. 2025) have received significant attention for generating samples in discrete state spaces, providing effective alternatives to autoregressive (AR) models. Additionally, several works aim to study how to guide a pre-trained discrete model towards desired distribution (Vignae al.) 2023 Schiff et af.] 20255 Nisonoff et al. 2025). However, there are some challenges to developing guid- ance mechanisms in discrete cases. The main challenge is that discrete guidance is often associated with a transition probability or transition rate, which requires extra computation for all possible tar- get positions after the transition. To reduce the number of forward passes through guidance models, existing methods treat the model as continuous function and use the first-order approximation to compute guidance efficiently (Vignac et al. 2023 Schiff et al. 2025 Nisonoff al. 2025). Unfortu- nately, this approximation might introduce negligiie approximation errors. Moreover, existing literature only consider some particular cases, like class conditional guidance or energy-weighted guidance, which might not be general enough for various tasks.
We seek to develop a general guidance framework for discrete diffusion and flow-based models, which is illustrated in Fig. [1] Leveraging the Continuous-Time Markov Chain (CTMC) framework, given a pretrained model for sampling from a source distribution and the density ratio between the target distribution and source distribution, sampling the target distribution. To the best of our knowledge, our guidance framework the first exact discrete guidance in general form. Since our guidance formulation is expressed by the density
method identifies the exact transition rate for

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Pre-trained Psfe(*1(1)
Conditional velocity field Mc(2.x]s))
Density Ratio x(x)== (xx)/p(x))
Guidance af(of.>)
Target model The (rf (x)
Target velocity field xf(
Ac+nje (2|x) (2(2)
+
14 (2,x)
C+
Framework of the proposed guidance discrete flow matching. Given pretrained discrete flow matching on source distribution Pi (x) and known density ratio r(z) (x)/ (r). our framework calcu- lates the target velocity field to generate target distribution (zx). Our framework general enough deal with energy-guide sampling with p(?) (ar) pi (x))e 7F(x), (x))= shown in Section 4.1 classifier guidance shown
Figure 1:
in Theorem and preference alignment in Section 4.2
Table I: Overall comparison with existing discrete guidunce. Our posterior-bused guidance provides exact for- mulation of guidance for generul target distributions with sumpling-efficien implementation. Existing methods are either limited to dlass-conditions generation or suffer from non-neg negligile approximation errors with multiple func- tion evaluations in each sampling step. A more detailed discussion is provided in Section 3
Guidance Method
Formula
Exact Guidance
# of function evaluations
Posterior-Basee (Ours)
Equation
Rate-Based Nisonoff Biteonof al. 2025
Equation 9
D-1;Dx (1S 1)
First- First-Order Approximated Nisonoff et al.
2025
Equation 13
Vignac et al. [2023]: Schitl et al. 2025
Note: In Nisonoff 2025
guidance requires D (|S] 1) + function evaluations. When implemented
using Algorithm the number of function evaluations reduces to D
ratio between the target distribution and source distribution, it also encompasses existing guidance methods as special cases. This is because the density ratio can be calculated by the energy function energy-guided sampling, the ratio of the classifier class-conditional generation, and preference alignment in reinforcement learning from human feedback (RLHF). In Table [ we compare the formulation of the guidance and the number of function evaluations in each sampling step. Our framework only requires one forward pass in each sampling step for any initial distribution.
To compute the proposed guidance, we employ the Bregman divergence to train a network for esti- mating the conditional expectation. To further utilize potentially available samples from the target distribution, we further propose a regularization term.
In summary, main contributions follows.
1. We introduce general guidance framework for discrete flow matching, achieving efficient sampling without approximation.
2. We propose to learn the guidance network by minimizing Bregman divergence. We further propose a regularization technique utilize samples from the target distribution.
3. We verify the effectiveness of the proposed framework energy energy-based sampling through simulations, preference alignment on text-to-image, and multimodal understanding bench- mark.
NOTATION. Let [N] (1, N) for a positive integer N. We use de to denote the time derivative function Kr with respect to f. For a D-dimensional vector =, let æd and z'd denote the d-th element of the vector and the (D (D-1)-dimeee -immenniiio vector (= ad-,,, 1,-11,20 2d41 For a positive integer p. let 1p denote p-dimensional all-one vector. We write f(h) = (hh f: RR-. + R satisfies fh)/h (z) satisfying (z) = if æ = z and 6, (z) = if x + 2. Let (a, 2) be the inner product two vectors v, Fuclidean space. We denote ||>|| || as the Fuclidean norm of vector s..
P)
0 as h + 0+ For two quantities
define the Kronecker delta
2

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2 DISCRETE FLOW MODELS
Discrete flow models (DFMs) aim to generate samples from a D-dimensional discrete state-space with probability mass function (pmf) 91, defined on the domain S where denotes the finite state space of each dimension. A natural example of this setting is text generation, where each coordinate corresponds to token drawn from the vocabulary S. begin with, we first introduce some background on discrete flow models (Campbell et al. [0224] Gat et al. 2024 (Sauul et al. 2025)
under the framework CTMC Norris 1998). which we describe in detail in Appendix C
2.1 TRANSITION RATE AND TRANSITION PROBABILITY
Given a CTMC with the marginal pmfss (00)) let uf be the associated transition rate of this CTMC time t, which is |S| P [S| matrix. For 2, E S with = * I, write uf (2,z) for the corresponding entry in this matrix representing the intensity of the transition from state r to state at time If ui (z, x) satisfy the following rate-properties:
u (z, x) 0, for any * I, and
Σ [1((2 0, ZES
and the Kolmogorov forward equation (also known as the continuity equation):
(x(x)
Σ ZESD
uf (xxx
Σ (xxx)))((2)
Σ
of (2,
incominn
outgoing
say uy can generate the probability path gt. Based the above Kolmogorov forward equation, for sufficiently small time step h, we have
(++ 2e (x) + (xr) + o(h) Σ
(zx(x) + n z)h + o(h) Ju(e)) (.(())
SESD
which implies that the transition probability from state = to state æ, for any x, = E Ss
(++))(x)22) 5x(x)) + uf (ax z)h h ohh).
Thus, the transition rate uf is also the generator of the CTMC {gtheeu,
Holderricth et al. 2025
2.2 MODELING TRANSITION RATE VIA MARGINALIZZTION TRICK
Given the transition rate ui (z, x) and current state æ at time we can generate sample from ge+h using the equation 2 with sufficiently small time step h. Therefore, we can learn transition rate uf that can transport from an initial noise distribution qu to a target data distribution Q1- To obtain such transition rate for sampling, natural method is to learn the conditional expectation of the conditional transition rate ui (z, (2)x that generates the conditional probability path 4011 (-1x))
interpolating from noise to the datapoint X1. Then uf (2, = ( (w) [u][ 26 x]]x can generate the target probability path qe (see Proposition 3.1 of [Campblll al.] (0024). i.e., it satisfies the
Kolmogorov forward equation] [ For completeness, we include the proof in Appendix D. We
are free to define the conditional probability path yo11 (-(x))))) and with the corresponding conditional transition rate needed.
is worth noting that learning a [S] P -dimensional vector-valued function (u, (2)x))) of current time and state is intractable when D is relatively large. To handle such high-ddieeniionll setting, a common approach is to construct a coordinate-ve conditional probability path and transition rate,
D II d=1
de (x|x1)
(o (of), and (2, = Σ (2 yu (24 114).
d=1
which means that the elements of the vector x, independent conditional on X1. Here, ug addd(" 2,4,00 a0 (x4)) is the conditional transition rate that generates the conditional probability path of
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A popular choice of probability path and the associated conditional transition rate used in the previ-
ous works (Campbell et al. 2024 Gat et al. 2024) is
Re (60g(=²) - (xd (4) (8)4(=")
qij (x" (xf) (1 Ke )e) (x*) nedng (x²); 2 1=11)) 1 ne
where [0, 1] + [0, 1] is a non non-decreasing function satisfying 0 and R1 = 1.
After defining the conditional path and rate, the marginal transition rate is given by
(2) Σ [u(
Σ
Σ (2" x² (099)11)())111)
d=1
D Sard (u
-
Σ
Egj' (1j(211 is the posterior. Therefore, we only need to learn a D | matrix-valued function U. (3:) (ug. (s, Then we can update each token in parallel during sampling.
where (x1))) -
Training Objective and Sampling Algorithm. Given the conditional transition rate, the uncondi- tional transition rate can be parameterized by learning the posterior o (z])z). Let ((O, 1]) be the uniform distribution on [0, 1]. A simple learning objective for discrete flow models is given by
D
Lg [l rag x1r (xx./b))))))))))))
(x1)
Σ
log Gije (x xx]]
(5)
dml
which is the cross-entropy between the true posterior 9ije and the estimated posterior of (Campbell
al. 2024 Gat et al., 2024 Wang et al. 2025).
Additionally, there are some alternative approaches to training the transition rate. including directly maximizing the ELBO terms the unconditional transition rate (Shaul et al., (025) and mini- mizing a Bregman divergence-type conditional generator matching loss (Holderrieth et al,] (025). It
turns out that parameterization through the posterior gives us a unified framework to formulate the discrete guidance not only for discrete flow-based models, but also for other generative models. We include more detailed discussions in Appendix E.I Moreover, details of the sampling implementa- tion for equation in Algorithm are provided in the Appendix A
DISCRETE FLOW GUIDANCE
In this section, introduce general guidance framework for modeling discrete data. It can be applied to both discrete flow models and discrete diffusion models (Campbell et al.) 2022] Sun al. 2023 Austin et al. 2021 [ou et al. 2024 Shi et al. 2024 [Shhoo al. 2024). More detailed discussions can be found in Appendix
3.1 PROBLEM FORMULATION
Consider a CTMC with a probability path pt associated with a transition rate uf. Suppose that the conditional transition rate and the conditional probability path are prespecified (see equation 4 for an example), and we are given a pre-trained posterior Piie such that the unconditional transition rate uf (2, x) Ex, (x))))))))))))
[u[ (z, x|]]]] can generate the probability path pt. We will call the distribution with the pmf p1 the source distribution. Our goal is to generate samples from the target distribution by rectifying the posterior or the unconditional transition rate corresponding to the
[u]
probability path Pt-
3.2 EXACT DISCRETE GUIDANCE
To begin with, first impose the following assumption the target distribution.
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The target distribution 9u is absolutely continuous with respect the source distri- bution Pt. i.e. the support of the target probability mass function is a subset of that of the source probability mass function: {x S 91 (z) >
Assumption 1.
0} S f SD pr p1)2) 0}.
This assumption admits a well-defined density ratio r(w) = Φi (z)/pi on the support of pt, which is crucial for our formulation of discrete guidance. the special case setting of conditional generation, the target distribution is the conditional distribution Qı (z) = pı (xly) where denotes the conditioning variable, and the Assumption | holds by construction.
Now, assume that the density ratio r(z) (x) (x) is known. The following theorem yields the rectified posterior Qıje by reweighting the source posterior Pijs with guidance term.
Theorem (Posterior-basee guidance). Suppose that the conditional probability path of the source distribution Pt. and that of the target distribution 91l1 are chosen to be the same for any E [0, 1]. the posterior regarding the target distribution has the following form:
Under Assumption [
E. Ex)" "xxxx xf =xf==" [r(x1)] (1))(=40))
qu|(=" x)) =
(6)
Ex Expepie [r(x))
In particular, if (x) = pı (xlg), we have
(x)(==>) = P11e(= plulxt = 2tt X1 =
p(y)x Xt x)) Pile (x)))
Here, only assume that the conditional probability paths the same. Theorem I provides a general guidance framework for both discrete diffusion and flow models, because both of them can be modeled via parameterizing posterior densities. In discrete diffusion models, if we further assume that the transition rates of both forward noising processes are equal, then we can generalize and recover the predictor guidance in Nisonoff al. (2025), which we state in the following.
Theorem 2 (Rate-based guidance). Suppose that the transition rates of the forward noising pro- cesses Qi 2) and (x, z) are equal for any t E [0, I]. Under Assumption 1 the backward transition rate (2, of probability q: has the following.form:
(2, Ex (x [r(x)]
(x [r(x)] uf (2,
(8)
where u2(,, x) and x2) is the time reversal of (2 2) and Qi (I, 2), respectively. Furthermore, if 9f (xx) Pi (x(l)) then we have
p(y)x 2) p(y) X; = x))
(f(2,, x oly)
"n (2,
which recovers Equation 2 in Nisonoff et al. (2025).
Theorem requires the source and target corruption rate matrices to be the same, which is stronger
than Theorem D Although Theorem 2 gives transition rate that can generate the target prob- ability path, the transition rate we get is not necessarily the same as what we want (2](2, x) = ))) However, Thcorem provides a guidance term for the posterior and leads [x [ug(2
[a)
the desired transition rate (2, See further discussion in Appendix E.3
In general cases, the guidance is time-dependent according to Theorem and Theorem How- ever, the masked diffusion and flow models with x; -iideependent scheduler, the guidance time- independent. To see this, note that in those cases the posterior is time-independent (see Theorem of Ou et al. (2025) and Proposition 6 of Gat et al. (2024) for details). Since the guidance the integral of the product of the density ratio and the posterior, the lime-independent posterior can lead to time-independent guidance.
3.3 TRAINING OBJECTIVE
According Theorem given the density ratio r(x), can obtain the exact guid- ance term by learning the conditional expectation of the density ratio (2(29 e
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xx]x=x(x [ xx=== "x* (rxx=))))(( [xx]]]] We parameterize the guidance he using neural network. A natural
xx training objective for learning conditional expectation is the Bregman divergence (Banerjee et al. 2005). which is defined as
Do(zlly) = F(x) - F(v) (VF(y), y),
(10)
where F(-) is a convex function. The most common choice of F is F(x) || /2, and then the
Bregman divergence the (o-loss. Unfortunately, as mentioned in Lou et al. (2024), the (2-loss does not work well for learning the conditional expectation of density ratio due to the positive nature of
the density ratio. In our work, we take F(x) (xr log z),, and the training objective given by the following.
Dr ((xx)) |n hx1, (xx1,
1d.00 (xx, - r(x1) h (xf, xi) +C (11)
(0) + C,
where r(x1) = r (x1, x:) ) = (he 1.0 (x) xx), h P. (xP. xz)) and C' is a constant
independent
The loss Lt.p only requires the source distribution data. Additionally, we can utilize samples from the target distribution if they are available achieve better performance. Similar to Ouyang et al. (024), we introduce a regularization term allowing us to learn the guidance h2 with sampling X1 ~ q1 during training. Notice that the minimizer of the objective in equation the underlying posterior Maje- Combining with Theorem I the exact guidance h; is the minimizer of the following objective.
D
n1.00 (xx, x.)
(0) =
(x1)
Σ log d=1
(12)
EEss nd.. (s. x1 (
Thus, our final training objective can be written as C, (0) = (0) + (0). where denotes the hyperparameter controlling the strength of the regularization.
3.4 SAMPLING
After the training stage, we can obtain the learned posterior-based guidance, which is a matrix- valued function Hi SD [0, 1] + R px|si with [H] (z))d, =he (s, 2). At the sampling stage,
given the current state Xt, we first sample x from
h1 (.,x,) (x:
9
(-1x)
pijp( (-1x))
h
for each d E [D), where 1.4 (x) = =xEEs h1 (s, x 11 (8) Then, we sample x by always- valid sampling procedure (see equation 14 in Appendix for details). Specifically, at each step, we input the current state x; into both the guidance network and pretrained posterior network, and then multiply the output to get the final transition probability the target distribution Git (-Ix;) to update the current state x like Algorithm [ The overall sampling algorithm can be found in Algorithm in Appendix A and the detailed design for the general rate-based method proposed in Theorem can be found in Appendix FF
Remark 1. The rate-based guidance involves multiple forward passes, which depend on the number of nonzero elements in the transition rate u. (2,) z 4 X given the current state xt. If use the always-valid sampling algorithm with rate-based guidance Algorithm 3 the guidance entails D + function calls in each sampling step, which is still computationally inefficient; see Fig. (d). For predictor guidance, Nisonoof el al. (2025) proposed to use the first-order approximation estimate the log-ratio log p(y)x;==) for efficient sampling. To be specific, we give the general form based on our proposed rate-based guidance:
p(yxx==))
log Exj-m1n((xaa)) (xx31) ze Ex (og (x z, V. log Ex, (xpepp(((x)o)) (x)x)))))))))]

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Plugging this into equation
2 the approximated guidance is
x,, v log to (o)((())))) ( 2) = exp (2-22,
(2, x:).
(13)
However, this approximation not only introduces approximation error but is also unreasonable for discrete data modeling. since the value of the right-hand side depends mainly on the location of 2 and in Euclidean space. Fortumately, our proposed posterior-baser guidance enables sampling low computational cost without approximation, requiring only a single forward pass. Table summarizes three rypes of discrete guidance.
3.5 GUIDANCE FOR PREFERENCE ALIGNMENT
Given the general form of guidance for discrete flow matching, our framework is able to deal with energy-guided sampling (Zhang et al. 2025 Lu et al. 2023) and also preference alignment in re- inforcement learning from human feedback (RLHF) (Rector-Brooks Brooks et al. 2025). In the following, provide the formulation for RLHF. Let denote the prompt with distribution Pa: Tref (o) /c) de- note the reference policy associated with pretrained model pip (oflc, oz). In the RL stage of RLHF (Ouyang al. 2022), consider to maximize
Ee-perol je-x R(c, o1) - lool( Tr/ (
where 7 is the temperature that controls the deviation from the reference policy. As mentioned in Rafailov et al. (2023), the above objective has the closed-form maximizer
" (01(c) = 3(c) Tref (o exp (R(c.01))
((c..o)) dog and -) is the reward function which can be rule-based or obtained by training a neural network the comparison dataset. Given the reward function, we aim to generate samples 0₁ ~ x= (o)0)/c) conditional on c. If we take pt = (:ef( obtain rectified posterior for sampling from л², i.e.,
where Z(c) = f exp
and q1 (.) xx (-Ic) in Theorem 1
would like use hς (01, C)
We leave the detailed training c) to approximate exp R((...))
objective in equation 22 in Appendix G
EXPERIMENTS
In this section, we present empirical evidence to verify the efficacy of the proposed framework. In Section 4.1 we conduct proof-of-concept experiments using energy-guided sampling. In Section we ilustrate the effectiveness of our method on reinforcement learning from human feedback. Built upon multimodal discrete flow-based model, demonstrate the effecttivness of our method on text-to-image generation and multimodal understanding benchmark.
4.1 SIMULATION RESULTS
Experimental Setup and Baselines. We consider 2-D setting similar to that in Zhang et (2025):
Lu et al. (2023). The sample space is S {0, 1,.... ., 32) We denote the source distribution as where log loy 1[x) is the energy function defined by given classifier p(y 1|x). Our goal is to sample from the guided distribution p¹²² by using either the source transition rate ui or the souree posterior Pıj-- We compare our proposed exact discrete guidance with the predictor guidance introduced in Nisonooff et al. (2025). Note that the density ratio
(xr and set the target distribution as its energy -uidd version p (x)(() x (I(x)e
y 2 0 is the guidance strength and E(x)
in this case is r(x1)= = (z)) (y(=11==1) where 2(7) is constant that does not depend on T1- 2(5)
(1)))
Thus, obtain the posterior baseed and rate- based guidance based Theorem [ and Theorem 2 respectively. For the predictor guidance proposed by Nisonoff et al. (2025). we use the predictor-
(2(2x # x, which is different from our proposed rate-based guidance; Appendix E.5 for further discussion.
guided rate u. ((7)
Bxi pa (111))p
Ex (e) xx)

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Experimental Results. We sample from the 2-D target distribution pi using different guidance
schemes, as shown in Fig. 2 (a-c). For 7 = 0,3, both rate-based and posterior-based guidance produce distributions close to the ground truth. When 7 = 10, 20, the distribution generated by the predictor guidance (Nisonoff et al., (025) differs from the ground-truth distribution. Overall, our posterior- based guidance achieves better performance than its rate-based counterpart, and also yields faster sampling, as illustrated in Fig. 2 (d). We include the experimental details in Appendix H.1 and more experimental results can be found in Appendix H.3
y=
Y=
y
y=
(a) Ground Truth (rings)
(c) Posterior Based (Uniform, Ours)
Soder Bod.
25
25 24
Rate-Based (Masked, Nisonoff etal. (2025))
(d) Sampling Time Comparison
Figure 2: Comparison of selected sampling results (64 steps) in 2-D experiments and sampling efficiency, (a) Ground truth distribution; (b) rate-bat rate-based guidance masked initial distribution (Nisonoff al.) (025); (c)
posterior-based guidance with uniform initial distribution (ours); (d) comparison of sampling times showing posterior-based guidance achieves 1.6x higher sampling speed than rate-based guidance.
4.2 RLHF ON MULTIMODAL BENCHMARK
Experimental Setup and Baselines. We develop the guidance based on state-of-the-art discrete flow matching model on multimodal tasks, FUDOKI (Wang et al., 2025). The detailed probability path for FUDOKI can be found in equation 21 and the experimental details can be found in Ap- pendix H.2 For text-to-image generation tasks, we adopt the same training prompts and reward function Liu et al. (2025b), which utilizes weighted sum of Pickscore (Kirstain et al.) 2023) and GenEval reward (Ghosh et al. 2023) as the reward function. We evaluate the performance on the widely used GenEval Benchmark (Ghosh al. 2023). For multimodal understanding tasks, we adopt the question and ground- truth answers from SEED (Li et al. (203a)) and the widely used LLM-as-a-judge for assigning reward. We adopt VLMEvalKit (Duan et al. (0224) to evaluate on several multimodal understanding datasets, including POPE (Li et al. (023b), MME-P (Fu et al.,
2023 MMB (Liu et al. 2023c), GQA (Hudson & Manning 2019), MMMU Yuu et al. 2023). and
MM- Vet Yu et al. (202)).
Experimental Results. Results on the GenEval benchmark for text-to-image generation are pre- sented in Table 2 Compared to the setting without guidance, our method achieves improvements on four out of six sub-tasks, highlighting the benefit of guidance in utilizing reward signals. We also provide the ablation studies of the regularization strength in Appendix H..)] The performance on multimodal understanding benchmarks is demonstrated in Table Our guidance framework consis-
tently improves performance. Qualitative results can be found in Fig. 3 Overall, the results confirm the effectiveness of our method for preference alignment across multimodal tasks.
Abeastitut modern weoden hrsse,, close to the talos, in the mountaiss of surriins unima style
Bus Times 555 6363 (b) Multimodal Understanding Results
Question: the phone number in the pieture "0131 525 snswer
0131
FUDOX)
Ouro
Arsiver:
No.
Yos.
PUDOR
Our
(a) Text-to- Texi-to-Image Generation Results
Figure 3: Qualitative comparison on text-to-image generation and multimodal understanding.
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Table 2: Visual Generation Performance on GenEval Benchmark.
Type
Method
Single Obj.
Two Citj.
Counting
Colors
Position
Color Attr.
Ovenal
LaamaGnn
Sun 1024
0.71
0.34
0.21
0.58
Fmun1-Cien Wanre m0
2124h
0.98
0..71
0.34
0.81
0.17
0.21
0.54
LDM (Rmm
0.92
0.29
0.23
0.70
0.02
0.05
0.37
0.35
Gen. Only
SDv2.1
PisArt-amii
Mor-hach 21
X7/4
0.98
0.96
0.51
0.44
0.80
0.85
0.07
0.17
0.50
DAUL-E
2073
0.94
0.66
0.49
0.77
0.10
0.19
0.52
(Podall
DALL-E3
[llkkrr
cl
0.96
0.87
0.47
0.83
0.45
0.67
SDD-MMediom [Asoo d8 ml
2024
0.99
0.94
0.72
0.89
0.33
0.60
0.74
SEBD
Ge et 2024
0.97
0.58
0.26
0.80
0.19
0.14
0.49
I.J.UIMF Waan
0.99
0.45
0.71
Tokenflmn
XL Mi 2m
0.95
0.60
0.41
0,81
0.16
0.24
0.55
Chamdlon
Team
0.39
(Wn
Janus-Pu Pu Che Che
0.98
0.82
0.51
0.89
0.73
Show-o
0.95
0.52
0.49
0.82
0.11
0.58
0.53
Trandfiscen 7Jmm 27
3074
0.63
D-DiT
0.97
0.30
0.54
0.76
FUDOKT Wang at
2023
0.96
0.85
0.56
0.88
0.68
0.67
0.77
Ours
0,94
0.36
0.53
0.89
0.70
0.77
0.78
Note: "Und"" UUnkrrstnddng. "Gon." Generation.
models integrating in external pretrained diflusion model. Vilues exceeding the baseline
FUDOKI highlighted gray
Table 3: Multimodal Understanding Performance on Various Benchmarks.
Type
Model
Params
POPE
MME-PP P.
MMB
GQA
MMMU
MM Voc
LLaVA-Pti-1
1.5 at 20z3b
1.38
84.1
1128.0
56.5
30.7
MobiteVLM Cha Dazsy
1.4B
84.5
1196.
53.2
56.1
MobileVLMM
2034
1302.
MobileVLM
39.6
MebaleV1 M-VY
2014
2.78
84.7
14405
63.2
61.1
LLaVA-Pts Zhe
2.78
85.0
1335
59.8
28.9
Und.
LLAVA
809.6
MLAA--T.
8509
15107
64.3
Instrc1BLIPP
78
36.0
49.2
26.2
Owen-V Vi
Chil
78
1487.5
60.6
57.5
IDFICS
38.4
85.2
58.5
60.5
InstreiBl
el 7174
13B
78.9
12128
49.5
25.6
LaVI! et 2123
78
46.8
MetaMerph [Toncet
75.2
Gemini Nazo de 204)
1.80
26.3
ILLUME Wanstell 2 (U66))
7B
88.5
1445.3
65.1
36.2
37.0
1545.9
LWM ei Mr2s3)
7B
75.2
41.8
9.6
VILA-U Wirgh MPA
7B
85.8
1401.8
60.8
33.5
Und. Gem.
Chameloon
2024
224
[Wu ec
1338.0
Janus-Tr-115 (Chm
Chn cl
1.58
86.2
1444.0
75.5
39.3
36.3
39.8
Show o
Xis d:
20125
1.30
73.8
948.4
48.7
25.1
Show-o:3 $11.100
1.30
80.0
1097.1
58.0
26.7
FUDOKT
1.58
86.1
1485.4
73.9
57.6
34,3
38.0
Ours
1.50
86.8
1492.7
74,2
58.2
35.4
38.6
Note: "Und"" UUndessanddng, "Gem." Generation.
models integrating un estemal pretrained diffusion model.
Values exoceding baseline
FUDOKI highlighted
RELATED WORKS
In this section, we review the main related works ours. literature can be found in Appendix B
comprehensive coverage of related
Guidance. line of work has studied guidance for diffusion and Now-matching models in con-
tinuous space (Dhariwal & Nichol, 2021: Lu et al.) 2023 Ouyang et al. 2024 Feng et al.) 2025
Subsequent efforts extend guidance to discrete space (Vignac et al. 2023; Schiff et al., 2025 Li et al. 2024a] Nisonoff al. 2025). but these methods typically rely first-order approximation
to reduce the computational cost. However, this approximation not appropriate in the discrete setting, which work seeks overcome.

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Published as a conference paper at ICLR 2026
Discrete Flow-based Models. Our work primarily builds line of discrete flow-based models (Campbell et al. 2024 Gat et al., 2024 Shaul et al. [025 Qin et al. 2025). These models construct transition rates that generate the target probability path through marginalizing conditional transition
rate, yielding a more comprehensive design space of conditional probability path than discrete dif-
2023] Vignac al. [023 Lou ct al., (224)) including masked diffusion models (Shi et al. 2024 [Shoo et al., 2014 Ou et al. 2025
fusion models (Austin et al.. 2021 Camphell al., 2022: Sun al.
Nie et al. 2025c Zhu et al. 2025 Zhao et al. 20255] Yang et al.
2025). This motivates us to propose posteroor baaeenoo--baeeddae guidance, which provides a flexible and unified perspective for generative modeling
through marginalization.
CONCLUSION
In this work, we introduce a novel guidance framework for discrete flow-based models. Our ap- proach provides exact guidance with improved sampling efficieney, offering unified perspective for constructing guidance across generative models. We further propose to train the guidance network by minimizing the Bregman divergence with regularization. The effectiveness of our framework is validated through experiments on energy-guided simulations, preference alignment for text-to-image generation, and multimodal understanding tasks.
ACKNOWLEDGMENTS
Hongyuan Zha's research is supported in part by Shenzhen Stability Science Program 2023, and National Natural Science Foundation of China (72495131). Fang Fang gratefully acknowledges search support from National Natural Science Foundation of China (72331005). Guang Cheng grate- fully acknowledges financial support from the National Science Foundation (NSF-CNS 2247795).
ETHICS STATEMENT
This work adheres the ICLR Code of Ethics. Our research does not involve human subjects, sensitive personal data, experiments that may pose harm individuals or communities. Publicly available datasets and benchmarks were used all experiments.
REPRODUCIBILITY STATEMENT
The code for our implementation can be found the supplementary materials. For our theoretical result Theorem I and Theorem 2 the assumption is stated in Assumption and the proof can be found in Appendix D.2 and Appendix D.3
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Weihao Yu, Zhengyuan Yang, Linjie Li, Jianfeng Wang. Kevin Lin, Zicheng Liu, Xinchao Wang, and Lijuan Wang. Mm-vet: Evaluating large multimodal models for integrated capa- bilities. ArXiv, abs/2308.02490 2023. URL https: ott.ttsss/ppp..smaatc.c...o.....
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Qinqing Zheng, Matt Le, Neta Shaul, Yaron Lipman, Aditya Grover, and Ricky TQ Chen. Guided flows for generative modeling and decision making. arXiv preprint arXiv:2311.13443, 2023.
Chunting Zhou, Lili Yu, Arun Babu, Kushal Tirumala, Michihiro Yasunaga, Leonid Shamis, Jacob Kahn, Xuezhe Ma. Luke Zettlemoyer, and Omer Levy. Transfusion: Predict the next token and diffuse images with one multi-modal model. In International Conference on Learning Represen- tations, 2025.
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//apii
Doc//ooppusDD:2675550
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A SAMPLING ALGORITHMS
Note that the transition probability (Equation 2 is valid only < [u](x,z] for each x E SP, when we just use its Euler discretization to generate samples. To remove this constraint, following the always- valid sampling procedure introduced in Shaul et al. (2025), given the current state x,, for each d E [D), we first sample x from the learned posterior Dije (-)xe), and then sample xi+h from
uc (++ x (x (1 0x)(.)),
(x (x)
(14)
q.d (xf x xx))
where we the first-order approximation
u (xf x(xx) = + hug (x,, xx1x + o(h).
Algorithm Sampling without Guidance
Require: pretrained posterior Uije. conditional transition rate u² ([)), initial value wo. step size
h 1: 4+0
2: X; f Iо
3: t do
4:
for 1....... do
parallel
5:
Calculate Uije (u+)xe)
Sample xf ~ 9ije d.0 ( | x2)
6:
7:
A +- Exzxe u1-t (8, x3 x+
8:
Sample Zd "jump 00 if Zd jump
hhd then
9:
u. (,, (xf) (1 8x9()
Sample xy
Ad
11:
end if
12: end for
13: + + h
14: end while
15: L L h
16: for d 1,.., D do
>> in parallel
17: Sample x ~ qije ( xx))
d.0
18: end for
19: return X1
B ADDITIONAL RELATED WORKS
Discrete Flow-based Models. Flow-based models for modeling discrete data were introduced by
Campbell et al. (2024): Gat et al. (2024); Shaul et al.] (2025): ; Holderrieth et al. (2025): Qin et al. (2025). Compared to discrete diffusion models (Austin
2021] Campbell al. [002]] Sun al. 2023 Lou et al.) (024), the main ad advantage of discrete flow models is the flexibility of designing the conditional probability path and the conditional transition rate without specifying corruption transi- tion rate; while discrete diffusion models require choosing forward transition rate with a simple form such that the conditional probability path can be computed easily. Similar to the framework of continuous flow models (Albergo & Vanden-Eijndee discrete flow models focus on the conditional path with probability interpolation, which is a convex combination of conditional probabilities. Shaul et al. (2025) proposed the mixture probability path with Fj dependent schedulers and the metric-induced path: the training objective they used is the negative EL instead of the cross entropy used in Campbell al. (024)) and Gat al.) (024
2023 Liu et al. (2022 Lipman et al.) 2023).
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Algorithm 2 Sampling with Fosterior-Based Guidance
Require: pretrained posterior Pilt. conditional transition rate (2, x|x1), initial value z0. posterior-based guidance H.. step size h
1: ++ 0
2: 20
3: while 1 + h do
4: for d = 1d=1,.... ,..... D
do
in parallel
4.8 Calculate the guided posterior Gije (af)x)) or H2 (()( (x1)xx)
5:
6:
qijc (. xx) Sample xf Ad
7:
Expoy up (8, x x
8:
Sample Z Zjump
9:
if 2
jump c hh h2" then
Sample x (,, (1 5xp(.)) end if end for
10:
td
11:
13: 6 / h
14:
end while
15: t 1-h
16: for d 1...,.... do
> in parallel
17: Calculate the guided posterior Life (x11x,) x H& (xx) (44) (x91)x))
d.0 Sample x ~ 4ije xx)
18:
19:
end for
20:
return X1
Recently, Havasi et al. (2025) introduced novel discrete flow matching framework with auxiliary process, which supports variable-length sequence generation.
Discrete Diffusion Models. Motivated by studies of discrete-time diffusion models in continuous
state spaces (Sohl-Dickstein al., 2015; Ho et al. 2020; [Song et al., [021aa] Karras et al. (022).
Austin et al. (2021) and Hoogeboom et al. (2021) proposed discrete-time framework for training diffusion models in discrete state spaces. Similarly, several authors have developed continuous- discrete diffusion models through continuous-time Markov chains (Campbell et al. 2022 [enton al. 2024). [Sun et al. (2023) proposed discrete score matching based on the idea of continuous score matching for modeling continuous-time diffusion models (Song et al. 202 Meng et al. [2022); Lou et al. (2024) introduced concrete score matching, which aims to identity the time- reversal of the noising process like Sun al. (2023). Empirically, the masked diffusion models (Shi ct al. 2024; Sahoo et al. 2024 Ou et al. 2025) outperform the diffusion models with other noise distributions, such as uniform distribution. Moreover, for masked diffusion models, Ou et al. (2025) showed that the training objective is equivalent to that of any-order autoregressive models (Hooge boom al. 2022 Shih et (002)). which is also equivalent the ELBO for discrete flow (see Ap- pendix D.I in Shaul et al. 2025). The main advantage of discrete diffusion models over (any-order) autoregressive models is parallel sampling. However, supporting variable-length sequence genera- tion is considerably more challenging than in autoregressive models. To overcome this limitation, sumrerrvtoeer models (Nie al., 2025c; Arriola et al. 2025) perform block-wise sampling and enable parallel sampling within each block, thus supporting flexible-length sampling.
Guidance. Classifier guidance (Dhariwal & Nichol 202T) and classifier-free guidance (Ho & Sali- Cenerationn(Schifii
mans| 2021: et 2023) are proposed for conditional generation, [Schiff et (2025) explored the classifier guidance tailored to discrete diffusion models by decomposing the reversed transition probability for practical purposes. Nie et al. (2025a) applied Slassifier-free guidance to masked dif- fusion models for unsupervised pretraining. In this paper, the posterior-based guidance provides a unified perspective for the existing guidance in previous work. including classifier guidance Dhari- wal Nichol (202T), energy-weighted guidance Lu et al.] (2023), guidance for transfer learning (Ouyang al. 2024). and flow matching guidance (Feng al. 2025) continuous state space. See
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discussion in Appendix E.1
Moreover, thanks the token-wise transition rate, discrete guid- ance only requires training one network, unlike the training-basee continuous guidance proposed by (Feng et al., 2025). which requires estimating normalizing constant by learning surrogate model.
Diffusion- and Flow-based Large Language Models. Recently a number of studies have focused on developing large language diffusion models. Several post-training approaches have been proposed to fine-tune discrete diffusion models aligned with human preference in a way similar autoregressive models via RLHF (Ouyang et al. 2022) DPO (Rafailov al. 2023 tech- niques. LLaDa (Nie et al. 2025c) leverages masked diffusion models and employs negative ELBO for supervised fine-tuning. It supports variable-length sentence generation by cemi-auutoegrrssivve, and utilizes deterministic low-confidence remasking technique to enhance performance similar to MaskGIT (Chang et al. 2022). LLaDa 1.5 (Zhu et al. 2025) applied DPO to LLaDa, using an empirical estimator with variance reduction techniques to approximate the log-probability. To extend GRPO masked diffusion models, d1-LLaDa (Zhao et al. 2025) proposes estimate log- probability via mean-field approximation and to do one-step unmasking to estimate token-wite wise log- probability with partially masked prompts. learn from multi-step denoising information, MMaDa Yang et al. (2025) perturbs sentence outcomes instead of prompts, and conducts policy optimization with a GRPO-type objective, which supports multi- modality. For the large language model based on discrete flow matching, WWnn al. (2025) developed multi-modal model with metric-induced probability path and kinetic-optimal conditional rate (Shaul et al.] (2025), achieving highly com- petitive performance compared advanced AR-based multi-modal models.
C BACKGROUND OF CONTINUOUS- -IME MARKOV CHAINS
Continuous-time Markov chains (CTMCs) are class of continuous-ting Markov processes on dis- crete state spaces. Given a discrete state space S a stochastic process x; is CTMC if (a) it has Markov property, that is, given the information of current time, the future and history are indepen- dent; (b) given the current time and state E SD, the transition probability is
P(x++ = =//x =x) = =z((z) + ur(2,x)h + o(h),
where h is the step size, and fying
is (time-dynamic) transition rate matrix satis-
ue(2, O, for any 2 # x, and Σ uz(2,z) 0 for any S
Intuitively, the transition rate un (z 2) measures the intensity of transition from the current state x to the target state at time t. This CTMC framework underlies recent advances in generative modeling. Building on the framework of CTMCs, the discrete diffusion models (Austin et al.) 2021 Campbell et al. 2022 Sun et al. 2023 Lou et al. 2024) and the discrete flow models (Campbell et al. [024] Gat et al. 2024 (Shaul et al. 2025] Qin et al. (025) can transport a simple distribution like uniform distribution just point mass to a complex data distribution through sequence of jumps.
D DERIVATION OF THE GUIDANCE
D. PROOF OF PROPOSITION 3. OF (CAMPBELL ET AL. 2024)
We include the proof of Proposition 3.1 of (Campbell et al. 2024 here for completion. We begin by taking the expectation over Pı on both sides of the Kolmogorov equation for P:p (x)x1) and u (,(,,,2201))
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(xp:(x) = Ex (x1)
Depap (x/x))] (definition of marginal over 21))
-Ex
uela 2/x1) Pep (2) [x]]]]
(Kolmogorov forward equation)
pi ((1 ( 2/11)
pi(1)) (Bayes' rule)
Σ (x p(()).
We observe that the final expression corresponds to the Kolmogorov equation for a continuous-ting Markov chain (CTMC) with marginals pe and transition rate (xxrrp (xxl))//< x[xx)]- This demonstrates that Ex (ppp((())))))) [Me (2, xr) x1)] generates p=(r).
D.2 PROOF OF THEOREM I
Proof. By Bayes' formula and the condition 4:/1 Ptj1+ we have
(1j((2/z) del1 (2)2))1(2))
(=)((122)1(2))
91 (z)
(x)))((()) Pije(2
(x(
qc(x)
()))) (1)
(i(e)
(1(+)
(=)
(1 (1 (1)((1))) (1]((//)
Σn,
22 Pa (1p((1
ge (x))
9i (mn Σn,
r(2)
Pxi (rxx)) [x(1]] P1)e(2|1r),
(15)
which implies that
r(z)
[]][(1/=== = Ex (x (x) p(x ddl = 2 >/d|x, = x, x = 2 (=
The result equation 6 follows by taking summation over z E SD for each E [D].
For classifier guidance equation setting (-)-) P1 (-ly) in the previous equation, we obtain that
(r(lv))p1 Pije (1]((2/2)) = plylxx x1 = =)
(1(2lv)p((z)
Pıja(2|0,))
p(yx; = x) P1((((a))
p(y|x1 = 2)
p(u1xe x)) p(x) =>> = x, x = z (=
Thus, suffices to show that ply|xi = 2) = p(y/>1 x1 = = Xx = x). To see this, note that
ply|x1 = 2) p(x) y.y v) p(x: 2/x1 = 2)p(x1 2.y y)
(2)
p(xi = X; =x) I)
p(x; = x X1 2, y)p(x1 2.y 1)
plyl x1 2, x, = =r),
p(x1 = x=xx=10
which completes the proof.
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D.3 PROOF OF THEOREM 2
Proof. By Bayes' formula and the condition ge/t+A Peje4hs we have
(e+h)e(2/2)) Or/e4- (2)=))))2))))1 (1)))(())))
Piceadeebppppeaeea--aelee (z)
(x)
(c(2)
(z)
(e(2) Pc+Mo(=|x).
(16)
By equation
we can obtain that
En, +
(++N)((2/x) =
(62(z) + uf(2, + +(hh)
En, ((((a)))(((ttx))))+++++++))+++a)
(((2)p(()) (z(z) + (2, o (x(2)q((x)
=d((2) + (x) (z, 22) o(h)
= =z) Excepi(xx (x11)
Ex (2, x)) o(h),
where the last equation the trick the proof of Theorem Then, the transition of the probability path is
(=/1)(=/1x) 6x(2)
Ex (x (xxx))((x1) up (2,
w
(2,x)=
lim h-
h
Ex, (x)l))
For classifier guidance, considering (21) p1 (xly), have
(2 x) u²(z, (1, = (x(2)))Pe(x))
(x()))(<<l))) u (z, p(u1x: = 2)
p(ulxi = x:) - (2,
which completes the proof.
IN-DEPTH DISCUSSIONS OF THE PROPOSED GUIDANCE FRAMEWORK
E. UNIFYING GUIDANCE THROUGH MARGINALIZATION TRICK
In this subsection, provide the guidance formulation for continuous matching, continuous flow matching, concrete score matching, and discrete flow matching. We assume that the conditional probability path of the source data distribution is the same that of the target data distribution.
E. CONTINUOUS SCORE MATCHING
We will use the convention of using t = 0 for data distribution and f = T for noise in continuous diffusion models. With little abuse of notation, denote r(2)
gu(z]
From equation 15 the score
Pa())
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function for sampling from qu is V log (x)
xoo (xul V log [x/x0]
2ofe V2 log 4: t0
20
r(zu)
Exo (xompo):((xal)) (r(xa) Poje Poje(20l
V. Pojo (2)(((
r(zo) - Bxx (x) [xx)]
v log +
(o)c(20o(2) V log Peoo (ol
E XxomPo (xal (xx))
ga(zo) (((x)
po( V ( dry
V. (x) +
Po(
Pe(m)
Exp Pooe [r(x)]]
P((x)
V.p,(=) V.(e)V.neoar
V. (x)) + Pe(m)
pe(ar))
E (oooppoo (x0|=))))))))))))))))))))))))))) (r xo)]
V. log (x) + V log Exp (xal)
where we use the facts 4:10 = Px and Ex Poj (xo (xol=>) (x)l)))) [x(x]]] = ad(ar)
p.(a)) This result recovers the
guidance given by Ouyang al. (2024). By setting (x)) (xrly), we also recover the classifier
guidance v log (x( V, Po + V. log ogpllyxx- x).
1.1. CONTINUOUS FLOW MATCHING
Denote v2 (x(r) and va(x x1) as the marginal and conditional velocity fields, respectively. Assuming the conditional kernels satisfy (x œ1) = ( æ1), and defining r(æ1) == qı (z11/P1((1) we have
2. (x) = Ex (x (x xx)]
U2fe (x1 x))
= x1)
Pije (x1 xx)
r(x1)
Ex - x1)
Ex (x)) (r) [r(x))
(f( + / (
r(ti)
ue(x (Ir a (1)( xx) dri.
(xx (r(x:))
which coincides the exact guidance (Feng et al.
2025) for continuous flow matching.
E. DISCRETE DIFFUSION MODELS
Denote the concrete score (Meng et al. 2022 Lou et al. 2024) sx(x) = (pe(s) (p((2) 2dottd _d z'd=aid"
which is a D([S) - 1 dimeensiio vector. Then, for any 2, x ST satisfying ad o xd z/dd i'd = xld.
have
(2 (x))
(x(x)) Ex (x)l))))))))))
4eji (x xx)
Pejz (2) (x)x)
(xsxppje(x111))
Pej (2c X1 (x)
r(x1)
D
Pi (2 (a4)
= =x((x)² Σ
II
pij (ad Pil (1)((a)
E Ex (x1) )]
d=1
B Rx /d xdd x.= (x((1))
Pi (=
=((x22 Σ
p (| (x9lx))
Expppi (x122) [x(x))
where we use = 1 for the data distribution. Following the discussion in Appendix E.3 the formu- lation above can be viewed special case of discrete flow matching.
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E.1.4 DISCRETE FLOW MATCHING
Similar to the previous formulation, we can obtain that
B ~p(x) bxt of x x1=2) [x(x)]]
(x4,)) (24 + +
(2,,))
1) ut (22 1x,, (xfl22).
Expeptt( (x) [x1)]]
To give specific example, consider the 21 -iddependent mixture path and its kinetic-optimal transition rate (see Appendix C of Shaul et al. 22025):
pin (I (x1) = di (2 keduy (xx) + -xe)po(=");
(24, = (2 ,Id
1- (6-4(2 - 82d (2
Then, we have
u (2 x = upia (2",,00 xx) + Kg Bx)
xptx)" -xf== xx=x) (xx==))((xx11 [r(x1)]
q.d
Kit Exi (x212)) [x(x]] 1))0)((=110
vd >p(xx) x 1x = " x =2m (r(xx1)
up (= x)
Bx Bxxxpin ppp (x1 [x1
E l >p(xx)" xxp= x,= [m(x)]]
11)0daa (2)))
Kic
Ex, Sx (x (c) [x]]]]
which implies that ul (2" .x) case.
(=",x)) an affine combination of uf (2 xx) and 1-xi 8gd (2 in this
E.2 DISCRETE GUIDANCE FOR DISCRETE-TIMG FRAMEWORK
In this subsection, we discuss the discrete guidance in the settings of the discrete-time Markov chain (e.g.. D3PM, Austin et al.) 2021), which is developed by /Vignac al. 2023 ; [Schiff et al. (2025);
Li et al.] (2024a). For convenience, we also use = 1 for data distribution and = 0 for initial distribution. Assume that Peje = Oajt for each in the sampling process is
By Equation 16 the transition probability of g
(=)/p()(())) qe (x) P1tn)(2222)
Ex Bxie ((+a))(2(x))
(x)
which is similar to the soft optimal policy introduced in Li et al. Pe4Mje(2la v), we recover the classifier guidance in Vignac et al. 2023 and Schiff et al. 2025 with
(2024a). By taking (++Me(=)o)) =
unit guidance strength.
Following the proof of Theorem the rate-based guidance can viewed as a limiting form of the discrete guidance in discrete time settings. In addition, the discrete-ting guidance has similar challenge to the rate-based guidance; that is, it requires multiple function evaluations in each sam- pling step. For solving this problem, |Vinaae al. (2023); |Schiff et al. (2025) employ the first-order approximation in practice. As pointed out in Remark such approximation- based approaches are inappropriate for discrete data intuitively.
E.3 COMPARISON BETWERN POSTERIOR- SEEEEIOR--BAEDD GUIDANCE, AND CATE-BASED GUIDANCE
Theorem D and Thcorem 2 present the discrete guidance based the posterior and the transition rate, respectively. The later one generalizes the discrete guidance of Nisonoff et al. (2025). which requires stronger condition than the posterior-based guidance in Theorem | To better understand the conditions in both theorems, we will analyze the relationship between discrete diffusion and discrete flow models. In the following, we consider two different states # z.
Both Guidance Yield the Same Transition Rates in Discrete Diffusion Models. Consider a dis- crete diffusion model with the corruption transition rate Q (2r, (ær,z). z). Here, assume that yi is the
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density of the data distribution. Following Campbell et al. (2022). the time reversal of Q7 (z, 2)
is
(2((2))))
Zej1 (2 x1) 4e (x|x²)
x2) (o),
4uj1 (crl)1) V1jo (x1 (x)) Ex, (on (x (x,=z) [o] 2)
(17)
Since u (2, 21 A (i 2) gels (2(21)
den (alpa)) is conditional transition rate that can generate the condi- tional path 9t11 (see Appendix H.1 Campbell et al. 2024), the discrete diffusion models can be viewed as special case of discrete flow models. Therefore, the condition Q2 2) = Qf (a, 2) as in Theorem 2 actually implies that we take the same specific conditional transition rate uy (2, (2,0
"/[ 2) = Qi (x, 2) (111())a1) ( in discrete flow models.
(a)la))
They Might Offer Different Marginal Transition Rates for Sampling from Target Distribution. Suppose that we are given conditional probability path q = 1 and conditional transition rate (2( = (2,2,, 22,,)_1))) What we want to know is whether there exists a common corruption transition rate Qr =) such that uy (2, x) and u. (2,,)) x) are the reverse transition rates of Qx z) under the data distributions FI and pi. respectively. If exists, by the time reversal formula (see, e.g.
Sun al. 2023 Lou al. 2024), we have
P((
me(ar) Emale)
up (2, E Pc(z)
pe(z) (2, (2. (111)))((11))))
Σ Pc(2) Qi (Ix, =|/11)
Peju (2 (x1)
Pej1 (ar ( Pi1e (21|=>)
m1 pc(z)
(x(x)
(1(x)) u/ (z, xi (011)())
ui (2)
Σ
(2)
(z())
Σ (((
aup uun (2
(x,,
del x mije (2))
(x)))
which implies that
Σ
= Σ 211
([]()))]) is the time reversal of uj (2, 2|21) = u2( 2/21). However, since the posteriors Pije and Qıj= are different, the above equation generally does not hold, unless (0(2,|az)) = Qi (z, 2|21) is independent of x1. If we use the rate-based guidance of Theorem 2 with the marginal transition rate uf( x) = Expepiiexx [s (2, x|x; )] that is used to sample from source data distribution, then we implicitly set the corruption rate Qi Qi 2) Exp<p1((<x/==) [u x xxxx)] (((2) in discrete diffusion models. In this see- nario, the reversed transition rate for generating probability path ge is
where (1(222/11)) = (1 2|(1) = u6 x (1) 9t (mmm1)
Expepo (x)1)) (2/ ((( (=)))((() Pe (x))
which is not equal to the desired transition rate Ex (n) (2) x|x1)] which we obtained based on Theorem [ in general. The main reason is that Ex (2, and Expepii(x (n) ju[ (2, x[x]] do not share the same time reversal as mentioned above. Moreover, if we choose conditional path (2 .r(=1) such that uz(2, 2x(11) (1](2]/1)) is independent of ZT, (2n()) then these two methods are equivalent. Formally, we have the following proposition.
(2)())))) is unrelated to Iı for two states x # 2. Let (mp(slm))
Proposition 1. Assume that u2 x(11)
(x)(2)))) Qc(x, (2, ar (2))) den (2[01) be the transition rate of noising process. Under the same con-
ditions of Theorem we have Qe(a, 2) pe(z) = Exiope 1xx (xx1))) (uf (2,(22, x(xx)) and Qe(e, 2) g)(e = Exicgup (x112) [u] x/1x) X1)]] which means that two methods are equivalent.
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Proof. We only consider the case of pt- Note that
Pxpe (x) (uf (2, x|xx)]]
Σ u2, (z, 21 21)
Paji (z)((1)
Σ
Q (z, 2)
Pep (z)22) Pije (21
(z(2)
Qt (z, 2)
pr (x(x)
which completes the proof.
In summary, the condition Q7 (z) Q (z. 2) in Theorem 2 is more restrictive than Theorem our goal is to use the desired transition rate u, (2, = Ex (. x)) [u]]] [2]] in the sampling stage.
From Posterior-based Guidance to Rate-based Guidance. For completeness, we also give simple proof of Theorem 2 based on the result of Theorem
Notice that, if Q (zx 2) Qi (a, 2)
(which implies that Pt|1
21/1)). by equation 17
we have
Oels [=(x]
( (2) xx]
(2, ug (2,
[x
(xr
Ex Ex (xx (x))
PLie( 22 18
E Excep1e(x11
E (x)p11((X)1)))
Pu (=
(x]])
Paja pe1
(x
E (xpep(((x)(
9x (2)))
((()//((x))
(c(=)/p((x)
Pell (=(b1)
(r(2)/p(xx) (c(2)/p(xx)
Ex (x)))((())
E xx>pp(11 [x(1)]
E Pxxpp(xiaa [r(x]]]
where the third equation we use (x1())))) q (xcs )p in equation 15 and the fourth equation we
Pije(Xi pi (x()) (x))(c
use the marginalization trick in the concrete score matching (Meng et al. 2022 Lou et al. 2024).
This completes the proof.
E.4 CLASSIFICATION OF DISCRETE GUIDANCE
As discussed in Appendix E.2 the rate-based guidance can be interpreted as the limiting case of the discrete guidance in the discrete-time framework, since all of them require the condition: (=)((x](23)) 4a (2a) ( for any < 8 < t < (here, we use = 1 for data distribution). In contrast, the posterior- posterior-based guidance (Theorem [T) only imposes the condition of conditional prob- ability paths, offering greater flexibility and interpretability for discrete flow models. Accordingly. discrete guidance can be divided into categories, summarized in Table 4
Table 4: Classification of discrete guidance. (Data distribution:
Assampiion
Time Pornwork
Dianno Guiebace
Targer Tranatioon Rate for Tilaznno Flow Modele
Phe (odlrr) (evil) for any 0 (S)
Contiranes
Posterior Rased Guidance (Thomne
te Nott s|xi]]
Ie fedde) in (a())) aerry 0 i (
Continnns
Rute Based Guidance (Theoro
NGoset et 3025
En Expepaai Sele, slx. (xk))))acccce)
3023 Saaiff
E.5 DISCUSSION ON THE GUIDANCE STRENGTH
In this subsection, we focus on the settings considered Zheng et al. (2033 : Zhang et al. 2025
Our goal is to generate data from the guided distribution y (x (1ry (y) pi (x) (yl ()), where 7
the guidance strength. Lemma 4.10 in Zhang et al. 2025 provides the comparison between
contrastive energy prediction Lu et al. (2023) and classifier guidance Dhariwal & Nichol (2021) Ho
& Salimans (202)) Here, we compare the rate- based guidance with the predictor guidance Nisonoff al. (2025) under different guidance strengths.
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The guided transition rate with predictor guidance strength y is
Pr (v(l)) [z)
4 (2,
ue (2()xx)
(18)
for z + Here, uy is transition rate that can generate pe (()).. In general, if 7 A the above guided rate u/ cannot generate (7) (xlv)- [wla) see this, we try to verify the Kolmogorov forward equation:
2x(x:7) where 2,1 = Pt pr())) (wlar). To
Pr (2)
Pr(2)p
Σ ( xly)
Σ
(0) 2lu)
Zx(177)
p
Pr (x)) (ulx)
(e(o)(z) 7
u1(2, 2) (c(=) (vl=)
Ue (2(2,
21(177)
pr(ulz) xxt
(v(77)
Σ 2r) px()) (ulz)
pp (w/lr)
Σ 2f2
Uu(
ur
(1)
2,(1: 7)
(ul=z)
[p7 (1)22 - p. (v|x:)
(epe(2) + pp(22)
Σ (I, 2)
7
(yllz) (zPe(z)+ Pc
pl (y|z)
* 7,
(,(17)
(zly).
Therefore, in this case, cannot sample from pfa) with the transition rate u, [ for empirical results, see the simulations in Section 4.1 When 7 1, the Kolmogorov forward equation holds, since
27 (ulz) pr(=)0,
Pe(zlyy) = (x)a)
(x(
(z(2)
d.p((2)
= 0pp((=l)) (=(1))
(+(2)
Σ (((z,z)p + pe(2ly)
Σ ()(x))
pr(2)
(00( P((2[y) v)
Σ
(c(2,z)))(
p+(2)
2t2
(p(()l)) - Pt [v/xr]
= Pr(2)
Σ
ux 2)
p(y)
where Q: the transition rate that can generate (rll))) and (xr) from L t= / = 0.
In contrast, as demonstrated in Theorem 2 the rate- based guidance
Pppicx (v(x1) (n |x uz(2,x)
u (2,xl))
(19)
allows to sample from the guided distribution p/1) with the probability path P; (+:7) (xlv) x
E., Pi (#1)p] (y) (x)(2))) The difference between the proposed rate-based guidance (equa- tion 19) and the predictor guidance (equation [8) in discrete state space is similar to that between the exact guidance and the classifier guidance in continuous state space; see Lemma 4. 10 of Zhang
et al. (2025).
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Pre-trained model (ip(afa)
Source velocity field up (x,x)
Conditional velocity field ux(2,x)))))
Density Ratio r(t,) = g(x)\/(xx))
Gutdance (x(R))(,(8)
Target velocity field uy (z..x)
(e*n)r(Z(x)) = (x(2) (2,x)) x)
Figure 4: Framework of the improved rate-based guidance on discrete flow matching.
F TRAINING AND SAMPLING FOR RATE-BASED GUIDANCE
F.1 TRAINING OBJECTIVE
The rate-based guidance is the following scaler-valued function
hx(2) - Exrep = Σ
hi x for any d E [D),
SES
which is also the minimizer of the following objective:
F ( = [[, (()x))x11__)) (x1--1(11))))xae.m.. (x=)) (x//x1) (x()) r(x1) log hy (xx))
Arate
(20)
In practice, we can directly use this training objective avoid additional summation operations.
F.2 SAMPLING
k((x)) (2 xxxx)x)) for ad * xy, z/d x'd and (x4, xf(x+) Ed ui (s", xf(x(), and then use it instead of ut in
We can construct new conditional quantity uf (2 x(1 = Mf(s)
equation 14
The sampling algorithm can be found in Algorithm and the overall framework is illustrated in Fig. 4
G APPLICATION TO THE RL STAGE IN RLHF
G.1 BACKGROUND AND MOTIVATION
In the diffusion language models (e.g. LLaDa, Nie et al. 2025c and flow-based language models (e.g. Fudoki, Wang et al. 2025). we can obtain the pretrained model. Recently, existing works have developed post-training methods improve preference through DPO (Rafailov al.! (023) or RLHF (Ouyang et al. 2022). The main challenge is that the joint posterior predictor cannot be easily decoupled like AR models. To overcome this issue, some approximation strategies have been proposed for estimating log probability. such as empirical ELBO approximation (Zhu et al.) 2025) and mean-field approximation (Zhao et al. 2025 Yang et al. 2025). However, these methods might introduce large approximation errors when the length of the sentence is large. Fortunately, our discrete guidance can provide novel framework for alignment of diffusion and flow-based language models without approximation.
G.2 ROADMAP OF ALIGNMENT FOR FLOW-BASED LANGUAGE MODELS
Designing conditional path and rate. Similar to Wang et al. 20055) in practice, we take metric- induced probability path with kinetic-optimal conditional transition rate Shaul et al. (2025):
(o off)= (o)1(ff(0f)
(21)
uf (24 2 (of) = uf (24 (14)) = Piu (z1 of) - d(2a, d(22,of))l for ad +x
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Algorithm 3 Sampling with Rate-Based Guidance
Require: pretrained posterior Pije- conditional transition rate (2,20))))) initial value sn based guidance hy, step size h
1: t 8+0 0
2: X f- ID
( h < 1 do 4: for = 1, 1..... do
3: while /
> in parallel
5:
Sample x pi |xx) xx)
(x) (
d-1 d+1
, x6 g xP)
6:
Calculate in! (8, x xx/
uf (8,xf1xx) xx1xx)) x
(x()))
7: Ad +- Exsxt 1.l (8, x((xx))
8:
Sample Zjmmp [00
9:
if jump S e w (xxx) (1 0x4(()
then
d.0
Sample x
Ad
11:
end if
12: end for
13: + 1 h
14: end while
t f h
16: for d= d 1..,, do
> in parallel
d,0 17: Sample x9 Pije (. (xx)
18: end for
19: return X1
where 3g is an increasing function of with 8o = 0, B1 = oo, and d(.,.) -) is metric.
Training guidance model. We can simply extend the unconditional training objective in the previ- ous section to the following conditional version:
Chirrer (0) + AC (0)
(of: 02, -xexp (R(c,01)) log h (of c).
Σ d=1
d,0 h
= - 0.
log Σ 12 (s, c)phi log h d.0 (o o2,
c)
(22)
Sampling. Given the prompt at the current time I, for each d E [D], we first sample of from (o c. o.) x n. (of. o,, (o ot),
d.0 T1j
and then sample of similar to equation 14
H ADDITIONAL RESULTS
H.1 IMPLEMENTATION DETAILS OF ENERGY-GUIDED SAMPLING
For fair comparison, train the guidance models (conditional expectation) using the Bregman divergence (Equation and Equation 20) without regularization. For the based guidance and predictor guidance, take the initial distribution Po p/²²) 8m for efficient sampling, where m is the mask state. For sampling with the posterior based guidance, we consider both masked and uniform initial distributions. We used the mixture probability path and the associated transition rate (Equation 4) with the cosine time schedule K, cos? [ t)] in our 2-D experiments, which is kinetic optimal as mentioned in Shaul et al. 2025
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Our pretrained model and guidance models SiLU networks with 3 hidden layers with dimension 256. For training, we use 100,000 datapoints from source data distribution. The optimizer Adam with learning rate le-4.
H.2 IMPLEMENTATION DETAILS OF MULTIMODAL TASKS
To match the conditional probability path used in Wang et al. (2025). we use metric-induced path with metric dx)( f())) || 211-. where ž, 2 normalized token embeddings which taken from the original text embedding layer of Janus-Pro-7B and the image embeddings of Llama-
Gen; for time schedule Bt, we set 3( - )0.9 suggested in Shaul al. (2025).
Our guidance model is network composed of 6 LLaDA blocks (Ne et al. 2025b) with hidden
dimension 768, time embedding mapping R to a hidden dimension of 768, a token embedding mapping each token into 768- 508-dimensional space.
H.3 EFFECTIVENESS REGULARIZATION A 2-D EXPERIMENT
In this subsection, we examine the effectiveness the regularization term in the training objective. Samples of the source and target distributions are shown in Fig. [S] and the two distributions differ substantially. We first pretrain a posterior model on the source data, subsequently pretrain density ratio model by minimizing the following training objective:
r (x1x) r (x)) +
(0) (xx)) log (x) +1)
Exira (xx) log
We then train posterior-bes based guidance with the proposed training objective (0) XCn.,(0), sweeping À from 0 to in steps of 0.2. Here, we take the uniform initial distribution. Finally,
As presented in Fig. 6 the posterior-based guidance trained with large hyperparameter offers distribution closer to the target distribution.
generate samples using Algorithm 2
128
Source
Target
96
32
128
d
Figure 5: Some samples of source and target distributions.
H.4 ADDITIONAL 2-D RESULTS
Additional results on 2-D simulations are provided in Fig.
Fig
Fig.
Fig.
TO
Fig.
and
Fig.
12
H.5
ABLATION STUDY OF THE REGULARIZATION STRENGTII
In Appendix H.5
we present an ablation study on the effect of the regularization strength introduced
in Section 3.3
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A 0.0
0-2
A 0.4
A 0.6
A 0.8
1.0
96
96
96
96
64
32
32
A 0.5
A=02
-,044
A-666
1-10
301
Figure 6: First row: samples generated by rosterior-based guidance (64 sampling steps) with different hyper- parameters in the training objective. The initial distribution is uniform. Second row: density estimation of the target distribution using posterior-based guidance with different hyperparameters.
(a) Ground Truth (rings)
(b) Rate Bused (Masked Nisomoll et
2025
(c) Rate Based (Masked, Ours)
(d) Posterior-Based (Masked, Ours)
(c) Posterior-Based (Uniform, Ours)
Figure 7: Comparison sampling results with different guidance schemes (64 sampling steps) in 2-D experi- ments. (a) The samples the guided distribution p(7) with different guidance strength 7: (b) data sampled by the predictor guidance (Nisonoff et al. 2025) with masked initial distribution; (c) the data sampled by the proposed rate-based guidance with masked initial distribution; (d) the data sampled by the proposed posterior- based guidance with masked initial distribution; (e) the data sampled by the proposed posterior-based guidance with uniform initial distribution.
Table 5:
Ablation study of the regularization strength.
Method
Single Obj.
Two Obj.
Counting
Colors
Position
Color Attr.
Overall f
FUDOKI
96.25
83.84
47.50
91.49
71.00
74.00
77.35
7 = 0.1
96.25
90.91
45.00
91.49
68.00
71.00
77.11
") 0.5
93.75
85.86
52.50
89.36
70.00
77.00
78.08
= 1.0
92.50
87.88
52.50
92.55
67.00
74.00
77.74
Note: Results percentages.
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2-3
1=10
Y-
Ground Truth (moons)
y=
y
Rate-Based (Masked, Nisonoft ot al
2025
Rate-Based (Masked, Ours)
Posterior-Based (Masked, Ours)
Posterior-Baze (Uniform, Ours)
Figure 8: Comparison of sampling results with different guidance schemes in moons.
Ground Truth (Sganssians)
Rate Based (Masked, Nisumoff et al
2033
Rate- Based (Masked, Ours)
Posterior Based (Masked, Ours)
Posterior Based (Uniform, Ours)
Figure 9: Comparison of sampling results with different guidance schemes in 8gaussians.
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V
M
14
21
Ground Truth (2spralss)
12
y-16
20
y
Y-
y- 20
24
16
24
21
32
21
2t
Rate-Based (Masked, Nisonofi
2025
Rate- Bate-BBsed (Masked, Ours)
y=0
y=
y- 20
V-
Y- 10
y- 20
24
16
16
Posterior Based (Masked, Ours)
Posterior Based (Unifarm, Ours)
Figure 10: Comparison of sampling results with different guidance schemes in 2spirals.
1=10
Ground Truth (checkerboard)
y-20
y=
Rate- Basee-BBeedd (Masked, Nisonoff et al.
2025
Rate-Based (Masked, Ours)
Posterior-Based (Masked, Ours)
Posterior-Base (Uniform, Ours)
Figure 11: Comparison of sampling results with different guidance schemes in Checkerboard.
Ground Truth (swisrolll
20
6
(Masked, Nisanoff
(Masked, Ours)
20
Y-
3
10
Y-
Ours)
Ouss)
Figure 12: Comparison sampling results with different guidance schemes Swissroll.
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LLM USAGE
In our multimodal understanding experiments, we follow the standard practice of using LLM-as- a-judge assess rewards and evaluations based on the question, ground-truth answer, and model response. LLMs are further used solely for polishing the writing. Importantly, no novel ideas, analyses, or discoveries are contributed by LLMs.
33


Reviews:
[{"review_id": "NulXtH3YZh", "reviewer": "Reviewer_375z", "summary": "Given a pre-trained discrete flow matching (DFM) model parametrized by a posterior $p_{1|t}$, and trained to generate samples from a distribution $p_1$ (somewhat confusing, it is denoted as the _source distribution_), the paper goal is to adapt the model to sample the target distribution $q_1=p_1(\\cdot | y)$ where $y$ is a conditioning variable. \n\nTo achieve this, the authors propose to add ans \"exact guidance\" which learn an additional model $h_t (x_1^d, x_t)$ such that $$q_{1|t}(x_1^d|x_t) = \\frac{h_t (x_1^d, x_t)}{\\sum_s h_t (s, x_t)} p_{1|t}(x_1^d|x_t)$$ is the posterior of DFM model trained to sample the target distribution $q_1$. \n\nFinally, the exact guidance method is evaluated mainly on the task of text-to-image.", "questions": "In Algorithm 2 Sampling with Posterior-Based Guidance it is state that the guided posterior needs to be calculated at each step, Since the sequence length and vocabulary size can potentially be large, how does this affect the sampling cost compared to the vanilla sampling in Algorithm 1?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The paper properly introduces its goal, the notation with a the required preliminaries work, and presents it method in a clear mathematical writing.\n\n2. The proposed method attempts to advance fine tuning for conditional sampling which is a very important problem in generative models.\n\n3. To the best of my knowledge the proposed method is innovative and constitutes a fair contribution.", "weaknesses": "While paper is interesting I still have a number of concerns:\n\n1. The \"RLHF ON MULTIMODAL BENCHMARK\" subsection is lacking:\n\n    a. The application of exact guidance to RLHF seems to be an important application of the method and should probably be introduced more clearly before the experiment section. In particular, explicitly state what object does $h_t^{\\theta}$ approximate in the this settings. Additionally, even though the actual loss for training in this setting appears in the Appendix G, would good if there was a direct reference from the paper.\n\n    b. In the main paper a single choice of probability path is introduced in equation 4. However going over the appendix it seems a different choice of probability path was used in the experiment, the metric induced path as in equation 21. Further, this probability path seems to use an additional metric which is not specified. While it is reasonable to leave these equations in the Appendix, it is expected to mention it in the experimental setup.\n\n   c. Almost no implementation detail is provided for experiment: which pre-trained model is used is not clear, architecture and model size are not specified, and number of training iteration as well as effective batch size.\n\n   d. It might be standard approach but I still find it disturbing that the GenEval reward model was used both in training and evaluation.\n\n2. The sampling section is lacking, going over the appendix it seems there is a strong connection between previous suggested sampling algorithm and the exact guided sampling algorithm. Stressing the difference in the main paper could help the reader.\n\n3. I find at least two mathematical claims that where introduced in the main paper but were not actually used by the proposed methods (negative claims): \n\n    a. Theorem 2 is introduced and in the next three lines it is already stated that the authors do not make any use of it.\n\n    b. Remark 1, this remark seems to claim again that the direct approximation of the rate would not yield the desired flow.\n\nThe paper is already heavy with alot of math to parse, and I would consider moving these to the appendix. Additionally, there is currently alot of important information in the appendix and it seems that rearranging the paper can possible make it alot easier to understand the main contribution and experiments of the paper.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "QPwNPdnjq5", "reviewer": "Reviewer_gFdm", "summary": "This paper proposes a framework for general guidance in discrete flow models. While these models support unified understanding and generation for images and text, the framework works to improve the preference of the results according to specific reward signals.", "questions": "1, Why do you consider your method with RL training as guidance?\n\n2, Could you supplement the intuitive explanation for all your theorems and tell how they help to approach your method as a conclusion?\n\nI would actively engage the rebuttal and would love to raise my score if my concerns are well solved.", "limitations": "", "rating": 4, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "1, The theory is solid. I am not a theory expert, but I validated the correctness of proof for theorem 1 and theorem 2. \n\n2, The overall advantage in both generation and understanding proves the effectiveness of the proposed framework. From both quantitative results on T2I and case studies on 2D data, I can see considerable improvements compared to existing methods.", "weaknesses": "1, The presentation and delivery is problematic. The word Guidance should be used to describe inference-time guidance. When using an external reward to post-train the model, the method should be defined as RL (as stated in the Experiment Section). Also, the most referred reference in the paper [A] is a training-free guidance method for discrete flow matching model. If the authors want to categorize the method as guidance, at least they should explicitly notify the reason and the difference compared to existing guidance method (training is needed here). \n\nAnother critical problem in presentation is the missing of reasonable transition. For example, there is no clear problem definition on what your guidance wants to do in empirical tasks, and for theorem 1, no explanation is available on why we need to consider D-1 vectors in the discrete flow matching guidance (When I refer to your reference literature, they do not mention this D-1 term). \n\n2, Due to the above misconception on RL and guidance, I am not sure whether it is suitable not to include other RL methods based on direct preference as baselines.\n\nReference:\n\n[A] Unlocking Guidance for Discrete State-Space Diffusion and Flow Models", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "2DFjgSeNV0", "reviewer": "Reviewer_N6g7", "summary": "The paper proposes Discrete Guidance Matching, a CTMC-based framework for exact, efficient guidance in discrete generative models like diffusion and flow matching. It derives precise transition rates via density ratio correction, requiring one model pass per step. Training minimizes Bregman divergence with regularization. It generalizes prior methods and outperforms baselines in tasks like energy-guided sampling, text-to-image preference alignment, and multimodal understanding, with faster sampling.", "questions": "1. Did you verify that the derived target transition rate satisfies CTMC’s conservativeness condition $ u_t^q(x,x) = -\\sum_{z \\neq x} u_t^q(z,x) $? If yes, please include the proof in the revised manuscript; if not, how do you address the resulting probabilistic inconsistency?  \n2. The coordinate-wise independence assumption for high-dimensional data ignores dimensional coupling. Do you have a strategy to model this coupling, for example, low-rank decomposition, or pairwise terms? Can you provide experimental results on high-dimensional tasks like $ D=512 $ text sequences?  \n3. How do you ensure the estimated density ratio $ r(x) $ remains non-negative during training and sampling? Have you observed negative density ratios in experiments, and if so, how did you resolve them?  \n4. For low-target-sample regimes like 10-shot or 1-shot preference alignment, does your regularization term still work? If not, do you have a weak-supervision or prior-based alternative to improve data efficiency?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 2, "presentation": 3, "contribution": 3, "strengths": "1. Unlike existing methods relying on first-order Taylor approximation, the paper derives the exact transition rate for the target distribution using density ratios, eliminating non-negligible approximation errors in discrete state spaces.  \n2. The framework encompasses existing guidance methods, such as class-conditional and energy-weighted guidance, as special cases and seamlessly applies to both discrete flow models and discrete diffusion models, including masked diffusion. \n3. Each sampling step requires only 1 forward pass vs. multiple function calls in rate-based or approximated methods, leading to faster sampling than rate-based baselines.", "weaknesses": "1. Theoretical Soundness Gaps in CTMC: The paper fails to verify that the derived target transition rate satisfies CTMC’s core conservativeness condition $ u_t^q(x,x) = -\\sum_{z \\neq x} u_t^q(z,x) $. This omission means the transition rate may not form a valid CTMC, breaking the probabilistic consistency of the generation process.  \n2. Unsubstantiated High-Dimensional Assumption: The coordinate-wise independent transition rate assumption, like for high-dimensional data, contradicts the inherent dimensional coupling in real-world discrete tasks such as long text and high-res images. The paper only validates its performance in 2D low-dimensional spaces and provides no analysis of high-dimensional performance, thereby limiting its practical utility. \n3. Missing Non-Negativity Constraints for Density Ratios: The Bregman divergence training with $ F(x)=<x, log x> $ does not guarantee non-negative density ratios $ r(x)=q_1(x)/p_1(x) $. Negative density ratios would lead to \"negative probability transitions,\" which violate CTMC physics; however, the paper lacks regularization or pre-sampling checks to prevent this.  \n4. Incomplete Experimental Scope: High-dimensional tasks such as 32×32 discrete images and low-target-sample regimes, which are critical for real-world few-shot alignment, are not tested, leaving key practical questions unanswered.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "MN9hnT7ASs", "reviewer": "Reviewer_nrvw", "summary": "The paper introduces Discrete Guidance Matching (DGM), a posterior-based guidance rule intended for discrete flow and diffusion models. The central theoretical statement asserts that, provided the density ratio \\(r(x)=q_1(x)/p_1(x)\\) is known and the source and target conditional paths match, one can recover the target posterior by reweighting the source posterior at each step. The manuscript positions this identity as a unifying lens on class-conditional guidance, energy-guided sampling, and RLHF-like objectives, and argues that the resulting sampler uses a single forward pass per step rather than the multiple passes required by rate-based or first-order schemes. Empirically, the paper demonstrates a two-dimensional toy “energy-guidance” setup and applies the approach to a multimodal discrete flow model finetuned with rewards, reporting modest improvements on GenEval and a composite multimodal score. Despite the tidy narrative, the practical validity of the assumptions and the evidentiary weight of the experiments remain limited; taken together, the contribution feels incremental, and I recommend rejection.", "questions": "The core theoretical promise of “exactness” rests on conditions that are rarely met in realistic discrete pipelines and on an oracle-like requirement that the terminal density ratio be known or accurately estimated; absent a robustness analysis, the practical force of the theorem is limited. In particular, the absolute-continuity requirement \\(q_1\\!\\ll\\!p_1\\) can be violated in RLHF or reward-driven regimes where the target allocates mass to sequences/events that the base model assigns essentially zero probability, leading to degeneracy of importance weights and ill-posed guidance; the manuscript neither diagnoses these failure modes nor proposes remedies (support repair, smoothing, clipping) or theoretical guarantees under misspecification. \n\nThe conditional-path matching assumption is likewise heavy: equating \\(p_{t|1}\\) and \\(q_{t|1}\\) (often with matched corruption schedules) is nontrivial outside the authors’ favored flow parameterization, and there is no sensitivity or stability bound quantifying how a small path mismatch per step compounds into terminal bias. Even within the proposed posterior scheme, the sampler relies on a first-order update while simultaneously critiquing first-order approximations for discrete data; the resulting approximation error is unanalyzed and unablated, which undercuts the “exact” branding. On the empirical side, the efficiency narrative is not audited end-to-end: the 2-D timing study confounds initialization (masked vs. uniform) and does not report system-level latency/throughput for the main tasks under fixed hardware, batch, and backend, while the additional training cost for the ratio estimator and guidance head is omitted—function-call bookkeeping is not a substitute for wall-clock and GPU-hour accounting. The evaluation design further risks reward leakage by optimizing with a metric (e.g., GenEval) used again at test time, blurring the line between genuine generalization and metric overfitting; moreover, reported gains are small, lack multi-seed variance or confidence intervals, and do not survive scrutiny without statistical characterization. Breadth is limited: beyond a toy 2-D setup and a single multimodal flow, there is no coverage of standard discrete diffusion baselines (e.g., multinomial/discrete diffusion, argmax flows) or of diverse discrete domains such as language-only generation, program synthesis, or tabular/graph settings; critical ablations are missing (quality of the learned \\(r\\), robustness to support violations, influence of the step-size/first-order update, and sensitivity to the initial distribution). Finally, the “target-data regularizer” presumes access to \\(x_1\\sim q_1\\), which is conceptually circular when the whole point of guidance is to approximate \\(q_1\\) in the first place. \n\nThese issues are compounded by substantial missing baselines across discrete diffusion/guidance and RLHF preference optimization: beyond a couple of handpicked methods, the paper omits widely used, directly comparable approaches in both families. Representative missing or under-engaged baselines that should be reported under matched hardware, batch size, and privacy/backend settings include Dhariwal & Nichol (2021, classifier guidance for diffusion), Ho & Salimans (2022, classifier-free guidance), Austin et al. (2021, D3PM for discrete state spaces), Hoogeboom et al. (2021, autoregressive/discrete diffusion), Chang et al. (2022, MaskGIT as a discrete generative baseline), Ouyang et al. (2022, RLHF for instruction following as a canonical PPO/SFT+RM baseline), Rafailov et al. (2023, DPO as a strong non-RL preference optimizer), Yuan et al. (2023, RRHF), Xu et al. (2023, ImageReward as a modern reward model for T2I), Kirstain et al. (2023, PickScore as a learned preference scorer), Wu et al. (2024, HPS v2 as a unified human-preference score), Dathathri et al. (2020, PPLM for posterior control of discrete text, closely related in spirit to guidance), and Bai et al. (2022, Constitutional AI / RLAIF variants showing strong preference optimization without explicit ratio models). Including these methods would stress-test the claimed unification and computational advantage, reveal where DGM sits relative to best-practice discrete diffusion/guidance and modern preference optimization, and help separate genuine algorithmic gains from metric leakage or implementation choices.\n\n**missing references:**  \nDhariwal, P., & Nichol, A. (2021). Diffusion Models Beat GANs on Image Synthesis. *NeurIPS.*  \nHo, J., & Salimans, T. (2022). Classifier-Free Diffusion Guidance. *arXiv:2207.12598.*  \nAustin, J., et al. (2021). Structured Denoising Diffusion Models in Discrete State-Spaces (D3PM). *NeurIPS.*  \nHoogeboom, E., et al. (2021). Autoregressive Discrete Diffusion. *arXiv:2103.12027.*  \nChang, W., et al. (2022). MaskGIT: Masked Generative Image Transformer. *CVPR.*  \nOuyang, L., et al. (2022). Training Language Models to Follow Instructions with Human Feedback. *NeurIPS.*  \nRafailov, R., et al. (2023). Direct Preference Optimization: Your Language Model is Secretly a Reward Model. *NeurIPS.*  \nYuan, W., et al. (2023). RRHF: Rank Responses to Align Language Models with Human Feedback. *NeurIPS.*  \nXu, K., et al. (2023). ImageReward: Learning and Evaluating Human Preferences for Text-to-Image Generation. *NeurIPS.*  \nKirstain, Y., et al. (2023). PickScore: A Large-Scale Human Preference Dataset and Scorer for Text-to-Image. *arXiv.*  \nWu, J., et al. (2024). HPS v2: A Unified Human Preference Score for Text-to-Image Evaluation. *arXiv.*  \nDathathri, S., et al. (2020). Plug-and-Play Language Models: A Simple Approach to Controlled Text Generation. *ICLR.*  \nBai, Y., et al. (2022). Constitutional AI: Harmlessness from AI Feedback. *arXiv.*", "limitations": "", "rating": 2, "confidence": 3, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "Conceptually, the work is clean and easy to follow: the authors articulate a clear posterior-reweighting view that helps connect several strands of discrete guidance under a single umbrella. The derivations are presented with serviceable notation, and the core loss used to fit the guidance term—cast via a Bregman-style objective with an optional target-data regularizer—is straightforward to implement in existing codebases. The paper also does a decent job motivating why rate-based or gradient-first-order methods can be computationally heavy in discrete settings, and it frames the proposed posterior-based rule as a way to reduce the number of model calls at sampling time. The organization is generally strong, with definitions and assumptions stated upfront, and the training/evaluation pipeline, while narrow, is at least described with sufficient detail to enable reproduction by a motivated practitioner.", "weaknesses": "The primary weakness is that the advertised “exactness” depends on an oracle-like requirement—knowing or accurately estimating the terminal density ratio—and on heavy modeling constraints—absolute continuity \\(q_1\\!\\ll\\!p_1\\) and equality of the conditional paths—that are rarely guaranteed in real discrete generative pipelines. There is no robustness theory that translates estimation error in \\(\\widehat r\\) into bounds on divergence between the induced terminal law and the desired target, nor is there sensitivity analysis for violations of absolute continuity or path mismatch; as a result, the central theorem’s practical relevance is unclear. The empirical section compounds this concern: the compute narrative around “one forward pass per step” is not cleanly audited, with timing comparisons confounded by different initial distributions and no end-to-end latency or throughput reported on the main RLHF tasks, and the additional cost of training a ratio model plus guidance head is not measured. Methodologically, the work criticizes first-order approximations for discrete data yet employs a first-order update in its own sampler without analyzing the induced bias, which undercuts the rhetorical stance. The evaluation design risks reward leakage by incorporating evaluation metrics into the reward itself, while the reported gains are small, lack statistical characterization (no confidence intervals, multi-seed variance, or significance tests), and are restricted to a narrow set of settings; broader discrete domains (language-only, program synthesis, tabular) and standard discrete diffusions are not explored. Finally, the “target-data regularizer” raises circularity concerns in scenarios where access to \\(x_1\\sim q_1\\) is precisely what guidance is meant to provide, and several strong baselines typical in modern RLHF or discrete-guidance literature are missing, making it difficult to assess how much the method truly advances the state of the art.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)