Greed is Good: A Unifying Perspective on Guided Generation
Authors: Zander W. Blasingame
Chen Liu
Venue: NeurIPS 2025 poster ()
Keywords: Flow matching
diffusion models
neural differential equations
guided generation
numerical methods
dynamical systems
Primary area: deep_learning
TL;DR: We present a theoretical framework which unifies posterior and end-to-end guidance for flow/diffusion models.

Abstract: Training-free guided generation is a widely used and powerful technique that allows the end user to exert further control over the generative process of flow/diffusion models.
Generally speaking, two families of techniques have emerged for solving this problem for *gradient-based guidance*: namely, *posterior guidance* (*i.e.*, guidance via projecting the current sample to the target distribution via the target prediction model) and *end-to-end guidance* (*i.e.*, guidance by performing backpropagation throughout the entire ODE solve).
    In this work, we show that these two seemingly separate families can actually be *unified* by looking at posterior guidance as a *greedy strategy* of *end-to-end guidance*.
    We explore the theoretical connections between these two families and provide an in-depth theoretical of these two techniques relative to the *continuous ideal gradients*.
    Motivated by this analysis we then show a method for *interpolating* between these two families enabling a trade-off between compute and accuracy of the guidance gradients.
We then validate this work on several inverse image problems and property-guided molecular generation.

Greed is Good:
A Unifying Perspective on Guided Generation
Zander W. Blasingame Clarkson University blasinzw@clarkson. edu
Chen Liu
Clarkson University cliu@clarkson, edu
Abstract
Training-fres guided generation is a widely used and powerful technique that allows the end user to exert further control over the generative process of flow/d- iffusion models. Generally speaking, two families of techniques have emerged for solving this problem for gradient-based guidance: namely, posterior guidance (i.e., guidance by projecting the current sample to the target distribution via the target prediction model) and end-to-end guidance (i.e., guidance by performing backpropagation throughout the entire ODE solve). In this work, we show that these two seemingly separate families posterior guidance as a greedy strategy of end-to-end guidance. We explore the theoretical connections between these two families and provide an in-depth the- oretical understanding of these two techniques relative to the continuous ideal gradients. Motivated by this analysis, we then show method for interpolating between these two families enabling trade-off between compute and accuracy of the guidance gradients. We then validate this work on several inverse image problems and property -uided molecular generation.
actually be unified by looking the
1 Introduction
Guided generation greatly extends the utility of state-of-the-ar generative models by allowing the end user to exert greater control over the generative process, ultimately making the tool more useful in wide variety of applications ranging from conditional generation, editing of samples, inverse problems &c. We focus particularly on subset neural differential equations that model affine probability paths, in other words, diffusion and flow models due to their widespread adoption in a large variety of practical tasks. E.g., audio (H. Liu et al. 2023; Schneider et al. 2024), images (Rombach al. 2022; Black Forest Labs 2024), biometrics (Blasingame and C. Liu 2024c), molecules (Hoogeboom al. 2022; Ben-Hamu et al. 2024), proteins (Watson al. 2023; Skreta et al. 2025), &c.
We can divide the guided generation techniques into two broad categories: conditional training and training- free methods. The former of these two requires the training of the underlying diffusion/flow model on additional conditional information, either as part of the training or at a later time as additional fine-tuning (Ho and Salimans 2021; J. Song, Meng, and Ermon 2021; Hu al. 2022). The latter category instead makes use of some known guidance function defined on the data distribution and incorporates this information back the model influence the generative process. These training free techniques can be further broken down into two sub-categories, i.e., posterior and end- to-end guidance. The former class of techniques uses a simple estimation of the posterior distribution that can be easily found in diffusion models (Chung, J. Kim, et al. 2023) and some flow models (cf. Lipman, Havasi, al. 2024, Section 4.8). This simple posterior estimate can then be fed into guidance function to construct gradient w.r.t. to the current timestep. We refer to this category as posterior guidance as they use this posterior estimate to perform the guidance process. This can then be used to update the ODE solve form of classifier guidance (Chung, J. Kim, et al. 2023; Yu et al.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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(t. Wang al. 2025)
Control signal optimization
( 2023)
End-to-end guidance
(Bes- Hamn al. 2024)
State optimization
Training free guided generation
groody strategy
(Bleiingame and C, Lin 20246)
(Yu al. 2023)
Posterior guidance
(Chung, J. Kim, al. 2023>
Figure 1: The greedy perspective as a unification of separate families in the taxonomy of training-free free guided generation. We provide a more detailed version of this in Figure 5.
2023). The latter class of techniques, in contrast, performs backpropagation throughout the entire sampling process the flow/diffusion model (Ben-Hamu et al. 2024; Blasingame and C. Liu 2024a). We refer to this category as end-to-end guidance as it performs backpropagation throughout the entire sampling trajectory.
The aim of this work is to bring these two seemingly disparate family of techniques together into a single unified view.
Our key insight is that we can bridge between techniques that use posterior sampling and techniques that use end-to-end optimization for guidance by viewing the former as greedy strategy on the latter.
Contributions. In light of this insight, we compare several state-of-the-art techniques from this perspective, showing how this perspective yields a unified and flexible framework for viewing guided generation with flow/diffusion models. We perform detailed analysis of this greedy strategy, showing that it is not only a unifying view, but that it actually makes good decisions under certain scenarios. We then show a perspective which allows one move between these two classes of guided generation techniques, opening up an exciting and novel design space. Lastly, we conduct some numerical experiments inverse image problems and molecule generation.
2 Preliminaries
Flow models (Lipman, T. Q. Chen, et al. 2023) are a highly popular class generative models that model the generative process neural ordinary differential equation (ODE) (R. T. Chen et al. 2018). Consider two valuee random variables: X,- ~ p(x) and X, q( demoting the source (noise) and targer (data) distributions, respectively. Then consider a time-dependent vector field u E C1.-('0, 1) x RR: RRd) with r 2 which determines a time-dependent flow 4, Cl.r ((0, 1) x Rd Rd) which satisfies the ODE
d
Pv(x) = dt
01(x)) = u(t, (x))).
(1)
This is known as C"-flow and this flow diffeomorphismm in its second argument for all & [o, 1]. For notational simplicity let u. (æ) + (u(,,z), A special case of flow models are known as affine probability paths and are defined as X, =a o,X1 with schedule (o o,). We provide more details on flow models in Appendix B.1.²
'For notational simplicity, we let Cla A.g..... (X1 x X2 X ... X X.;Y Y) denote the set of continuous functions that k--times differentiable in the i-th argument mapping from (X1 Xa ... x X.) to Y. Y omitted, then R. 'Without loss of generality we consider flow models which subsume the ODE formulation of diffusion models.

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(kel)
V.
State optimization
Control signal optmiizaioo
guidance
Posterior guidance
Figure 2: Visual comparison of different training-free free guided generation techniques.
An overview of training-free guidance with gradients
We explore techniques for solving training-free guidance problems- this is in contrast with techniques like classifier (Dhariwal and Nichol 2021; Y. Song, Sohl-Dickstein, et al. 2021) and classifier-fre (Ho and Salimans 2021) guidance- which use some off-the-shelf guidance function C E C2 (12d) defined on the output of the flow model. I.e., we wish to optimize the ODE solve such that the output 21 minimizes Suppose we have numerical scheme (Euler, RK4, DPM-Solver, &c.) denoted
12 X 12d
C(R x Rd Rd) R2
(2)
+ (tns tn+1, Inny u.) + In+1-
For simplicity we will omit the explicit dependency of the numerical scheme on ul and assume implicitly; likewise, let (tm..... more formally below Equation (3).
(tt,,ttt1; where h = tn-1 tn. We write this objective
Problem statement. Given some t E [0, 1) and step size regime {4 < l2 < IN = 1} solve:
Find sequence
{x N n=1
which minimizes L(IN),
subject to
In+1 (t tn. Fn).
Next, we will detail two popular families of techniques for solving the problem mentioned above. We illustrate the relationships between these different families in Figure 1, taxonomy of training-free guidance methods. We note that these two seemingly separate branches can be unified back into single branch, by the viewing posterior guidance techniques a greedy strategy of the later. Likewise, we provide a visual overview of the guidance mechanisms in Figure 2.
3.1 Posterior guidance
A popular technique for training-free guidance is what will term posterior guidance (Chung,
J. Kim, et al. 2023; Yu et al. 2023). The key idea behind this strategy is to use the parameterized target prediction model a ( i.e., the expected value of the posterior distribution given X, æ,
to provide a guidance gradient of the form V=L(=/ (x)) for some guidance function L C* (TR.). For literature working with score-based generative models (Y. Song, Sohl-Dickstein, et al. 2021), this interpretation arose from the famous Tweedie's 's formula (Stein 1981; Efron and N. R. Zhang 201 I). Thus, for each v, in the ODE solve, add guidance to it in the form of posterior guidance gradient.
3.2 End-fo-end optimization for guidance
Another popular class of techniques is what we will term end-to-end guidance (Ben-Hamu et al. 2024; Blasingame and C. Liu 2024a), i.e., techniques which perform guidance by optimizing the initial condition 2p w.r.t. the guidance function such techniques require performing backpropagation through neural ODE. Fittingly, we will import notations and terminology from the study of neural differential equations (Kidger 2022) to discuss these techniques. The first technique for performing this kind of guidance is known as discretize-teen-optiie (DTO) where the numerical

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scheme (cf. Equation (2)) is part of the computation graph of the model reverse-mode automatic differentiation (Linnainmaa 1976) applied, i.e., vanilla backpropagation. The memory cost of such techniques, however, is (n), prompting researchers to explore the second method known as optimize-then- -ixcrretiee (OTD) which instead solves another ODE in reverse-time which models the continuous-time dynamics of reverse-mode cifferentiation, this is called the continuous adjoint method (R. T. Chen et al. 2018; of. Kidger 2022, Section 5.1.2).
Given a flow model Up E C ([0, 1] x Rd; Rd) that is Lipschitz continuous in its second argument and the solution x [0, 1] - R", z + (t), let a₂ == aC/Ox, denote the adjoint state. Then a
can be found by solving the continuous adjoint equation:
Oc
daæ (1) a (1)
Oul
O.e: (1)
(are).
dt
On
N.B., this technique was first proposed by Pontryagin et al. (1963) and popularized for neural differential equations by R. T. Chen et al. (2018). This approach has constant memory cost O(1); however, this comes with the cost of several drawbacks related to the numerical scheme. While these issues are not partieularly relevant to theoretical analyses, note them in Appendix E for the ML practitioner.
4 A greedy perspective on guidance
Now returning back our problem statement from Equation (3), the end-to-end guidance techniques amount to optimizing the initial condition 2o in light of the entire solution trajectory admitted by the numerical scheme. A natural question we consider for problems of this form is that rather than finding the full sequence {æ,, }, can we make use of local information instead? L.e.,
Key insight 1. Rather than solving the full ODE from 2t, what if we greedily took a locally optimal step at each æ; instead?
Formally, we define a greedy strategy is the following augmentation to the numerical scheme from Equation (2) as
Φ Gtn, Fn u. 11+1 = T 1nt1, :)
(5)
(6)
where G is the greedy action which makes its decision from only information available at time t.-
Now in particular we are interested in a specific greedy action, Lee., posterior guidance. We define this greedy action as the solution to the following iterative process with initial value (0) En which solves
(I++)) xn
Z VL Z1),,, (ax
for some sufficient number l > and learning rate rj > 0.
By construction this greedy action is the popular strategy of posterior guidance. The rest of this section is then devoted exploring the connections between this greedy action and end-to-end guidance schemes. More, succinctly we state our insight below:
Key insight 2. Posterior guidance can be viewed as Euler schemes within the DTO or OTD backpropagation schemes.
To make our analysis simpler, let us write the flow from 8 to I. in terms of the target prediction model. The flow from time s time & can then be expressed as the integral of the right-hand side of Equation (19) over time. Thus, the flow is now expressed as a semi-linear integral equation with linear term a,æ and non-linear term b (x). Due to this semi-linear structure, we apply the same technique of exponential integrators (Hochbruck and Ostermann 2010) that has been successfully used to simplify numerical solvers for diffusion models (Lu et al. 2022a; Q Zhang and Y. Chen 2023; Gonzalez et al. 2024). N.B., the full derivations and proofs for this section can be found in Appendix B.

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Let == au/a; denote the signal-to-noie noise ratio (SNR), then Vt is a monotonically increasing sequence t, due the properties of 0t) (ef. Equation (17)) and thus has an inverse t, such that (x)) = With abuse of notation, we let æ = (t)(1) and i (+) = (n) (+) As such, we can rewrite the solution to the flow model in terms of y by making use of exponential integrators, which we show in Proposition 4.1 with the full proof provided in Appendix B.3.
Proposition 4.1 (Exact solution of affine probability paths). Given an initial value of a at time s E 00, 1] the solution Dt at time t E [0, 1] of an ODE governed by the vector field in Equation (18) is:
0+ ws + at f" q i (2:7)
Tr
Ft
d%
d,
Remark 4.1. This result bears some similarity to Lu et al. (2022b, Propostion 5.1); however, they integrate w.r.t the log-SNR; their result can be recovered, mutatis mutandis, with the identity A log Tt.
4.1 Greedy guidance as an Euler scheme
Now equipped with this simplified form, begin draw connections between end-to-end guidance and our greedy strategy. In Proposition 4.2 we show that the greedy action in Equation (7) can be interpreted backpropagation via DTO scheme with an Buler step of size h 71 7t-
Proposition 4.2 (Greedy explicit Euler scheme within DTO). For some trajectory state It at time t, the greedy gradient given by V ((())))) is the DTO scheme with an explicit Euler discretization with step size h 7/1 -/--
Now we examine greedy action from the perspective an OTD scheme. In Proposition 4.3 we show that a greedy strategy can be viewed as the first iteration of a fixed-point method of an implicit Euler discretization of the continuous adjoint equations.
Proposition 4.3 (Greedy an implicit Euler scheme within OTD). For some trajectory state 2t at time t, the greedy gradient given by V V., F. (x.)) is an implicit Euler discretization of the continuous adjoint equations for the true gradients with step size h = 71 - 7t..
Proof sketch. First, we use the technique of exponential integratoor to simplify the continuous adjoint equations. Then we perform first-order Taylor expansion around Yts which is equivalent to an implicit Euler scheme, as we calculate the gradient flow from 1 to t. The full proof is provided in Appendix B.5.
5 Is greed good?
A natural question to ask in light of this discussion taking this greedy action is why even bother backpropagating through the ODE solve at all for guidance? After all, we could simply run the optimization process directly in the data space (cf. Equation (7)). So why perform end-to-end guidance or this greedy action at all? N.B., the full derivations and proofs for this section may be found in Appendix C.
We begin by examining the structure of the gradient V=L(#f11 the following:3
By the chain rule we observe
V (x) = V V L (of, (x1))
The question then is what is the behavior of V 40 (x)? We answer this in Theorem 5.1 below, providing integral equation for V. क6 (x:)
Let Ve, be shorthand for the gradient the output (x).
5

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Theorem 5.1 (Jacobian matrices of affine Gaussian probability paths). For the standard affine Gaussian probability path with flow model & e, (2),, the Jacobian matrix & (,,(a)) as function of x is given as the solution to
V 2
0t 10
tu
Tu Variju ( (92, (x) Vn & (a: du,
(10)
0s
oa
where
[arij((x)) ())(ee)
[(0] æ (2)) -
(11)
we observe the Jacobian-vector product v, & (x)) v corresponds to an integral of covariance projections applied to v.
Remark 5.1. From Theorem 5.1
Thus, we see that the continuous- time backpropagation process through the flow model is a projection of the loss by covariance matrix into the directions of highest variance, i.e., the guidance encourages the state to evolve within states on the data manifold. We elaborate on this more in Appendix C.2. While this is a nice observation we cannot solve such an integral in practice. What about our greedy strategy, how does it impact the loss function?
5.1 Dynamics of gradient guidance
We now consider how the output of the flow model will change under greedy guidance. In particular, we are interested in how T% (x) changes under the following gradient step
='=z-nV
x' z'= x 17 V (z (z))((z))
(12)
To do this, we make use of the Gateaux differential (Gâteaux 1913) which allows us define the differential that describes how the output of the flow model æ evolves with changes to æ at time t. We present the result this question in Proposition 5.2 below.
Proposition 5.2 (Dynamics of greedy gradient guidance). Consider the standard affine Gaus- sian probability paths model trained to zero loss. The Gateaux differential of x at some time t te [0,1] in the direction of the gradient V v T (x))
is given by
= -
( V V ())
(13)
Remark 5.2. Recall that from Theorem 5.1 and (Ben-Hamu et al. 2024, Proposition 4.1) we know that both V Φ6 (a=) and Va (ar) consist of covariance matrices, thus the dynamics of greedy gradient guidance are governed by this covariance projection of the loss.
Next, we ask what is the difference between the idealized gradient V Va (z) and the greedy gradient (=))) Intuitively, we find that is bound by the local truncation error, i.e., O(h)) which we show below.
Theorem 5.3 (Dynamics of gradient vs greedy guidance). The difference between the dynamics of gradient guidance in Proposition C.4 and greedy gradient guidance in Proposition 5.2 for point x at time with guidance function L E C2 ( is bounded by ((h)) where h == 71 7t.. i.c.
t - =0
(14)
An important question is whether a greedy strategy makes good decisions each timestep. l.e., if we make a good decision at time f, does that ensure that an optimal solution was made in the sense of Φ6 1/e (z)). natural way to examine this question is consider whether convergence in the local case implies convergence of the whole solution trajectory. We find that up to bound dependent on the
'Readers familiar with the work of Ben- et al. (2024) may notice some similarities between our result Theorem 5.1 and Ben- Hemu et al. (2024, Theorem 4.2). We discuss this more Remark C.3.

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step size, convergence in the greedy solution implies convergence in the flow, which we state more formally Theorem 5.4.
Theorem 5,4 (Greedy convergence). For affine probability paths, if there exists sequence of (n) a at time such that converges to the locally optimal solution (q)((x)")) + æi.
states
Then the solution, (1(())))) converges to a neighborhood of size O(h2) centered at oi.
Beyond Euler
Motivated by this connection between the powerful, but expensive, end-to-end guidance techniques and posterior guidance techniques, ask is there middle-ground between them? A natural extension would be to consider something beyond the Euler scheme from the previous section, e.g., applying the midpoint method two Euler steps. To motivate this discussion more rigorously present Theorem 6.1, which shows that for any explicit single-step between the ideal gradient and this estimated gradient is on the order of the local truncation error of the underlying numerical solver.
Lunge-Kuuthe solver, the error
Theorem 6.1 (Truncation error of single-step gradients). Let T be explict Runge- Kutta solver of order a > of a flow model with with $6 &,,, (x)) Then for any E [0,
V T f (x)
V. O1h+t))
(15)
where h 1
We can use a higher-order solver to move between posterior and end-to-end guidance exchanging compute and gradient accuracy.
Key insight 3.
This theoretical tool enables move between posterior and full end-to-end guidance choosing whichever point between compute and accuracy happens to be most suitable, hopefully opening a larger design space for solving interesting problems. Additional discussions and the full derivations are found in Appendix D.
7 Experiments
Motivated by the theoretical connections from the previous sections we apply the greedy posterior strategy (Fuler) to several problems using flow/diffusion models, as well as several methods lying in the in between space of end-to-end guidance and posterior guidance, namely, single-step midpoint scheme and 2-step Euler scheme.
7.1 Inverse problems for images
common application of posterior guidance has been in solving inverse problems (Y. Song, Sohl- Dickstein, et al. 2021; Chung, Sim, and Ye 2022) (ef. Appendix H). As such, we explore several inverse problems in the image domain. In particular, we explore a set of inverse image problems on subset of 100 images from the FFHQ (Karras, Laine, and Aila 2019) 256 x 256 dataset. We make use of the pre-trained diffusion model from Chung, J. Kim, et (2023) trained on the FFHQ dataset.
Inverse problems and metrics. Following (B. Zhang et 2025) we conduct experiments on the following linear tasks: super resolution, Gaussian deblurring, motion deblurring. inpaintining (with box mask), and inpainting (with a 70% random mask); along with three non-linear problems: phase retrieval, high dynamic range (HDR) reconstruction, and non-linear deblurring. We use the standard (PSNR), structural similarity index measure (SSIM),
evaluation metrics of peak Learned Perceptual Image Putch Similarity (LPIPS) (R. Zhang et al. 201 8), and Fréchet Inception Distance (FID) (Heusel et 2017). solve the probability flow ODE with the midpoint scheme and 20 discretization steps; further configuration details are reported in Appendix I.I.
7

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Figure 3: Qualitative visualization of using posterior guidance to solve inverse problem the task of inpainting with 70% random mask. Top row the ground truth, middle row the measurement, and the bottom row is the reconstruction.
Table 1: snapshot of the quantitative results for solving inverse image problems on FFHQ. We
report the mean performance (PSNR, SSIM, and LPIPS) across 100 validation images along with the
FID. All tasks are using noisy measurement with noise level 8y 0.05. The full results are found
in Table 6.
Task
Method
PSNR (t)
SSIM (t)
LPIPS (L)
FID (L)
Greedy (Euler)
30.87
0.823
0.141
40.73
Greedy (midpoint)
31.03
0,816
0.139
38,80
Inpaint (random)
Greedy (2-step Euler)
30.80
0.811
0.144
39,23
DAPS
31.12
0.844
0.098
32.17
DPS
25.46
0.823
0.203
69.20
Greedy (Fuler)
28.01
0.766
0.182
57.04
Greedy (midpoint)
28.36
0.776
0.185
58.55
Gaussian deblurring
Greedy (2-step Euler)
28.18
0.774
0.181
57.18
DAPS
29.19
0.817
0.165
53.33
DPS
25.87
0.764
0.219
79.75
Results. We present some qualitative results on reconstructing images from a random mask in Section 7. Quantitatively,
present snapshot of our full results (ef. Table 6) on the inpainting with random mask and Gaussian deblurring tasks. For reference we include the standard DPS (Chung, J. Kim, al. 2023) and the recent state-of-the-art DAPS (B. Zhang et al. 2025). We observe that the posterior guidance strategy works well performing closer to DAPS than DPS. Interestingly, on these tasks the extra compute and smaller truncation error of the midpoint and 2-step Fuler did not lead to any noticeable performance gains. We report further results in Appendix J.2 along with additional analysis and discussion.
Ablations on discretization steps. As we discussed in Section we can improve performance by taking step sizes. We preform a more involved ablation of this design axis the high-dynamic range (HRD) reconstruction experiment detailed in Table 2. Additionally, we also report the results from the RED-diff (Mardani et al. 2024) algorithm. Following B. Zhang al. (2025) for the non- linear inverse problem of HDR reconstruction we perform 4 runs per algorithm and report the mean and standard deviation. We notice that the Greedy (5-step Euler) performs very well beating the SOTA DAPS algorithm, even the 2-step and 3-step perform variants well or better than DAPs on this problem. Increasing the number of discretization steps leads better performance (ef.. Theorem 6.1). Interestingly, the standard deviation decreases well with the results becoming more consistent. We also compared a full DTO run with vary step sizes, i.e., end-to-end optimization with a vary number of steps used in calculating the gradient (the full 20 are used for the final sampling). We do observe similar trend of increasing performance as we increase number of steps, however, far under-performs the greedy strategy for similar compute budget.

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Table 2: Further ablations on the number of discretization steps on the non-linear HDR inverse problem.
Method
PSNR (1)
SSIM (1)
LPIPS (
(4)
DAPS
22.112
0.752
0.162
42.97
DPS
22.73 73.007
0.
4..
112.82
RED-diff
12.1623..4 41
0.512
0.25810.088
108.32 32
Greedy (Fuler)
25.07 114 2:.22
0.776 0.116
0.173
43.25
Greedy (2-step Fuler)
26.32=4.34
0.802=0.111
0.173:2.065
38.64
Greedy (3-step Euler)
27.17 24.21
0. 2.820
0.154410002
36.07
Greedy (4-step Euler)
27. 27.89 4..10
0.828=0.00
0.151
36.94
Greedy (S-step Euler)
28.27 4-4.01
0.831
0.149 +00009
35.35
DTO (1-stcp)
13.16
1.521+0.u5p
108.39
DTO (2-step)
14.91
0.372
0.483
98.93
DTO (4-step)
16.37=1.se
0.455=00.082
0.457
93.52
DTO (8-step)
16.37
0.455=0.082
0.457:0.006
93.52
48.43
53.64
60.05
73.95
78.02
82.75
88.45
91.30
Qualitative visualization of controlled generated molecules for various polarizability (a) levels. Top row is generated using end-to-end guidance with DTO scheme and the bottom row is generated using greedy guidance.
Figure 4:
7.2 Molecule generation for QM9
We also illustrate the core ideas with some experiments controllable molecule generation on the QM9 dataset (Ruddigkeit et al. 2012), a popular molecular dataset containing small molecules with up to 29 atoms. Following Hoogeboom al. (2022) and Ben-Hami et al. (2024), we perform the conditional generation of molecules with specified quantum chemical property values. In particular, we target the following properties: polarizability a, orbital energies Еномо, ELUMO and their gap Ae, dipole moment, and heat capacity C., The property classifiers were trained following the methodology outlined in Hoogeboom et al. (2022). The underlying flow model is an unconditional equivariant flow matching model with conditional optimal transport path (Lipman, Havasi, et al. 2024, Section 4.7; of. Tong, Malkin, et al. 2023; Tong, FATRAS, et al. 2024), i.e., the EquiFM (Y. Song, Gong, et al. 2023) model. We solve the ODE with Euler's method and 50 discretization steps; further configuration details are reported Appendix I.I. Further details are provided in Appendix I.2.
Metrics. evaluate the guided generation we calculate the mean absolute error (MAF) between the predicted property value of the generated molecule by the property classifier and the target property value (Satorras et al. 2021). Additionally Appendix J. we report the quality of the generated molecules by evaluating the atom stability (the percentage of atoms with correct valency) and molecule stability (the percentage molecules where all atoms are stable).
In Section 7.1 we present visual comparison between molecules generated targeting different polarizability a values using DTO end-to-end guidance scheme (essentially D-Flow) and the posterior guidance scheme. Notice that increases the compactness of the molecules
Results.

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Table Quantitative evaluation of conditional molecule generation. The MAE reported for each molecule property (lower is better).
Property
A meV
ÉHOMO meV
SLUMO meV
μ
CN
Unit
Bohr?
D
K-mol
Greedy (Euler)
11.282
1265
725
1092
1,559
6,469
Greedy (midpoint)
5.313
1196
599
1057
1,417
2.967
Greedy (2-step Euler)
5.667
1205
695
1222
1.491
2.767
Greedy (3-step Euler)
5.098
1152
600
1152
1.384
3.229
Greedy (5-step Fuler)
4.177
1083
571
939
1,328
2.332
DTO (1-step)
13.049
10
681
86.512
1.666
15.144
DTO (2-step)
6.113
1359
666
1199
1.533
3.757
DTO (4-step)
6.115
1294
668
1190
1,406
2.829
DTO (8-step)
4.549
1070
608
1078
1.247
2.594
DTO (16-step)
3.454
817
608
939
1.177
2.003
DTO (32-step)
2,912
750
410
666
0.721
1,566
DTO (40-step)
2.384
625
372
556
0.719
1.425
DTO (50-step)
1.404
401
176
373
0.372
0.866
EquiFM
9.525
1494
622
1523
1,628
6.689
Lower bound
0.10
64
39
46
0.043
0.040
generated by DTO scheme decreases. This trend is less noticeable for the posterior guided samples. We report quantitative results in Table 3. We report the unguided EquilM generated molecules as an upper bound and include the theoretical lower bounds from Ben-Hamu et (2024). It is here that we notice a sharp decrease in performance from using posterior guidance. In particular the greedy (Euler) strategy is is highly unstable even performing worse than the unguided model on the property. The introduction of additional step the form of either midpoint 2-step Euler does seem to improve performance; although the significance varies property to property. We observe that the midpoint method seems perform slightly better than the 2-step Euler. We performed experiments with Ralston's 's third-order method and a fourth-order Runge-Kutta scheme but noticed steps, posit this is due to the large step
significant instability in comparison just taking size and that a hybrid scheme like that employed by Moufad et al. (2025) might be reasonable solution such problems, but ultimately we leave that question up future work. Moreover, observe that increasing the number of steps generally improves performance with greedy (5-step Euler) performing the best among all the greedy guidance strategies. The gradients of DTO (50-step) strategy are the ideal and perfect gradients w.r.t. the flow model as the numerical solver takes 50 discretization steps and thus form the upper bound of performance. We that as the DTO strategy incorporates more discretization steps the results converge to the upper bound along this particular design axis; in particular, we notice that the greedy strategy does well in the the regime of less discretization steps.
8 Conclusion
In this paper we present a unifying view of two different families of guided generation: end-to-end guidance and posterior guidance from the lens of greedy algorithm. We present numerous theoretical connections tying these two families together. Our theoretical analysis shows that there might be some reason to believe that such cheap approximation of the gradient tasks. By exploiting the theoretical connections we created, we investigate guidance techniques which lie in between these two families giving rise several experiments on inverse image problems and on controlled molecule generation to illustrate this new design space. We hope that our findings help future researchers find the optimal spot between computational cost and accuracy of gradients for guidance problems.
be reasonable for certain
exciting novel design space. We then conduct
10

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Organization of the appendix
In Appendix A discuss previous approaches by exploring posterior guidance and end-to-end guidance in greater detail to provide a more comprehensive overview of how this greedy perspective connects these various works. Appendix the main paper. Likewise, Appendices C and D is devoted to proofs and derivations from Sections and 6 respectively. In Appendix we discuss some important practical issues when using OTD for guidance, which we believe several to be useful background for the reader. We provide some additional connections between posterior guidance and control signal optimization in Appendix F that we were unable to include in the main paper. Appendix H is devoted to providing brief background on inverse problems. Likewise, Appendix is devoted to discussing the implementation details the numerical experiments in Section 7 and providing a background for the experiments. In Appendix J we include additional results that we could not fit into the main paper. Lastly, in Appendix we discuss the limitations and broader impacts of this research.
devoted the proofs and derivations from Section 4 in
Appendices
Related works
26
A.1 Posterior guidance.
26
A.2
End-to-end guidance
27
B greedy perspective
27
B.1 Additional details on flow models
28
Assumptions
28
B.3 Proof Proposition 4.1
28
B.4 Proof Proposition 4.2
29
B.5 Proof Proposition 4.3
30
C
Dynamics of guidance
31
C.I Proof of Theorem 5.1
32
C,2 Dynamics of gradient guidance
33
C.3 Proof Proposition C.4
34
C.4 Proof Proposition 5.2
34
C.5 Proof Theorem 5.3
35
C.6 Proof Theorem 5.4
36
D Beyond Euler
37
D.I Proof Theorem 6.1
37
D.2 A useful reparameterization of the flow model.
................................
38
E Notes using OTD in practice.
................................
39
F On control signal optimization
..........
41
F.I Continuous adjoint equations for control signals.
.
41
G Implementation details
..........
42
G.1 The construction the greedy guidance schemes
................................
42
G.2 A PyTorch pseudocode illustration
43
H brief introduction inverse problems
43
H.l Inverse problems and diffusiion models.
44
Experimental details
44
1.1 Inverse image problems
44
1.2 Molecule generation for QM9
45
1.3 Numerical schemes
46
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1.4
Hardware and compute cost
46
Further experimental resulty.
46
J.1
Molecule generation for QM9
47
J.2
Further results inverse image problems.
47
J.3
Sampling trujectoriss for inverse problems
49
J.4 More qualitative samples for inverse problems
49
J.5
More qualitative samples for controlled molecule generation
53
K
Discussions.
53
K.I Broader Impacts
53
K.2 Limitations
53
Overview of theoretical results
For convenience provide list of theorems to make navigating the theoretical results easier.
4.1
Proposition (Exact solution of affine probability paths)
4.2
Proposition (Greedy as an explicit Euler scheme within DTO).
4.3
Proposition (Greedy implicit Euler scheme within OTD)
5.1
Theorem (Jacobian matrices of affine Gaussian probability paths)
5.2
Proposition (Dynamics of greedy gradient guidance)
5.3
Theorem (Dynamics of gradient greedy guidance)
5.4
Theorem (Greedy convergence).
6.1
Theorem (Truncation error of single-step gradients)
4.1
Proposition (Exact solution of affine probability paths)
29
4.2
Proposition (Greedy as an explicit Euler scheme within DTO)
29
4.3
Proposition (Greedy as an implicit Euler scheme within OTD)
30
C.1
Lemma (Gradient of target prediction model)
C.2
Lemma (Dynamics of Jacobian matrices for flows)
31
5.1
Theorem (Jacobian matrices of affine Gaussian probability paths)
32
C,4
Proposition (Dynamics of gradient guidance)
34
C.4
Proposition (Dynamics of gradient guidance)
34
5.2
Proposition (Dynamics of greedy gradient guidance)
34
C.4.1
Corollary (Dynamics of gradient vs greedy guidance)
35
5.3
Theorem (Dynamics of gradient vs greedy guidance)
35
5.4
Theorem (Greedy convergence).
36
D.I
Theorem (Local truncation error of discretize- settretize-then--ppe then-optimzze gradients)
37
6.1
Theorem (Truncation error single-step gradients)
38
D.1.1
Corollary (Convergence of a a-th order posterior gradient)
38
D.1.2
Corollary (Dynamics of a-th order posterior gradient)
38
D.2
Proposition (Reparameterized for the target prediction model of affine probability paths)
39

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F.1
Theorem (Continuous adjoint equations for the control term)
41
25

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(t. Wang al. 2025)
Control signal optimization
( et 2023)
(Mazion et al. 2025)
End-to-eud guidance
(Res-Hamm al. 2004)
(Blneingame and C, Lin 2004g)
State optimization
(Clark et al. 2024)
(Karunratanakol et al. 2024)
A groody strategy
Training free guided generation
Zhang al. 2025)
(Nowack al. 2024)
Song, Valadar, et 2023)
(Pan, Liew, 2024)
(Yu al. 2023)
(He al. 2024)
(Moutad al. 2025)
Posterior guidance
(Kadkhodliic and Simoncelli 2021>
(Y. Wang, Ya, and Zhang 2023> (Chang Kim, 2023)
(Bansal al. 2023)
Figure 5: A more detailed taxonomy of training-free guided generation methods from Figure from the main paper.
A Related works
We provide brief summary of previous work exploring either posterior guidance or end-to-end guidance strategies. In Figure 5 we provide more detailed taxonomy of training-free methods for gradient-based guided generation based on Figure from the main paper.
A.1 Posterior guidance
Recent work in flow/diffusion models has explored the guidance using this strategy: we highlight a few notable examples. Diffusion Posterior Sampling (DPS) (Chung, J. Kim, et al. 2023) is guidance method that uses Tweedie's 's formula (Stein 1981) to estimate the gradient of some guidance function defined in the output state w.r.t. the noisy state, i.e., E X, æ:). Likewise, the work of Bansal et al. (2023), He et al. (2024), Y. Wang, Yu, and J. Zhang (2023), and Yu et al. (2023) explores similar concepts by employing Tweedie's 's formula for diffusion models. Most of these works have explored using the SDE (or Markov chain) formulation of diffusion models rather than the ODE formulation, which is what we primarily focused on our analysis.
Correcting the guidance trajectory. Several works have explored extensions to the DPS framework by using multiple steps of an SDE solver to correct errors made by the guidance steps. particular, FreeDoM (Yu et al. 2023) explores the usage of time-reversal strategy repeated for set number of times each sampling step to correct possible guidance errors. Likewise, recent work by B. Zhang et al. (2025) explored modeling Langevin dynamics on top of diffusion ODE to correct measurement errors in inverse problems. significant number of the proposed methods which use posterior guidance arise from solving inverse problems (cf. Daras et al. 2024).
26

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Scheduled hyperparameters. Researchers realized that extra performance can be gained in such problems by scheduling hyperparameters like the learning rate (or guidance strength) at different timesteps in the numerical scheme (Moufad et al. 2025; Yu et al. 2023).
Beyond Fuler. Recent work by Moufad al. (2025) explores an extension to (Chung, J. Kim, et al. 2023) by using a two-step method to estimate the guidance gradient. This is mostly closely related to the greedy (2-step Euler) method from the main paper, although they use a stochastic sampling method, so it would be more akin to taking two Euler-Maruyama steps.
A.2 End-to-end guidance
Within the last year, many researchers have explored backpropagation through flow/diffusion models for controllable generation. As mentioned in the main paper, the two main strategies for solving such a problem is a DTO or OTD scheme (cf. Appendix E).
Diseroiize-teen-optiiiize FlowGrad proposed by X. Liu et al. (2023) uses a DTO scheme to optimize additional control signal (more details on this later) to perform guidance with flow models. Although the analysis of Ben-Hamu et al. (2024) makes use of the continuous adjoint equations, in practice they use the generally preferred approach of DTO with gradient checkpointing." Likewise, Clark et al. (2024), Karunratanakul et al. (2024), and Novack et al. (2024) all use gradient checkpointing with DTO perform backpropagation through the flow/diffusion model.
Another stream of work has explored the use of continuous adjoint equations to perform the backpropagation. The advantage of such approaches is the (1) memory cost, and we enumerate the drawbacks in Appendix E, but suffice to say there are several. To the best of our knowledge, the first work to explore this was Nie et al. (2022) which used OTD with SDEs for the adversarial purification task. More general work came later by Ben-Hamo et al. (2024), Blasingame and C. Liu (2024a), and Pan, Liew, et al. (2024). More specifically, Pan, Liew, al. (2024) and Pan, Yan, et al. (2023) explore bespoke solvers for the continuous adjoint equations of diffusion ODEs. Blasingame and C. Liu (2024a) extends these works by developing bespoke solvers for diffusion ODEs and SDEs and performs more theoretical analysis of the problem in the SDE setting. Marion al. (2025) explore using the continuous adjoint equations as a part of a larger bi-level optimization scheme for guided generation. The work of Ben-Hamu et al. (2024) extends the analysis of continuous adjoint equations for diffusion models to flow-based models and provides an alternative perspective to the analysis performed in the earlier works. Recent work by L. Wang et al. (2025) explores an extension of Ben-Hamu et al. (2024) to Riemannian manifolds which incorporates control signal to the vector field and optimizes both the solution state and co-state, they call their approach OC-Flow.
sedimize-then-disrree
Parallel to these works (conceptually) is the work of Wallace, Gokul, Ermon, et al. (2023) who uses EDICT (Wallace, Gokul, and Naik 2023), an invertible formulation of diffusion models, to perform backpropagation through the diffusion model. Although not presented or viewed this way in the original work, the later work by Blasingame and C. Liu (2024a) showed that this approach can be viewed as specific discretization scheme of continuous adjoint equations. We note that the EDICT solver, while reversible, is a zeroth-order solver and has poor convergence properties (ef. F. Wang et al. 2024).
Control signal optimization. We discuss this in more detail in Appendix F, but there several works that explore the optimization of an additional control signal =(1) rather than the solution trajectory atI); namely, X. Liu et al. (2023) and L. Wang et al. (2025).
B A greedy perspective
We present the proofs and derivations associated with Section 4.
*See attps://doot kidger. waldger.sit/ddie http///fffa//p///aaaaaaaa for an excellent summary of such design considerations and why DTO is generally preferable over OTD.
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B.1 Additional details on flow models
Applying this flow to the random variable Xu we define a continuous Markov process {Xx with mapping X, (Xo). The goal, then, is to learn flow 4 such that X1 = D1 (Xo) ~ q(æ). This procedure amounts to learning neural network parameterized vector field u E (l. 1] x ;R;R)) this learning procedure can be performed efficiently through simulation-free training process known as flow matching (Lipman, R. T. Q. Chen, et al. 2023) or more generally generator matching (Holderrieth et al. 2025).
Throughout the rest of this paper we will assume standard flow model trained to zero loss and we denote the parameterized flow model via (z)). We let (2,(2)) = (Øc D (x)) denote the flow from time to time t, s, E 1].
Affine probability paths. A special subset of flow models, are flows which model an affine probability path, i.e., given a schedule (ax, στ) the random process {X,} described via the affine equation
= = a + o,x0
(16)
((0, [); 1J) which satisfy where oe,, ae E c
Cug 01 0,
an = ou 1, Vt E (0, [à > 0, o, < 0].
(17)
The marginal vector field can then be expressed as the following conditional expectation:
uz (x(x))
El ā; Xo| X; = mr)
(18)
This nice form of the marginal vector field enables use to rewrite the vector field in the forms of either source (Ho, Jain, and Abbeel 2020) or target (Kingma et al. 2021) prediction as
Be
ue (x)x)=
=
x
+
f((z),
(19)
Be
31
=a;
=b.
where B3 aa for source prediction with f((x:) = Fojr x = 22) and Br ag for target prediction with f:( 11 (xx || Xx - =/]; and ats are useful shorthands to be used later.
Remark B.1. The probability flow ODE formulation of diffiusion models (Y. Song, Sohl-Dickstein, al. 2021) is subsumed by flow models, and represents model with affine Gaussian probability paths (AGGP), i.e., (Xo, X1) ~ no.i (zo, æ.) = p( (ro (x)))()))))))))))))))))))))))() with p(æ) = N(xx) 0, I (Lipman,
Havasi, et al. 2024). Thus without loss of generality we consider flow models of affine probability paths.
B.2 Assumptions
Throughout the norm I| corresponds the Euclidean norm || 112- Additionally, we make the following (mild) regularity assumptions:
Assumption B.1. The function at at
ès is integrable in 1).
Assumption B.2. The toial derivatives
d.y (r]),((a)) exist and are continuous for 0 S S k 1.
d=
Assumption B.1 is necessary for the simplification that we perform with exponential integrators and Ben-Hamu al. (2024) make the same assumption in their analysis of the continuous adjoint equations for affine probability paths. Assumption B.2 is to ensure that we can take a Taylor expansion of (f,(()) (z).
B.3 Proof of Proposition 4.1
We restate Proposition 4.1 below.
'Clearly, diffusion models which solve the severse-time SDE different and require separate analysis.
28

---

Proposition 4.1 (Exact solution of affine probability paths). Given an initial value of æ at time 8 E 0, 1] the solution Ds at time t E [0, 1] of an ODE governed by the vector field in Equation (18) is:
3t Ig + ot { i (x,,) dr.
0t
T
(8)
Proof. Recall that we uniquely define flow model through the vector field E ((,, 1| IR,; Rd). The vector field which models the affine conditional flow with schedule (ar, oe), is defined as
u (x) 1 + t X z).
(20)
With some simple algebra, we can rewrite the vector field in terms of ø1/0.
(x) ag + beal (2:),
oz
σz
(21)
b, ai
ae
Now using this definition we can rewrite the solution for æ; from as in terms of é Lijt.
at F. + J:" ut (x+T) dr,
(22)
Φ = xs + f(" a, + by b, (x+) dr.
(23)
Note the semi-linear form of the integral equation. We can exploit this structure using the technique of exponential integrators, (see Gonzalez et al. 2024; Lu et al. 2022a: Q. Zhang and Y. Chen 2023), simplify Equation (23), under Assumption B.1,
[ eli au du 2y (xx) dr.
Øt el au du x. +
(24)
Now, the integrating factor simplifies quite nicely to
Ox eli en e/ As du eoo 1 do
(25)
such that Equation (24) becomes
by 2
v. + 0:
(x+) dr.
(26)
oy
We can simplify b to find:
δgo: a.o,
d
at
of dl
0x
di
Y
(27)
where "/ == a1/or, iee, the signal-to no-ise ratio. As such, we can rewrite Equation (26) with a change of variables By Fo (x) Ct
81 - + 0e 1 2i (xn d%
(28)
concluding the proof.
B.4 Proof of Proposition 4.2
Proposition 4.2 (Greedy as an explicit Euler scheme within DTO), For some trajectory state D at time t, the greedy gradient given by V=L(= (x)) is the DTO scheme with explicit Fuler discretization with step size
71 7t-
29

---

Proof. From Proposition 4.1 we see that using the target prediction model to estimate is akin taking first-order approximation of the flow. More specifically, under Assumption B.2 we can construct (k 1)-th Taylor expansion of Equation (8) with:
Tt 3t
0g
0
dy
d
x
1/
(xn)
9=9 7
( 17-78 7s)
n!
dy + O(h+t))
(29)
k-1
1
1a+1
0t
+
0
117 (20g)
+ O(hhtt)
(30)
0x n
dy
(n+1) 1)!
7=
where = 7/- " is the step size. Then it follows that for the first-order discretization the flow, omitting high-order error terms becomes,
0:003 I. + (as ds
)æ (z.).
(x.).
(31)
In the limit as t + 1 have ã; 211. (x..). Thus, the greedy gradient is DTO scheme with explicit Euler discretization with step size h ?1 7t-
B.5 Proof of Proposition 4.3
We restate Proposition 4.3 below.
Proposition 4.3 (Greedy as an implicit Euler scheme within OTD). For some trajectory state 2; at time t, the greedy gradient given by V Va, (x.e(e))(22)) is an implicit Euler discretization of the continuous adjoint equations for the true gradients with step size h = 71 7t-
For clarity we restate the definition of the continuous adjoint equations. Let us C ((O, 1] x Rd 12)) be a model that models the vector field of some ODE and be Lipschitz continuous in its second argument. Let æ [J + Rd be the solution the ODE with the initial condition æo Rd. d; (t, 22). For scalar-valued loss function L C² (RR)) in æ1, let ag OC/0z, denote the gradient, Then an and related quantity ag == 0C/00 can be found by solving an augmented ODE of the form,
OC
da
Oug
ag
(1)
(t) ax)
(t, xz),
Ox
dt
dæ
(32)
da
dt (t) ag (1) Ous
a. O,
(1, .t).
00
Now present the proof.
Proof. The adjoint state can be simplified by rewriting the vector field in terms of the target prediction model to find
da (1) -,,a (t) zz (1)
Gx fje (x:)
(33)
Ox:
We can express this backwards-in--iiee ODE as an integral equation the form of
(s) = ax(()) - S: a, + b))x(7)
dxl (x)r
dr,
It, dr.
= a() + ( araa(()) + (T)
(time-reversal)
(34)
Oxt
NNote that despite + 0 the asymptotic behavior is well-defined (see Ben- Hamu al. 2024).
30

---

Using the technique of exponential integrators we rewrite the integral as
Ox 0a (x:+)
age (x)()) = edi a, du ax (1) +
eli dub 0, (T))
dr.
dac
0 (t()) + σe
b, a (T)
Ox 1|1 (207) (277)
dr,
Ox, dx ax (t) σe {" axl
Or
117 (x0
dr.
(35)
dx
By Assumption B.2 it follows that the vector-Jacobian product has (k - 1)-th total derivatives, allowing define first-order Taylor expansion around 7
ag (s)
o, Tt (t( + (at
0t as) am))) d.
(s)
de (
+ O(h)).
(36)
dx.
Thus, the first-order approximation of the adjoint state at time t with step size of h = n1 i is the implicit equation
On (1) az (l)(1))
(37)
Now solve the implicit equation we can use the fixed-point iteration method. Let a (1)(0)) a. (1), then the first iteration has
1
a (t) ax (1)
Vz, (z)))
(38)
dz.
Thus, we have shown that the greedy gradients are equivalent to the first iteration of an implicit Euler discretization of the continuous adjoint equations.
C Dynamics of guidance
In this section we detail some of the formalisms omitted in the main paper concerning the dynamics of the gradient flow and greedy gradients.
We begin by re-establishing some useful prior results. Ben- Hamu et al. (2024, Proposition 4.1) showed that the gradient of the target prediction model is proportional to the variance of the random variable defined by Pi (x) (az),, restate their result below.
Lemma C.1 (Gradient of target prediction model). For affine Gaussian probability paths, the gradient of the targel prediction model iie (2) w.rl. æ is proportional to the variance of Pipp (3) (æ)), i.e.
ag Varije
(39)
where
Var (x E (pp((m)a) [0] x (x)) =1)
(40)
Remark C.I. This can written more generally in terms of the (pushforward) differential D (x) where the underlying spaces smooth manifolds and 2ir Hamu al. 2024). In this section, only consider flow models defined in Euclidean spaces, and so we opt not to elaborate on this generalization.
j is smooth map between them (Ben-
We restate a well-known result below in Lemma C.2 regarding the continuous time analogue to forward-mode autodifferentiation, in other words, forward sensitivity.
Lemma C.2 (Dynamics of Jacobian matrices for flows). Let Do E Rd and let f E C2.1((0,T] Rd, Rd) be uniformly Lipschitz in x. Let x [0, T + Rd be the unique solution to
x
dæ
x=0)= Zo.
dt (t) ftt,x(1))
(41)
31

---

Let (2, s,t E [0,T] denote the flow associated with Equation (41). Then let (t) = v. denote the Jacobian matrices, where J : [s,, T + R2 solve the differential equation
dJ
J,(s) =1,
dt (1) = V (.,,
(42)
where ot,(t,,)) refers to the gradient w.r.f the second argument.
Remark C.2. This result well known and has been extended controlled differential equations (Friz and Victoir 2010, Theorem 4.4) and rough differential equations (Friz and Victoir 2010, Theorem 11.3). Kidger (2022, Theorem 5.8) discusses this result for neural ODEs.
C.1 Proof of Theorem 5.1
We restate Theorem 5.1 below.
Theorem 5.1 (Jacobian matrices of affine Gaussian probability paths). For the standard affine
Gaussian probability path with flow model 4 (2)),, the Jacobian matrix V x &a (x) as function of x is given as the solution to
V 2
(x)
I 1
Vargju ( (x))) V du,
(10)
on
where
(a) = E (p)(a))
[x] - x (x)) =()((=)))))))))
(11)
This proof follows a similar technique to that used by Blasingame and C. Liu (2024a) to simplify adjoint equations for diffusion models using exponential integrators.
Proof. Now recall Lemma C.2 which discusses the dynamics of Jacobian matrices for flows, rewriting this an integral equation yields:
s. x = I / Vn., u. ( s.u (x) ) V $ (x) du.
(43)
Now recall the definition of the marginal vector field in terms of the target prediction model (ef. Equation (19)) which
to rewrite Equation (43)
の 8.0 (x) = I + /: Va, On T
V by b ( Va Va T (x) du, a.u (x) Va t0 s.u (x)
x ( n, (x)) Va To (x) du, I + / Va T s.u (z) + bn V
(44)
where (i) holds by V., 69 (x:) Next we can make use of the popular technique of exponential integrators to simplify Equation (43) in combination with Equation (19). Thus, the integral equation Equation (44) becomes
V td (x)( Aa(s. (aas))== S: Aalu,, Va, (", Ta (x) du,
(45)
where (s, = exp 1sd an du is the integrating factor. This simplifies to As t) = 01/0.. Using this, Equation (45) can be simplified to
oo L
b V (4). (x) du.
V 4
(46)
Now can apply Lemma C. further simplify Equation (46) find
V. 8 5 (x)
at I + 0z
au bn Var (
(x)) V T s.u (x: du.
(47)
32

---

Next we simplify the coefficient a. a.bu/ou in the integral term. Let 7/s == ar/o; equal the signal-to- noise-ratio. Then we observe
di at
bb
01
of
01
0
0,00
0
o
(
d
01
01
1
dt
0+
0e
9 Y
(48)
where (i) holds by the quotient rule and (ii) holds by definition of n.- Using this simplification can perform a change-of-variables to simplify the gradient resulting in
V 2 (x)
at "+++0 +tt [ Os
tu Tu Var ( s.u
(z))
(x) du.
(49)
Remark C.3. Readers familiar with the work of Ben-Hamu et al. (2024) may notice some similarities between our result Theorem 5.1 and Ben-Hamu et al. (2024, Theorem 4.2). The difference between the two is that the former is simplified integral equation; whereas, the latter is the exact solution and no longer requires solving an ODE. However, this later solution does require solving time-ordered exponential which requires formal truncated series expansion, e.g., Magnus expansion.
Theorem 5.1 is closely related to Ben-Hame et al. (2024, Theorem 4.2) which we restate below within the context of our notational conventions." 8
Theorem C.3. For the standard affine Gaussian probability path, the differential of $6, (2) as of function of x: is
[
o exp
V T TT
2 >/ 2ar Parraa (xx) dt
(50)
where T exp denotes the ordered exponential.
The time-ordered exponential
(Grossman and Katz 1972) is defined as
T Texp J." A(s) ds
1
ds dsl.... f( ds, T(A(s)) F(A(s))..AAAs))) A(sa),
Σ n=0
J1
(51)
ds2 ds2... f:r dsn A(s:)A(sz) A(sn).
dsi
0=
and the solution be found the Dyson series (Sakurai and Napolitano 2020) Magnus expansion (Magnus 1954), which are truncated in practice. The meta-operator T denotes the time-ordering (Dyson 1949), e.g., consider the time-ordering of two operators A, B:
(s1) B(s2) + ((2) A(s1)
if > 82,
B(s)) =
(52)
otherwise.
For more details refer the reader Weinberg (1995).
C.2 Dynamics of gradient guidance
We state this more formally below in Proposition C.4.
With abuse notation let of denote the time derivative of of
This is closely related to the Peano- Peano-Baker Baker series (see Frazer, Duncan, and Collar 1938, Section 7.5).
33

---

Proposition C.4 (Dynamics of gradient guidance). Consider the standard affine Gaussian probability paths model trained to zero loss. The Gateaux differential of a at some time E [0, in the direction the gradient & (D/ (x)) is given by
8m ( - - V2
(ar) f(x1).
(53)
Thus the behavior of œ1 when guided by L is determined by the operator v (x) which iteratively projects the gradient of the loss function by the covariance matrix Varij((r)) Put another way;
Performing gradient guidance with L at time t < 1 amounts to guidance which follows the target distribution p( ()) by projecting Va, ((x)) onto to the target distribution via the local covariance matrix.
It is for this reason that it is undesirable simply perform guidance in the data space as we are likely to deviate from this target distribution. From Equation (53) we know that applying the gradient at earlier timesteps causes the initial gradient vz, ((x1) to be projected into high variance directions of the target distribution causing the guided sample to stay closer to the true target distribution.
The next question is: how does 11 change when æ: is updated with our greedy guidance strategy?
C.3 Proof of Proposition C.4
We restate Proposition C.4 below:
Proposition C.4 (Dynamics of gradient guidance). Consider the standard affine Gaussian probability paths model trained to zero loss. The Gateaux differential of x at some time e [0, in the direction the gradient V&L (4h [ff,1(2)) (x)) is given by
8x 4 (x) - V 4
(x)
Va, ((x1).
(53)
Proof.
This can be shown from straightforward derivation:
d
2x の
Φ (a 1 (0
d/
1=0
(8))
Va (x V/ V (x))) VaL (of,
==
V V V (x) (x) Va, C(x1).
(54)
where (i) holds by the definition of the Gateaux differential, (ii) holds by the chain rule, and (iii)
holds by a substitution of Equation (9) with the simplification of œ Φ6 (2))
C.4 Proof of Proposition 5.2
We restate Proposition 5.2 below.
Proposition 5.2 (Dynamics of greedy gradient guidance). Consider the standard affine Gaus- sian probability paths model trained to loss. The Gateaux differential of x at some time t [0, in the direction of the gradient v (2*).. (x)) is given by
& = - の V (2:) Tv,, L(x1).
(13)
34

---

Proof. This can be shown from straightforward derivation:
8
1,1
(z)
16,1 (x (x-nVel
VV (z (x])((=))
dr
1=
(())
Te (a- (x))) vol l (z (2f)((2))
1==0
V
x (x) (
(13)
V Vo (x) Vn ((x1),
(55)
where (i) holds by the definition the Gateaux differential, (ii) holds by the chain rule, and (iii) holds by the chain rule.
We note an interesting corollary below.
Corollary C.4.I (Dynamics of gradient vs greedy guidance). The difference between the dynamics of gradient guidance in Proposition C.4 and greedy gradieni guidance in inrPo Proposition 5.2 for point time with guidance function C²(((. is
11, 02 (x)
V 10 (Volf - Voal(((z)) Vas
(56)
C.5 Proof of Theorem 5.3
We restate Theorem 5.3 below.
Theorem 5.3 (Dynamics of gradient vs greedy guidance). The difference between the dynamics of gradient guidance in Proposition C.4 and greedy gradient guidance in Proposition 5.2 for point a at time with guidance function L. E C. (TR')) is bounded by O(h)) where h == 7/1 7/0. i.e,
V 16
V V (x)( = 01h)).
(14)
Proof. From Corollary C.4.1 1 it is clear that the difference between 6, T (a) and 0 the difference between the true gradient and gradient of the target prediction model. Recall Theorem 5.1 which enables to write the gradient as the solution to an integral equation:
t,1 (x) amounts
I.
V 2 (x)
a1 "Ita
tu Th Var (x)) C. (x) du.
(57)
Now as + as + 1, we can simplify the integral equation
V T [ (x) σ1 [
%/ In Var ( (x) du,
(58)
au
and then by rewriting the integral in terms of dy = du we find
V De
Var
1y m (a:) "h (x) dy.
(59)
Next we take first-order Taylor expansion of Varii yields:
68 Ne. (x)) $ Te,7 (x) centered 7, which
Var (2 V
7. (x) Tt (a) + O(7 - 7
(60)
For this analysis, it actually more convenient to include the term as part of the Taylor expansion rather than computing it in closed form in the integral. Now plugging Equation (60) into Equation (59)
35

---

yields
11
1
V th (x)
=01
7
Varij(()) + 0(7 71) dy
0,
7t Varij((0)
(r
n
d7 +7+0(13)
a;
75
Tt Variu( (z) (71 - 71) + -(1)),
σ1
(61)
where holds with h 71 - 7t- Then, with a little algebra we have
V (x) =
at
(71 - 7)) Varij(()) + O(h)),
di
a2
Q1
at
Varlu(()) + O(h2),
a +0())))))0(())
Q1
σ1
o7
Varij (a) O1h2),
(62)
where (i) holds by the boundary conditions of the schedule (cf. Equation (17)). Now recall Lemma C.1 which states:
(x) at
Varzj((a)).
(63)
of
Thus from Equation (62) and Equation (63) it is easy to see that
V (f) Vaalie (x) O(11),
(64)
holds and thus
- (x))] 0(h3).
(65)
C.6 Proof of Theorem 5.4
We restate Theorem 5.4 below.
Theorem 5.4 (Greedy convergence). For affine probability paths, if there exists a sequence of states (n) at time such that it converges to the locally optimal solution ({(((())) + xj.
Then the solution, (1((()))) (n)), converges to a neighborhood of size O(h²) centered at xi.
Proof. By Assumption B.2, we can take a (A: 1)-th order Taylor expansion around yt of the flow in
Equation (8) to obtain
= (x))
01 Er
Qt
+
01
d
x
1177 (x:.)
(7 ( -m)
n!
+ + O(h*t),
n=0
d
(x-7)"
01
11/ (xn)
0
dy O(1+tl),
Tt.
n=0
dh
7= 71
n!
@
0t 01
k-1
d
(xx)
ha+1
+ O(hh+1))
(66)
d
(n + 1)!
7=70
where h = - 7/ is the stepsize. Let k = then we have:
+ (x)) = xr + +o "1 + (hh?).
01
(67)
010t )zj ) (xx) + O(h)).
xr, (an
(68)
σ
36

---

By definition and d1 = 1, then
(1)(()0) = (2))(())) + O(h2)
(69)
which is equivalent
(x)) 21)1(100| Cih²,
(70)
for some constant C₁ 0. Since ad (x, ) + कं know that for any e 0 there exists some
n 2 N such that || - 2 (a) <. Thus,
$ (x(() - a S $ [ (3)) (x) (z) + |0 2t (a((f
<<+ G.
(71)
Therefore, ) converges to a point inside a neighborhood centered at æ with radius O
D Beyond Euler
In this section we provide the full proofs and derivations for Section 6 in the main paper.
D.1 Proof of Theorem 6.1
Before showing Theorem 6.1 we show more general version below.
Theorem D.1 (Local truncation error of discretize-thes optimize gradients). Ler an explict Runge- Kutta solver of order >> 0 to the ODE
dæ
x:
vo
dt (1)=us( ((,))
(72)
on [0, I1 which satisfies the regularity conditions for the Picard-Lindelte theorem. Let (x) denote the flow from to I, for any 8, E T) admitted by the ODE. Then,
V 2 (x) V 2 s. (x)] = (1
(73)
Proof. Consider an explicit k-stage Runge-Kutta method given by
Un. = us tn + cyh, on + >
j=
(74)
1=1
Ea+1 In + h by
(75)
where aj رy. oj are all given via the Butcher Tableau (Stewart 2022, Section 6.1.4). Now, we consider a single step from time to time t with initial value and step size h == t s. Then, the gradient is
8+ E ag,
V t (x)) = Va h
by V 210
j=l
= I+hh
by
V us(s +cgh,
I+ Σ
ag V Vaul
(76)
1=1
1=1
where we let
8 = x + h
aj. Ui
(77)
37

---

Next, recall Lemma C.2 which gives the following ODE
dJ dt
J=(6)=1 -
(L) uo, (1, ((,)))),))))))_())
(78)
Next, we augmented the ODE above with the underyling ODE for the solution state, .(t) = ug (t, ((t). We now apply the same Runge-Kutta solver to this augmented ODE for the Jacobian matrices which yields
)
=-=I+h
bj Va, us (s + oght
I+ h
Σ o3 Vau.
(79)
Clearly, Equation (79) and Equation (105) are equivalent. Now as the underlying numerical solver has local truncation error O(hat1) we find that
V T (x))))))))))))))))(())))))) V -hh+tt).
(80)
Remark D.I. This result is intuitive differentiation is linear operator. However simple, believe the insight is useful on discussion using DTO/OTDD/postrrio methods for guidance and thus include it here.
Remark D.2. Theorem D.I shows that DTO and OTD are really just two sides of the same coin and that one of the main differences is the choice of end points when discretizing.
Remark D.3. Onken and Ruthotto (2020, Appendix A) made similar observations; however, it is for only of the case of Fuler.
Theorem 6.1 (Truncation error of single-step gradients). Let & be an explict Runge- solver of order 0 of a flow model with flow (x,(()) Then for any t 00 1}.
V 4h ( V2 = O(h+t1,
(15)
where h = -
Proof.
This follows corollary of Theorem D.1.
Corollary D.1.1 (Convergence of a-th order posterior gradient). For affine probability paths, if
(m) at time such that it converges to the locally optimal solution there exists sequence of states at
&6 (x(")) + aj. Then solution, 46 D1 (a)( (m)),, converges neighborhood size O(ha+1) centered at ai.
Proof. This follows as a straightforward derivation from Theorem D.I.
Corollary D.1.2 (Dynamics of a-th order posterior gradient). Consider the standard affine Gaussian probability paths model trained to zero loss. Let & be an explicit Runga- Langa-Kutta Kutta solver of order
of a flow model with flow 2. (x)). The Gateaux differential of x at some time t E [0, in the direction of the gradient V.L ( (x)( is givev by
0 ( = V bb ( 4
Va, L(x1).
(81)
Proof.
This follows straightforwardly from Proposition 5.2 and Theorem 6.1.
D.2 useful reparameterization of the flow model
We present useful reparameteriization of the flow model, which is parallel result to Proposition 4.1.
38

---

Proposition D.2 (Reparameterized for the target prediction model of affine probability paths). The ODE governed by the vector field in Equation (18) can be reparameterized as
dy dy
aoaels ( 0
yc)
(82)
0u
where y:
do ct 0e
Proof. The ODE governed by the vector field in Equation (18) can be written as
dæ,
df a,æ; + b,o% (x))
(83)
Now we can use the technique of exponential integrators to rewrite the ODE as
d
e -a du 23
eli ee- du b, 1lc (z)).
(84)
dt
The exponential term can simplified to
elo au du
(85)
We introduce a change-of-variaales, y: dy: ba yt
do vt. Thus, the ODE becomes a
(86)
dt at
Next, recall that b = to (ef. Equation (27)) which enables a change of integration variable:
dy on
(
[
yr
(87)
dy
on
Remark D.4. Recall that, often, for affine probability paths we let dg 1, further simplifying Proposition D.2 to
dy: 2 (0-87).
(88)
dy
Remark D.5. Proposition D.2 is a tangential result to the prior result Pan, Liew, et al. (2024, Equation (1))) which was for diffusion models and was developed w.r.t. the source prediction model rather than the target prediction model and was solved in eeverse-time..
This parameterization in Proposition D.2 can be combined with Theorem D.1 to construct a DTO approximation of the gradient with truncation error (Ye 7s) 0++
E
Notes on using OTD in practice
While the OTD approach has become quite popular after the work of R. T. Chen et al. (2018), several later works have noticed several key issues that we wish for ML practitioners.
Recall our prototypical neural ODE (or flow model) the form
dæ
di (1) = ug (1, xx))),
(89)
and assume it is defined on the interval [0, T and the flow model statifies the usual regularity conditions. Then, the continuous adjoint equations (Kidger 2022, Theorem 5.2) are:
ØL da
Oua
an (T) OeT di (l) am (1)
(1, (()),
da
(90)
da (t) am (t)
Oug
ag (T) = 0,
(t, x(t)),
dt
a0
where (t) == Ot/Ox(t) and as (0) == ac/00.
"Tchhically forward-time due the conventions of diffusion models.
39

---

Table 4: Comparison of different strategies for performing backpropagation through flow models. For the complexity analysis n denotes the number of discretization steps and d the dimensionality of the state. Note, for accuracy we mean there are no truncation errors. Note that whilst in general the stability of reversible solvers is quite poor, there are some solvers which have a non-trival region of stability.
Method
Time
Memory
Accurate gradients
Stability
DTO
(
O(ndP)
DTO recursive checkpointing
On n)
Odd Hollogn)
OTD + stored trajectory
(nn)
O + d²)
/
OTD + reversible solver
O(n)
(dd)
/
OTD
((n) (cd2)
x
Truncation errors. One area of concern is the potential mismatch between the forward trajectory (ax, J and the backward trajectory {#r,, I when performing the backwards solve. E.g., consider an explicit Euler scheme
14, = T4 + (fi+1 - tz) (t,, xt,).
(91)
The same scheme when applied to solving the backward trajectory would yield,
St, 1 (4: (1+ ) (li+1, (1,11))
(92)
Clearly, there is no guarantee that these two trajectories match during the forward and backward solve introducing a source of error. One potential solution is to use an algebraically reversible solver (see Blasingame and C. Liu 2025; Kidger et al. 2021; McCallum and Foster 2024) which guarantees that the forward and backward trajectory match perfectly. Another option is to store the forward trajectory {æt, J.y in memory and use interpolated adjoints if the backward timesteps do not perfectly align with the forward timesteps (see S. Kim et al. 2021).
Stability concerns. Consider the simple ODE, v(t) (y(() defined on E [0, with y(0) yo and X < O. Clearly, most ODE solvers with a non-trivial region stability (see Harier and Wanner 2002, Definition 2.1) will solve this ODE without issue, as the errors will decrease exponentially with À < O. However, in the backwards in time solve from y()) the errors will grow exponentially. It can be shown that the adjoint state suffers from similar stability issues. The local behavior of differential equation is described through the eigenvalues of the Jacobian of the vector field (see Butcher 2016). For x; this is given by
dus and for ag this is given by Bi
ia. (-
à (-
az (t)
Ous
dx (1, (1.2(())
Dug (1, (t))
(93)
dx
Clearly, the Jacobians for ag and 2; solved in reverse-time are identical. meaning the stability of the backward solve pushed onto the solve for the adjoint state (see Kidger 2022, Section 5.1.2.4) for more details. Reversible solvers eliminate truncation errors, but tend to suffer from poor stability, e.g., the region of stability for reversible Heun applied to neural ODEs is the complex interval [-i, i] (Kidger et al. 2021). Recent work by McCallum and Foster (2024), however, has shown strategy for constructing reversible solvers with a non-trivial region of stability.
Recommendations. In light of these concerns we propose we consider to be best practices for deciding what scheme to use.
Generally, the best choice is DTO when memory allows as it is the most accurate in terms of the forward discretization. If memory an issue then using a clever checkpointing scheme (Griewank 1992; Griewank and Walther 2000; Stumm and Walther 2010) can help alleviate such issues in exchange for additional compute time. The recursive checkpointing strategy in combination with DTO is actually the default (and recommended) implementation in the Diffrax library. Alternatively, one could store the forward trajectory in memory and then apply the OTD scheme on these stored states (not activations). This strategy of caching the forward trajectory is quite popular and was used by Blasingame and C. Liu (2024a) and Domingo-Enrich the continuous adjoint equations. Another option is to use an algebraically reversible solver in
al. (2025) in practice when solving
40

---

conjunction with OTD. Lastly, one could use vanilla OTD, which we should mention can actually work reasonably well depending on the application despite the concerns listed above.
In Table we summarize the discussion of this section and hope it is helpful to the reader.
F On control signal optimization
Rather than optimizing the trajectory of the solution the initial condition, several works (X. Liu et al. 2023; L. Wang et al. 2025) have explored the guidance from the perspective of optimal control (Kirk 2004). In essence this technique first injects additional control signal, C (TR; IR.). the vector field, uh, such that
dx, u (2x)) + =(1).
(94)
dt
Thus, instead of optimizing fare krej,.) directly, this control signal can instead be optimized, serving as one of the key insights in (X. Liu et al. 2023; L. Wang et al. 2025). Lee. suppose we have a neural ODE with vector field u, (z)) then we can write the optimization problem as
L-" |z(()]| dt,
min ((xT)+
(95)
s.t. Fr = zu + [ T
(xx1 + (t) dr.
The next natural question then is to ask about the behavior of a greedy strategy applied to =(). To simplify the analysis, we now consider a control signal applied to the posterior model æ such that it is replaced by =fje (2xe =tt) which amounts to simply rescaling (t) from Equation (94) with b From this construction, it should be clear that the greedy gradient for the control signal is merely Ve, [(ē1). ). If using the original formulation where the control signal is applied to the vector field, rather than the denoiser, the gradient simply scaled by weighting function dependent on time. Note that this approach is similar to the greedy approach taken by Blasingame and C. Liu (2024b); however, they inject the control signal the source prediction model rather than the target prediction model.
F1 Continuous adjoint equations for control signals
We model the gradient for this signal by augmenting the continuous adjoint equations with the adjoint state (1) == OC/Oz((). In Theorem F.1 we show that this gradient is simply an integral of the adjoint state a.e (1).
Theorem F.1 (Continuous adjoint equations for the control term). Let ul E C T| x Rd=; Rdz)) ) be a parameterization of some time-dependent vector field of a neural ODE that is Lipschitz continuous in its second argument, and let z E C. ([0, 1]; Rd) be an additional control signal such that the new dynamics are given by Equation (94). Let a aL/az(t), then
(1 = 1" (x) ds.
(96)
Our proof follows the structure of the modern proof of Pontryagin's original result (Pontryagin et al. 1963) presented by (R. T. Chen et al. 2018); and is similar to the form used by Blasingame and C. Liu (2024a, Theorem 2.2).
Proof. For notational clarity, we use the notation x=t) = xt.. We define the augmented state on [0, T] as
d æ (1)
(o, (t)) + xtt)
dt
faug
dz
de (L)
(97)
and the augmented adjoint state as
aaug
= (t)
ax
(t).
(98)
02
41

---

The Jacobian of fang has form
0 mgg
Crag (t.mf())
1
(99)
0(x, =]
The evolution of the adjoint state is given by
dlaang
fau (1).
(1)
[ax
az) (4)
(100)
dt
0 ar. 2
Therefore, (t) evolves with
ddu (1) -z (1),
au (T) O,
(101)
dl
thereby finishing the proof.
G
Implementation details
We discuss how to implement the greedy strategy.
G.1 The construction of the greedy guidance schemes
Recall that the general Butcher tableau for a k-stage explicit RK scheme (Stewart 2022, Section 6.1.4) written
C2
0221
C3
03,1
03,2
(102)
0k.1
0,,2
a(k- 1).k
b1
b2
bk-1
Thus a single-step given by
Un.j ue
tu + cgh, +
Σ
J = 1,2,.....
(103)
b Int1 = En + h Σ byun...
(104)
where aj,á, by.c; ej all given via the Butcher Tableau (Stewart 2022, Section 6.1.4). Now, consider a single step from time s to time t with initial value æ and step size h == t - s. Then, the gradient is
(s 8 1stc
V (x) = Va +
Σ by V us
Σ
1=1
1=1
3
]
I
by
Ve + cch,
I I+h>
aj.s V Vau
(105)
1=1
i=1
where we let
2 x + h
(106)
Which can easily be found through standard reverse-mode mode autodifferentiation frameworks; likewise, the gradients for multiple Euler steps can be found.
42

---

G.2 A PyTorch pseudocode illustration
Below in Code G.1 we provide an example PyTorch implementation of the greedy guidance strategy with an Euler scheme for sampling the main ODE.
Code G.1: Example implementation of greedy guidance
posterson
pottriior_ttpp( x, a_atopan1) t1ist [0, 1]
1liat[:] L
nethod "culer':
Baler step xtdel(t, x) dt elif nothod
Medpoint step dt model(s 0.5 dt, x 0.5 zodel(t, x))
clif nethod -- 'ralston"":
Ralston's third
k2 modell (1 0.5 ét, x 0.5 d: k1)
13 atdel(t 0.76 dt, 0.75 dt 22)
1/3 4/9 23) olif nethod н* 'boun3':
di (2/9
Heun therd order
aedel
1/3
model (t 2/3 St, 2/3 as k2)
(1/4 3/4
alif sethod -- 1234
R5-4 2rep
modal(t (t. D.Bedt, 0.E+dc42)
k3 model (t 0.Sedt, x 0.5edt4k2)
44 model(t
dtak3)
(1/8
1/3 1/3
olif nethod en "mulciple_edd ssler!s
steps
tliat(:]
dt Dt D. statsp tarch. itspacnt tliat[i), n_atapa
for in te: rodal (t, t) de
postertor gusdance psendocode asmeed and loss deftned
for tinwatapa:
quick fiz
xt, opt (aaeeeieeeeeeeeeeeeeeeee. grodd (Tros)
'egi":
optimizer tarah..ptin.S501(44 opt],
else:
optimizer torch.optin LUFOS ops], Aer_iterrear_1t0r lrstt, line siae__earch_ttt toe strronggvolree
stop rango((pp__eopp)) optinizer zero grad((
posterior_ctep(s xt_cpt)
loso(xthat) ()
optimizer.com stop()
no :o_grad()
xt xt_opt mtdel(t, xt_opt)
H A brief introduction to inverse problems
Inverse problems cover a large class of scientific problems (Chung, Kim, et al. 2023) that encompass scenarios where partial measurement y is made of æ. When the mapping + not injection,
43

---

recovering æ from y becomes an ill plseed inverse problem. Generally, the relationship between the underlying sample æ and the measurement y is given by
= A(z) + 77 y r E Rd, x E Rdd,
(107)
where A: Rd. + Rd is the forward measurement operator and n (0, 12) is the measurement noise.
The inverse problem then find p(zly).
More details on these types of problems can be found in Chung, J. Kim, al. (2023), Moufad et al. (2025), and B. Zhang et al. (2025).
H.1 Inverse problems and diffusion models
Recall that the ODE formulation of diffusion models is just a particular type of affine Gaussian probability path (Lipman, Havasi, et 2024). Following the conventions of the EDM model (Karras, Aittala, et al. 2022) we write this ODE formulation, known in the literature as the probability flow ODE, below
dx, = - Vx, æ, logp(a, xt) dt, where p( ..)) is the joint distribution of æ; noise level dz. N.B., for diffusion models dt is a negative timestep and we integrate in reverse-time from T to 0. These models are also called score-based generative models due to learning the score function Va, log, 22).
(108)
One of the insights Chung, J. Kim, et al. (2023) and Y. Song, Sohl-Dickstein, et al. (2021) is to apply Bayes' theorem for inverse problems to score-based generative models, i.e.,
p(aly) p p(()
(109)
p(y)
V p( = Vn ((x) + Var loyl(y))
(110)
Adapting this diffusion models, assuming is defined on By (the output), we have
Va, log (y) Va, log, xx. Va, logp(yy (e.
(111)
The unconditional score is the regular score function learned by diffusion models and thus is appropriately learned; however, the other term is much more difficult to work with. The approach of Chung, J. Kim, et al. (2023) is use an approximation of
pyy [ σt) Em ((oopp(molm))) m. [p zo. ou)],
(112)
via Tweedie's s formula (Stein 1981) to write
p kt, o7)) ss p [x.])
(113)
The approximation error can be quantified by the Jensen gap (Chung, Kim, et al. 2023, Theorem 1).
I Experimental details
We provide additional details of the experiments performed in Section 7. N.B., for all experiments we used fixed random seeds between the different software components ensure fair comparison.
I.1 Inverse image problems
Inverse problems. The inverse problems are implemented in the same way as in B. Zhang et al. (2025). We reiterate some of the important settings below. For Gaussian and motion deblurring made use of kernels of size 61 x 61 with standard deviations 3.0 and 0.5 respectively. The box inpainting task makes use of random box of size 128 x 128 to mask the original images, while the random inpainting task randomly masks each pixel with a probability of 70% following (B. Song al. 2024). The measure for the high dynamic range reconstruction problem defined as
A (clip(aazo - 1), 1),
(114)
with = 2.
This is not the the from the scheduler (ae, σx) used in the main paper.
44

---

Diffusion model. We make use of the pre-trained diffusion model from Chung. J. Kim, et al. (2023), trained on the FFHQ 256 256 dataset. We focus on the probability flow ODE formulation popularized by Karras, Aittala, et al. (2022) known as EDM described as
dæ = - V m, logp(o,, Vt) dl.
(115)
Following Ben- Gen-Hamu et al. (2024), we employ a midpoint scheme to solve this ODE reverse-time with N 20 steps. We use the noise schedule o; t which means oN 1. The discretized noise schedule fon N is given by the following polynomial interpolation
Fmax ))²
n
On
Omax
(116)
N -
We use p - 7, max 100, and é min = 0.01 for all experiments and integrate over [e, T. N.B., truncating the integration domain rather than 0 is quite common in diffusion models (Y. Song, Dhariwal, et al. 2023).
Hyperparameters. Unlike previous works (B. Zhang et al. 2025) we did not adjust the hyper- parameters per task and left them the same throughout. The learning rate was set 1 for all experiments, and we performed Rope = 50 optimization steps with the stock implementation of the torch. optim............m.. SGD method for each step of the ODE solve. We set By 0.05 for all tasks.
Ablation study. For the ablation study in Table 2 we used the L-BGFS optimizer over the standard SGD optimizer used in the main experiments (for the greedy guidance runs). For full DTO we used SGD as it provided better performance over L-BGFS in that scenario. N.B., due to compute limitations we couldn't run DTO for step sizes larger than 8. Importantly, we the maximum number of optimization steps between the greedy and DTO strategies; for greedy we take 5 optimization step per step the ODE solver, so 100 total. Likewise, for DTO we take 100 optimization steps in total.
I.2 Molecule generation for QM9
We follow the experimental methodology taken in previous work (Ben-Hamu et al. 2024; L. Wang et al. 2025) and follow the conditional generation pipeline used by Hoogeboom al. (2022). An cquivariant graph neural network (GNN) was trained for each property on half the QM9 dataset, serving as classifier-this model was then used guidance function during the experiments. The EquiFM (Y. Song, Gong, et al. 2023) model was trained on the whole QM9 training set and was used as the underlying flow model for the experiments. Following L. Wang al. (2025), the test time properties were sampled from the whole training set; contrast to Ben-Hamin et al. (2024).
Following Ben-Hamu et al. (2024) we used the L-BFGS algorithm (D. C. Liu and Nocedal 1989) with optimizer steps and inner steps with a linear search, in particular we used stock PyTorch implementation torch... www.opt......... LBFGS. For the DTO experiment we used a learning rate of 1. We tried this for the posterior guidance experiments but encountered severe instability. We found that a learning rate of 7 0.001 seemed to work better.
Recall that Proposition 5.2 states that the greedy gradient is scaled by the covariance projection. This effect is lessened as t + 1. thus in later timesteps the greedy gradient is more likely to push samples off the data manifold. We observed this, with exploding losses even at small learning rates. To remedy this, we took inspiration from other works (Chung, J. Kim, et al. 2023; Moufad et al. 2025; Yu et al. 2023) and annealed the learning rate. We chose the following simple scheduler:
11(1 -1) 1) 1> 0.5
=
t 0.5
(117)
where 7) 0.001 is the base learning rate.
45

---

Runge-Kutta& 4. Additionally, we ran some experiments using RK4 but ran into insurmountable stability issues. Recall that RK4 is given by
1 ug
n,
(118)
h
h
12
Un
+
ki
(119)
h
Un + h
kg u0
nn
k22
(120)
2"
k1
up
(tn
h, In + hk3)
(121)
h
xn+1 +
(k1 + 2k + 2k3 + k1).
(122)
6
Using the step size h = t we encountered large stability issues with the ką term due to being evaluated at the endpoint of the flow model trajectory. We tried mixed-solvee scheme were we would start with Euler and then switch to RK4, but that did not help. We also tried the common diffusion trick of truncated the time interval [0, 1 c] for some small > O, but this did not solve the stability issues either. Ultimately, we abandoned it for this work and left such explorations for future work. seems reasonable to suppose that schemes which don't evaluate on the endpoint, e.g., Ralston's method, Heun's s third-order method, or Ralton's 's third-order method may fair better.
I.3 Numerical schemes
We detail the numerical schemes used for posterior guidance beyond Euler.
Midpoint.
The midpoint scheme used in both experiments is implemented as
h h 2t + uo, z2)) 2
= + hus
++
(123)
2
with step size (. 1.12
2-step Euler. This scheme used in both experiments is implemented as
h
T a++ = 3e + (1 see;
(124)
σ1 = x144 + us
t+ 2 21
(125)
2
with step sizes h =
I.4 Hardware and compute cost
Inverse image problems. The inverse image problem experiments were run a single NVIDIA H100 80GB GPU. took roughly minutes and GB of VRAM to generate images for each inverse problem. As such each experiment took about an 40-50 minutes. Experiments which used the midpoint method, unsurprisingly ran about 90% slower.
Molecule generation. The molecule generation experiments were run single NVIDIA V100 16GB GPU. It took about 3 minutes and 1.5 GB VRAM to generate molecule leading to the experiments taking the order of 300 minutes to complete. Experiments which used the midpoint method, unsurprisingly ran about 90% slower.
J Further experimental results
We present additional experimental results that we could include in the main paper for the sake of space.
This is appropriately adjusted for diffusion models with terminal time of 0.
46

---

J.1 Molecule generation for QM9
In Table 5 we present the atom stability percentage (ASP) and molecule stability percentage (MSP) per property for each guided generation model. Interestingly, despite their poor quantitative performance MAE (cf. Table 3) the greedy (midpoint) and (2-step Euler) strategies have slightly better stability than DTO.
Table 5: Stability reported in ASP/MSP per property.
Property
Ac
ELUMO
μ
Ce
DTO Greedy (Ealer)
94.90/65.00
96,20/44
95.90/67.00
96.0066.000
94.60/61.00
95.0067.000 .00
74/6688888
96.40/7.000
97.40/79.00
98. 98.40/84.80
97.6088.00
85.55/21.22 220
Gready (midpaint)
97.46/80.00
97.51/83.00
97.91/81.00
97.77/83.00
97.70/81.00
97.09/80.00
Greedy (2ostep Euler)
97.67/82.00
96.95/74.00
98.18/84.00
96,29/72.00
97.4093.00
97.75/84.00
EquiFM
98.88/89.00
J.2 Further results on inverse image problems
To put the results from Section 7.1 into context we present some detailed comparisons to other works from the domain of inverse problems with diffussion models, namely:
1. DAPS (B. Zhang et al. 2025),
2. DPS (Chung, J. Kim, et al. 2023),
3. DDRM (Kawar et al. 2022).
4. DDNM (Y. Wang, Yu, and J. Zhang 2023),
5. DCDP (Li al. 2024),
6. FPS-SMC (Dou and Y. Song 2024),
7. DifIPIR (Zhu et al. 2023), and
8. RED-diff (Mardani et al. 2024).
We present the full comparison in Table 6.
47

---

Table 6: Additional results for inverse image problems on FFHQ 256 x 256.
Task
Method
PSNR (f)
SSIM (f)
LPIPS (L)
FID (U)
Greedy (Buler)
27.94
0.728
0.217
66.64
Greedy (midpoint)
27.98
0.727
0,224
70.96
Greedy (2-step Fuler)
27.95
0.728
0.230
68.93
DAPS
29.07
0.818
0.177
$1.44
DPS
36.86
0.753
0.269
Super resolation 4
DDRM
26.58
0.782
0,282
79.25
DDNM
DCDF
28.
0.807
0.178
53.81
FPS. SMC
28.42
0.813
0,204
49.25
DiMPIR
26.64
0.260
65.77
Greedy (Euler)
Greedy (midpoiat)
24.08
0.724
0.186
44.55
Greedy (2-step Euler)
23.88
0.720
0,188
44.09
DAPS
24.07
0.814
0.133
43.10
Inpaint (box)
DPS
22.51
0.792
0.209
61.27
DDRM
22.26
0.207
78.62
DDNM
24.47
0.837
0,235
46.59
PPS-SMC
24.86
0.823
0.146
48.34
Gready (Baler)
30.87
0.823
0.141
40.73
Greedy (midpoint)
31.03
0.816
0.139
38.80
Greedy (2 step Euler)
30.80
Inpaint (rundom)
DAPS
31.12
0.844
0.098
32.17
DPS
25.46
0.823
0,203
69.20
DDNM
29.91
0.817
0.121
44.37
DCDP
30.69
0.842
0.142
52.51
FPS-SMC
28.21
0.823
0.261
Greedy (Ealeer)
28.01
0.766
0.182
$7.04
Gready (midpoint)
28.36
0.776
0.185
58.55
Greedy (2-stop Euler)
28. 18
0.774
0.181
57.18
DAPS
29.19
0.817
0.165
53.33
Gaussian deblurring
DPS
25.87
0.764
0,219
79.75
DDRM
DDNM
28.
0.804
0.216
57.83
DCDP
27.50
0.699
0.304
86.43
FPS FPS-SMC
26.54
0.773
0.253
67.45
DiAPIR
27.36
0.236
59.65
Greedy (Baler)
29.35
0.748
0.207
63.05
Greedy (midpoint)
29.73
0.762
0,207
66.21
Greedy (2-step Faler)
29.64
0.764
0.203
63.99
Motion deblurring
DAPS
29.66
0.847
0.157
39.49
DPS
24.52
0.801
0,246
65,23
DCDP
25.08
0.512
0,364
125.13
27.39
DitPRR
26.57
0.255
65.78
Gredyy (Baler)
15,10
0.282
0.598
298.06
Greedy (midpoint)
15.10
0.286
0,595
299.45
Greedy (2 step Euler)
15.07
Phase setrieval
DAPS
30.63++3.
0.851=0.00
6.139
42.71
DPS
17.64422.0
0.441=0.120
0.410=0.000
104.52
RED-diff
15.6044..46
0.398
0.506
167.43
DCDP
28.65 8.09
0.781
0. 0.203 0.203 10.16
68. 13
Greedy (Baler)
24.767
0.551
0.327
79.06
Greedy (midpoint)
25.09
0.558
0,332
76.73
Greedy (2-step Enler)
24.81
0.547
0.330
76.26
Nonlinear deblur
DAPS
28. 18.29441.77
0.783
0.
155-+00022
49.38
DPS
23.39442.0z
0.623 .022
91.31
RED-diff
30. 30.866 166.1..55
0.795 .o2
0.160
43.84
27.92 64
0.779 067
0.183_0..01
Greedy (Filler)
24.16
0.767
0.181
43.59
Gready (midpoint)
26.62
0.809
0.160
37,86
Greedy (2-stop Euler)
25.70
0.797
0.165
37.97
High dynamic range
07.1243..5 53
0.752 o41
0.162_0..72
42.97
DPS
22. 92.7344..00
0.591
0.261=0.11
112.82
RED-diff
22.16+3.41
0.512-0.0s8
0.258-00.089
108.32
48

---

J.3 Sampling trajectories for inverse problems
We present the solution trajectories the different guidance algorithms below for solving the HDR inverse problem. Note that the midpoint and 2-step Euler, unsurprisingly, have better approximations of æ1.
Figure 6: Sampling trajectory for greedy (Euler) solving the HDR inverse problem. Top row is L (zt) and the bottom row is Vt.
Figure 7: Sampling trajectory for greedy (midpoint) solving the HDR inverse problem. Top row is midpoint estimate and the bottom row is Zt-
Figure 8: Sampling trajectory for greedy (2-step Euler) solving the HDR inverse problem. Top row 2-step Euler estimate and the bottom row is xt.
J.4 More qualitative samples for inverse problems
We showcase some examples generated by the greedy gradient strategy (Euler) the different inverse problems.
49

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Figure 9: Qualitative visualization of using greedy guidance to solve the super resolution 4x inverse problem. Top row is the ground truth, middle row the measurement, and the bottom row is the reconstruction.
Figure 10: Qualitative visualization of using greedy guidance to solve the super resolution 4x inverse problem. Top row is the ground truth, middle row the measurement, and the bottom row is the reconstruction.
Figure 11: Qualitative visualization of using greedy guidance solve the Gaussian deblurring inverse problem. Top row is the ground truth, middle row is the measurement, and the bottom row is the reconstruction.
50

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Figure 12: Qualitative visualization of using greedy guidance to solve the motion deblurring inverse problem. Top row is the ground truth, middle row is the measurement, and the bottom row is the reconstruction.
0
S
Figure 13: Qualitative visualization of using greedy guidance to solve the Phase retrieval inverse problem. Top row is the ground truth, middle row the measurement, and the bottom row is the reconstruction.
Figure 14: Qualitative visualization of using greedy guidance solve the nonlinear deblurring inverse problem. Top row is the ground truth, middle row is the measurement, and the bottom row is the reconstruction.
51

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Figure 15: Qualitative visualization of using greedy guidance to solve the HDR inverse problem. Top row is the ground truth, middle row is the measurement, and the bottom row is the reconstruction.
52

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/
0.014
0.166
1.388
2.322
2.804
3.878
4.922
7.951
Qualitative visualization of controlled generated molecules for various dipole moments (n). Top row is generated using end-to-end guidance with a DTO scheme and the bottom row is generated using posterior guidance.
Figure 16:
J.5 More qualitative samples for controlled molecule generation
In Appendix J.4 we present some qualitative results for cooperty-guides molecule generation. In particular, we target different dipole moments.
K
Discussions
K.1 Broader Impacts
Controllable generation be used for many tasks both benign and malicious. The insights from this paper could be used to develop more effective adversarial attacks, generation of harmful content, other malicious applications.
K.2
Limitations
As this work is mostly theoretical, our experimental illustrations are limited, serving more to illustrate the key concepts rather than advancing the state-of-the-ar within the particular problem. We believe that future work can use these insights to make informed design choices when developing solutions to guided generation problems.
In our controllable molecule generation experiments, we take naïve strategy for annealing the learning rate leaving performance on the table. Moreover, we don't consider mixed accuracy schemes, i.e., using Euler for certain steps closer to the target and midpoint for steps further away (ef. Moufad et al. 2025).
53


Reviews:
[{"review_id": "LbKDBUmsGF", "reviewer": "Reviewer_XpGA", "summary": "This paper presents a novel and elegant theoretical framework that unifies two major families of training-free guided generation: posterior guidance (e.g., DPS) and end-to-end guidance (e.g., D-Flow). The central contribution is the formal demonstration that posterior guidance can be viewed as a \"greedy,\" first-order Euler approximation of the more computationally expensive end-to-end guidance.\n\nThe authors provide a rigorous mathematical analysis to support this claim, showing the equivalence of the posterior guidance gradient to a single explicit Euler step in a DTO scheme and to the first iteration of a fixed-point method for an implicit Euler discretization in an OTD scheme. Furthermore, the paper provides a formal error analysis bounding the difference between the \"greedy\" gradient and the \"true\" gradient from end-to-end backpropagation.\n\nBuilding on this unified view, the authors propose a framework, suggesting that higher-order numerical solvers can be used to interpolate between the two guidance families, offering a potential trade-off between computational cost and gradient accuracy.", "questions": "Stabilty of results: In Table 2, the euler results, seem very unstable, can the authors confirm that this is true and not originated from some bug?", "limitations": "i dont believe there are potential negative societal impact", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "**Strengths**\n\nTo the best of my knowldege, the unification of posterior and end-to-end guidance is novel. The \"greedy\" perspective is intuitive, and likely to have a good view of the field. The theoretical claims seem valid and are supported by the proofs in the appendix (I didnt fully validated the proofs). Lastly, the proposed framework is a native extenstion of the theory. \n\n**Weaknesses**\n\n* Explore more solvers: I wonder how the results will look like with higher-order solvers. Does the trend being maintined? \n* Limited scope of experiments: I think that the experiments can benefit from a deeper analysis of task complexity vs. method being used. \n* Limitations: I couldnt find any refernfce to limitation of the work. \n\nI believe the theoretical contribution of this paper is strong and the community will benefit from this work.", "was_revised": true, "final_justification": "I had some concerns, they were addressed during the rebuttal.", "rebuttal": "We thank reviewer XpGA for their interest in our work and thoughtful feedback. We are encouraged that they find the unification framework to be novel and supported by strong theoretical contributions. We are especially excited that reviewer XpGA believes that *\"the community will benefit from this work.\"* We now address the questions and suggestions raised by reviewer XpGA below.\n\n> Explore more solvers: I wonder how the results will look like with higher-order solvers. Does the trend being maintained?\n\nWe ran some more experiments for controlled molecule generation which we summarize in the table below.\n\n| Property                | $\\alpha$          | $\\Delta \\varepsilon$ | $\\varepsilon_{\\mathrm{HOMO}}$ | $\\varepsilon_{\\mathrm{LUMO}}$ | $\\mu$ | $C_v$                                                |\n|:-------------------------|:-------------------:|:----------------------:|:-------:|:-------:|:-------:|:------------------------------------------------------:|\n| Greedy (Euler)          | 11.282            | 1265                 | 725                           | 1092                          | 1.559 | 6.469                                                |\n| Greedy (midpoint)       | 5.313             | 1196                 | 599                           | 1057                          | 1.417 | 2.967                                                |\n| Greedy (2-step Euler) | 5.377             | 1275                 | 560                           | 1204                          | 1.563 | 2.975                                                |\n| Greedy (3-step Euler)   | 5.098             | 1152                 | 600                           | 1152                          | 1.384 | 3.229                                                |\n| Greedy (5-step Euler)   | 4.177             | 1083                 | 571                           | 939                           | 1.328 | 2.332                                                |\n| DTO                     | 1.404             | 401                  | 176                           | 373                           | 0.372 | 0.866     |       \n| EquiFM                  | 9.525             | 1494                 | 622                           | 1523                          | 1.628 | 6.689                                                |\n| Lower bound             | 0.10              | 64                   | 39                            | 46                            | 0.043 | 0.040                                                |\n\nWe note that generally, taking more discretization steps resulted in a more *useful* gradient. This corroborates with recent work which explored taking two Euler steps [1] (*N.B.*, they approach the problem from a different perspective and use different terminology, but the equivalence is valid).\nIn addition to our discussion on using Runge-Kutta 4 in **Appendix H.2** we also tried Ralston's third order method. However, we found similar issues.\nIntuitively, this make sense as by **Theorem 6.1** we found that the error of the gradient lies within the $\\mathcal O(h^{\\alpha + 1})$ neighborhood. Thus for large step sizes (like in the beginning of posterior guidance) such methods are *likely* to be unsuitable.\nFor future work one could explore a more clever scheme using a mix of different order solvers. We discuss this more in **Appendix J.2.**\n\n> Limited scope of experiments: I think that the experiments can benefit from a deeper analysis of task complexity vs. method being used.\n\nWe agree that we could have discussed this part better. The main analysis for these two aspects lies within **Theorem 5.1**, **Proposition 5.2**, and **Theorem 6.1**. \n\n**1.** In **Theorem 5.1** and **Proposition 5.2** we show that the greedy guidance strategy results in a covariance projection of the loss function; however, this projection is weaker and less significant as $t \\to 1$. Thus for later time steps the greedy gradient is more likely to push samples off the data manifold.\nWe posit this is why there was little stability concerns for the inverse image problems; whereas the the controlled molecule generation problem is less resilient to being pushed away from the data manifold. This makes intuitive sense as the very small changes could result in invalid molecules.\nThis corroborates with the analyses of prior work [2].\n\n**2.** Then in **Theorem 6.1** we show that the accuracy of the gradients is bound by the local truncation error of the underlying numerical scheme; however, for large step sizes higher-order solvers may be *less* suitable.\nThis however, depends on the the straightness of the underlying ODE, if the true underlying ODE is perfectly straight (highly idealized, but presented for illustrative purposes) then all the schemes would have no error.\n\nThis analysis is further supplemented by an additional corollary of **Theorem 6.1** and **Proposition 5.2** which we state below.\n\n**Corollary.** *Consider the standard affine Gaussian probability paths model trained to zero loss. Let $\\boldsymbol \\Phi$ be an explicit Runge-Kutta solver of order $\\alpha > 0$ of a flow model with flow $\\Phi_{s,t}^\\theta(\\boldsymbol x)$. The Gateaux differential of $\\boldsymbol x$ at some time $t \\in [0, 1]$ in the direction of gradient $\\nabla_{\\boldsymbol x} \\mathcal L \\left(\\boldsymbol \\Phi_{t,1}(\\boldsymbol x)\\right)$ is given by*\n\n$\\begin{equation}\n\\delta_{\\boldsymbol x}^{\\boldsymbol \\Phi} \\Phi_{t,1}^\\theta(\\boldsymbol x) = -\\nabla_{\\boldsymbol x} \\Phi_{t,1}^\\theta(\\boldsymbol x) \\nabla_{\\boldsymbol x} \\boldsymbol \\Phi_{t,1} (\\boldsymbol x)^\\top \\nabla_{\\boldsymbol x_1}\\mathcal L(\\boldsymbol x_1)\n\\end{equation}$\n\nThis corollary provides similar analysis tools for studying the higher-order solvers.\n\n> Limitations: I couldn't find any reference to limitation.\n\nWe discuss the limitations in **Appendix J.2** (page 42 of the submission).\n\n> Stability of results: In Table 2, the euler results, seem very unstable, can the authors confirm that this is true and not originated from some bug?\n\nWe discuss the instability more in **Appendix H.2**, but to summarize it relates to **Proposition 5.2** and the greedy gradient being pushed off the data manifold for $t \\gg 0$. Additionally, our code is provided in the supplementary material.\n\n[1] Moufad, B., Janati, Y., Bedin, L., Durmus, A. O., Douc, R., Moulines, E., and Olsson, J. (2025).\n“Variational Diffusion Posterior Sampling with Midpoint Guidance”. In: The Thirteenth Interna-\ntional Conference on Learning Representations. URL: https://openreview.net/forum?id=\n6EUtjXAvmj\n\n[2] Ben-Hamu, H., Puny, O., Gat, I., Karrer, B., Singer, U., and Lipman, Y. (2024). “D-Flow: Differ-\nentiating through Flows for Controlled Generation”. In: Forty-first International Conference on\nMachine Learning. URL: https://openreview.net/forum?id=SE20BFqj6J\n\n## Closing remarks\nWe thank reviewer XpGA for their questions, which gave us the opportunity to clarify  our contributions and improve our work.\nIt is our hope that our answers fully address all the important questions raised by the reviewer, and we are happy to answer any additional questions and consider further suggestions.\n\nWe kindly ask the reviewer to consider increasing their score if our responses address their concerns satisfactorily."}, {"review_id": "cwaAS2AKVO", "reviewer": "Reviewer_RAY9", "summary": "The authors explore the theoretical connections between these two families and provide an in-depth theoretical understanding of these two techniques relative to the continuous ideal gradients. Motivated by this analysis, they then show a method for interpolating between these two families enabling a trade-off between compute and accuracy of the guidance gradients. they validate this work on some inverse image problems and property-guided molecular generation.", "questions": "1. How the theoretical analysis close the gap between practice and theory? The experimental results leave room for improvement.\n2. Could more state-of-the-art methods be unified under the proposed framework to better demonstrate its practical relevance? Additionally, how does the framework concretely indicate potential directions for future advancements?", "limitations": "Yes, the authors have discussed the limitations and future work in the Appendix", "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\n1. The theoretical part of the paper is structured clearly and presented in a concise and readable manner.\n2. The paper presents substantial theoretical analysis on posterior guidance and end-to-end optimization, which could potentially inspire future advances in method design.\n\n### Weakness\n\n1. Including more illustrations and pseudocode would help readers better understand the overall workflow of the paper and how the proposed theoretical insights lead to concrete improvements.\n2. While training-free guidance methods are known to face substantial limitations—including limited controllability, off-manifold generation, longer sampling requirements, and sensitivity to hyper-parameters—the paper does not provide further analysis or exploration toward addressing these challenges, which could have strengthened the contribution.\n3. While the paper outlines future directions involving more hyperparameters and case-specific design choices, it is worth noting that training-free methods are already known to be highly sensitive to hyperparameters, which raises concerns regarding the feasibility and robustness of such approaches.", "was_revised": true, "final_justification": "I appreciate the theoretical efforts of this paper and the detailed explanation from the authors. I have increased my score accordingly.", "rebuttal": "We thank reviewer RAY9 for their time, feedback, and helpful suggestions in more clearly communicating our ideas.\nWe are happy that the reviewer found the theoretical part of the paper to be \"structured clearly and presented in a concise and readable manner\", in addition to appreciating our \"substantial theoretical analysis\".\nBelow, we address the reviewer's questions and concerns.\n\n> Including more illustrations and pseudocode would help readers better understand the overall workflow of the paper and how the proposed theoretical insights lead to concrete improvements.\n\nYes, we agree that including more illustrations and pseudocode would strengthen the paper.\nWhilst, we cannot attach new figures in the rebuttal we can illustrate some pseudocode\n```python\n# posterior step\ndef posterior_step(t, x)\n    tlist = [0, 1]\n    dt = tlist[1] - t\n\n    if method == 'euler':\n        # Euler step\n        x = x + model(t, x) * dt\n    elif method == 'midpoint':\n        # Midpoint step\n        x = x + dt * model(t + 0.5 * dt, x + 0.5 * dt * model(t, x))\n    elif method == 'ralston3':\n        # Ralston's third order\n        k1 = model(t, x)\n        k2 = model(t + 0.5 * dt, x + 0.5 * dt * k1)\n        k3 = model(t + 0.75 * dt, x + 0.75 * dt * k2)\n\n        x = x + dt * (2/9 * k1 + 1/3 * k2 + 4/9 * k3)\n    elif method == 'heun3':\n        # Heun's third order\n        k1 = model(t, x)\n        k2 = model(t + 1/3 * dt, x + 1/3 * dt * k1)\n        k3 = model(t + 2/3 * dt, x + 2/3 * dt * k2)\n\n        x = x + dt * (1/4 * k1 + 3/4 * k3)\n\n    elif method == 'rk4':\n        # RK-4 step\n        k1 = model(t, x)\n        k2 = model(t + 0.5*dt, x + 0.5*dt*k1)\n        k3 = model(t + 0.5*dt, x + 0.5*dt*k2)\n        k4 = model(t + dt, x + dt*k3)\n\n        x = x + dt * (1/6 * k1 + 1/3 * k2 + 1/3 * k3 + 1/6 * k4)\n\n    return x\n\n# posterior guidance pseudocode\n# assumed dt and loss are defined\n\nxt = x0\n\nfor t in timesteps:\n    # quick fix\n    xt_opt = xt.detach().clone().requires_grad_(True)\n\n    if optim == 'sgd':\n        optimizer = torch.optim.SGD([xt_opt], lr=lr)\n    else:\n        optimizer = torch.optim.LBFGS([xt_opt], max_iter=max_iter, lr=lr, line_search_fn='strong_wolfe')\n\n    for step in range(opt_steps):\n        optimizer.zero_grad()\n\n        x1hat = posterior_step(t, xt_opt)\n\n        loss(x1hat).mean().backward()\n        optimizer.step()\n\n    with torch.no_grad():\n        xt = xt_opt + model(t, xt_opt) * dt\n\nreturn xt\n```\n\nAdditionally, in **Appendix H.3** we outline the numerical schemes used for guidance.\nWe would also like to note that our code is provided in the supplementary material.\n\n> While training-free guidance methods are known to face substantial limitations—including limited controllability, off-manifold generation, longer sampling requirements, and sensitivity to hyper-parameters—the paper does not provide further analysis or exploration toward addressing these challenges, which could have strengthened the contribution.\n\nWe agree that these are all pertinent topics of great interest and appreciate the suggestions.\nHowever, we would like to respectfully argue that we did address some of these issues---albeit we need to make this more clear in the writing.\n\n**1. Limited controllability.**\nIn **Theorem 5.4 and Corollary D.1.1** we show put a bound on *controllability* in the sense of how well the gradient can steer the output of the flow model towards a desired solution.\n\n**2. Off-manifold generation.**\nWe address this concern in **Proposition 5.2**, wherein we show that the guidance strategy applied to earlier time steps $t \\ll 1$ applies a covariance projection to the guidance signal encouraging samples to stay on the data manifold.\n\n**3. Longer sampling requirements.**\nWhile not fully explored in this work, we believe our analysis on exchanging the compute for calculating the guidance gradient and the accuracy of the gradient does shed some **new light** on this discussion.\n\n> While the paper outlines future directions involving more hyperparameters and case-specific design choices, it is worth noting that training-free methods are already known to be highly sensitive to hyperparameters, which raises concerns regarding the feasibility and robustness of such approaches.\n\nYes, our work highlights another potential design axis introducing more hyperparameters.\nWe believe the benefit of this work is **unifying** two widely used families for designing guided generation algorithms into a single family.\nThus we inherit the robustness and feasibility of the two families.\nWe believe our \"substantial theoretical analysis\" sheds some light on this.\n\n> How the theoretical analysis close the gap between practice and theory? The experimental results leave room for improvement.\n\nThe experimental results are meant to illustrate the **connection between posterior and end-to-end guidance** rather than outperforming the SOTA.\nThus the theoretical results is meant to explain the performance gap between the two techniques and then show how the performance can be bridged by taking more steps in approximating the gradient.\nWe would like to emphasis that this perspective is **novel** and could help frame future research into the topics.\n\n> Could more state-of-the-art methods be unified under the proposed framework to better demonstrate its practical relevance? Additionally, how does the framework concretely indicate potential directions for future advancements?\n\nSeveral state-of-the-art models (D-Flow [1], AdjointDEIS [2], OC-Flow [3], and MGPS [4]) are all subsumed by this analysis, see **Appendix A.2** for D-Flow and AdjointDEIS, **Appendix F** for OC-Flow, and **Appendix A.1** for MGPS.\n\nAs the difference between the inverse image problems and controlled molecule generation experiments illustrated, certain problem are more sensitive to samples being moved off the data manifold *cf.* **Proposition 5.2** which would indicate a DTO-type of strategy would be more promising.\nMoreover, one could explore more sophisticated posterior guidance schemes which use higher or lower order solvers dependent on the step size $h$.\n\n[1] Ben-Hamu, H., Puny, O., Gat, I., Karrer, B., Singer, U., and Lipman, Y. (2024). “D-Flow: Differentiating through Flows for Controlled Generation.” Proceedings of the 41st International Conference on Machine Learning. URL: https://openreview.net/forum?id=SE20BFqj6J\n\n[2] Blasingame and Liu, C. (2024). “AdjointDEIS: Efficient Gradients for Diffusion Models”. In:\nAdvances in Neural Information Processing Systems. Ed. by A. Globerson, L. Mackey, D. Bel-\ngrave, A. Fan, U. Paquet, J. Tomczak, and C. Zhang. Vol. 37. Curran Associates, Inc., pp. 2449–\n2483.\n\n[3] Wang, L., Cheng, C., Liao, Y., Qu, Y., and Liu, G. (2025). “Training Free Guided Flow-Matching\nwith Optimal Control”. In: The Thirteenth International Conference on Learning Representations.\nURL: https://openreview.net/forum?id=61ss5RA1MM\n\n[4] Moufad, B., Janati, Y., Bedin, L., Durmus, A. O., Douc, R., Moulines, E., and Olsson, J. (2025). “Variational Diffusion Posterior Sampling with Midpoint Guidance”. In: The Thirteenth International Conference on Learning Representations. URL: https://openreview.net/forum?id=6EUtjXAvmj\n\n\n## Closing remarks\nWe thank reviewer RAY9 for their suggestions on how to improve this paper.\nWe hope that our responses fully address all the questions and concerns raised by the reviewer and we are happy to consider any additional questions.\n\nWe kindly ask the reviewer to consider increasing their score if our responses address their concerns satisfactorily."}, {"review_id": "HZhDXkvV1q", "reviewer": "Reviewer_oWM7", "summary": "The paper provides a unifying theoretical framework for two seemingly distinct families of training-free guided generation techniques used in flow and diffusion models: posterior guidance and end-to-end guidance. Both techniques control the generative process, often by injecting gradients from a target prediction model to steer the generation towards desired properties. The authors analyze both approaches and show that posterior guidance can be framed as a greedy gradient step within the end-to-end scheme.", "questions": "- Some reference to prominent approaches are missing. Can the authors explain the connections between the analysed methods and e.g., \\[1\\] and \\[2\\]?\n\n\\[1\\] [\\[2311.16424\\] Manifold Preserving Guided Diffusion](https://arxiv.org/abs/2311.16424)\n\n\\[2\\] [Pseudoinverse-Guided Diffusion Models for Inverse Problems](https://openreview.net/forum?id=9_gsMA8MRKQ)", "limitations": "yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 3, "originality": 3, "strengths_and_weaknesses": "**Strengths**\n\n- Theoretical Unification: The paper's primary strength lies in the theoretical framework that unifies posterior guidance and end-to-end guidance. This provides a deeper understanding of their relationship and is a significant conceptual contribution to the field of guided generation.\n\n**Weaknesses**\n\n- Experiments:  \n  - While the authors show results of experiments for 3 variants in the interpolation space (Euler, midpoint, 2-Euler), I find the experimental setup lacking. Given the theoretical connection that were drawn between the posterior guidance approach and the end-to-end approach, a more elaborate ablation on the various interpolation intermediate steps between the two would greatly improve the understanding of the practical implications of the theoretical findings. Do the experimental results behave as one would expect? Is there anything surprising, as many theoretical results do not transfer to practice in high dimensional regims, and with neural networks.   \n  - One of the claims is that this interpolation allows to trade quality and efficiency, it is not clearly demonstrated in the paper.    \n- It would be valuable to better position this paper in existing literature, see e.g., [\\[2410.00083\\] A Survey on Diffusion Models for Inverse Problems](https://arxiv.org/abs/2410.00083) \n- Clarity: it is hard to parse what the different methods are, adding algorithm boxes with pseudo code would improve ease of understanding.", "was_revised": true, "final_justification": "The authors have been responsive and addressed my concerns. \nWith the suggested modifications and additions to the manuscript (add pseudo code in main paper, additional experiments exploring further the interpolation space between the methods, etc.) I think the paper present a valuable analysis.", "rebuttal": "We thank reviewer oWM7 for their detailed feedback and constructive suggestions.\nWe are happy that appreciated that the \"paper's primary strength lies in the theoretical framework that unifies posterior guidance and end-to-end guidance.\"\nBelow we provide the suggested ablations and experiments along with answering the reviewer's questions.\n\n>  While the authors show results of experiments for 3 variants in the interpolation space (Euler, midpoint, 2-Euler), I find the experimental setup lacking. Given the theoretical connection that were drawn between the posterior guidance approach and the end-to-end approach, a more elaborate ablation on the various interpolation intermediate steps between the two would greatly improve the understanding of the practical implications of the theoretical findings. Do the experimental results behave as one would expect? Is there anything surprising, as many theoretical results do not transfer to practice in high dimensional regims, and with neural networks.\n\nWe agree with the reviewer that such ablations would be a valuable addition to the paper.\nWe ran some additional experiments on the controlled molecule generation task with high-order solvers and found that increasing the number of steps did improve the performance of the model at the cost of more compute.\nSee our discussion with reviewer XpGA for a more detailed analysis of these results.\n\n\n**Table 1.** We compare the effect of taking more discretization steps in the greedy guidance framework.\n\n| Property                | $\\alpha$          | $\\Delta \\varepsilon$ | $\\varepsilon_{\\mathrm{HOMO}}$ | $\\varepsilon_{\\mathrm{LUMO}}$ | $\\mu$ | $C_v$                                                |\n|:-------------------------|:-------------------:|:----------------------:|:-------:|:-------:|:-------:|:------------------------------------------------------:|\n| Greedy (Euler)          | 11.282            | 1265                 | 725                           | 1092                          | 1.559 | 6.469                                                |\n| Greedy (midpoint)       | 5.313             | 1196                 | 599                           | 1057                          | 1.417 | 2.967                                                |\n| Greedy (2-step Euler) | 5.377             | 1275                 | 560                           | 1204                          | 1.563 | 2.975                                                |\n| Greedy (3-step Euler)   | 5.098             | 1152                 | 600                           | 1152                          | 1.384 | 3.229                                                |\n| Greedy (5-step Euler)   | 4.177             | 1083                 | 571                           | 939                           | 1.328 | 2.332                                                |\n| DTO                     | 1.404             | 401                  | 176                           | 373                           | 0.372 | 0.866     |\n\nWe also ran an additional set of experiments wherein we explored a differing number of discretization steps within a DTO guidance scheme, ranging from 1 (*i.e.* posterior guidance only applied to $\\boldsymbol x_0$) to the full 50 used in the D-Flow model.\nThese results further showed that we can exchange compute cost with performance (and accuracy of gradients).\n\n**Table 2.** We compare the effect of a different number of discretization steps for optimizing $\\boldsymbol x_0$.\n\n| Property                | $\\alpha$          | $\\Delta \\varepsilon$ | $\\varepsilon_{\\mathrm{HOMO}}$ | $\\varepsilon_{\\mathrm{LUMO}}$ | $\\mu$ | $C_v$                                                |\n|:-------------------------|:-------------------:|:----------------------:|:-------:|:-------:|:-------:|:------------------------------------------------------:|\n| DTO (1-step)            | 13.049            | $989 \\times 10^{12}$ | 681                           | 86.512                        | 1.666 | 15.144                                               |\n| DTO (2-step)            | 6.113             | 1359                 | 666                           | 1199                          | 1.533 | 3.757                                                |\n| DTO (4-step)            | 6.115             | 1294                 | 668                           | 1190                          | 1.406 | 2.829                                                |\n| DTO (8-step)            | 4.549             | 1070                 | 608                           | 1078                          | 1.247 | 2.594                                                |\n| DTO (16-step)           | 3.454             | 817                  | 608                           | 939                           | 1.177 | 2.003                                                |\n| DTO (32-step)           | 2.912             | 750                  | 410                           | 666                           | 0.721 | 1.566                                                |\n| DTO (50-step)                     | 1.404             | 401                  | 176                           | 373                           | 0.372 | 0.866                                                |\n\n> One of the claims is that this interpolation allows to trade quality and efficiency, it is not clearly demonstrated in the paper.\n\nWe justify this claim in **Theorems 5.3 and 6.1**.\nWe hope that our additional experiments discussed above help illustrate this further experimentally.\n\n> It would be valuable to better position this paper in existing literature, see e.g., [2410.00083] A Survey on Diffusion Models for Inverse Problems\n\nWe thank the reviewer for suggesting the survey paper, it is quite interesting.\nWe wish to note, however, that beyond already referencing several of the papers mentioned in the survey, that the focus of our paper lies beyond just diffusion models or inverse problems as we are interested more generally in flow models.\nWe also wish to remind the reviewer that we did positions our paper in light of several papers on posterior guidance and end-to-end guidance, see **Appendix A** and **Figure 5** in the paper.\nWe would be happy to add this survey to our literature review if accepted.\n\n> Clarity: it is hard to parse what the different methods are, adding algorithm boxes with pseudo code would improve ease of understanding.\n\nWe thank the reviewer for this suggestion and will incorporate this feedback if the paper is accepted.\nWe wish to note that we already detailed the numerical schemes in **Appendix H.3.**\n\n> Some reference to prominent approaches are missing. Can the authors explain the connections between the analysed methods and e.g., [1] and [2]?\n\nWe thank the reviewer for highlighting these works.\n\n**1. MPGD.**\nFrom [1, Theorem 1] we see that *Manifold Preserving Guided Diffusion* (MPGD) update rule can be written in our notation as\n\n$\\begin{equation}\n\\boldsymbol x_{1|t}' = \\boldsymbol x_{1|t}^\\theta(\\boldsymbol x_t) - c_t \\nabla_{\\boldsymbol x_1}\\mathcal L\\left(\\boldsymbol x_{1|t}^\\theta(\\boldsymbol x)\\right).\n\\end{equation}$\n\nThus we see that MPGD applies the gradient to the output of the denoiser.\nWe can then apply this to the data prediction formulation (*cf.* **Proposition 4.1**, *n.b.*, MPGD is originally discussed in the context of the discretized diffusion SDE but we focus on flow models in our work).\n\nWe actually discuss nearly the same algorithm in **Appendix F**, but from another perspective (and of course in the context of ODEs).\nThis can be viewed as an instance of greedy *control signal optimization* which we discuss in **Appendix F.**\nIn our discussion in **Appendix F** we discuss how applying this technique to the denoiser results in rescaling $\\nabla_{\\boldsymbol x_1}\\mathcal L(\\boldsymbol x_1)$ by $b_t$.\nWe will update our discussion to include a reference to MPGD.\nWe thank the reviewer to pointing out this connection.\n\n\n**2. Pseudoinverse guidance.**\nFrom [2, Eqs. (6) and (7)] we see that [2] focuses specifically on solving inverse problems and uses a known measurement matrix $\\boldsymbol H$ to project the Jacobian $\\nabla_{\\boldsymbol x}\\boldsymbol x_{1|t}^\\theta(\\boldsymbol x)$ to time $t$. That is, this a specialized formulation which amounts to projecting the loss by the Jacobian of the denoiser, this is quite similar to the greedy gradient discussed in our paper.\n\nWe would also like to remind the reviewer that we discuss [2] in **Appendix A / Figure 5.**\n\n**Summary.**\nWe wish to note that both works are focused on learning the score function $\\nabla_{\\boldsymbol x_t} \\log p(\\boldsymbol y|\\boldsymbol x_t)$ whereas our work takes an **entirely different** perspective and considers flow models in addition to the ODE formulation of diffusion models.\n\n## Closing remarks\nWe thank the reviewer of the detailed feedback and questions.\nWe especially appreciate the additional related work the reviewer highlighted.\nWe hope our responses helped answer the questions and clarified the strengths of our contributions.\nWe would be happy to consider any additional questions or further suggestions.\n\nWe kindly ask the reviewer to consider increasing their score if our responses address their concerns satisfactorily."}, {"review_id": "kwO1IAbfd9", "reviewer": "Reviewer_Hyh7", "summary": "The paper shows that posterior guidance is a greedy approximation of end-to-end guidance, and presents a unifying framework for training-free guided generation in flow/diffusion models. The authors derive error bounds (Theorems 5.4, 6.1) linking greedy and exact gradients, and propose higher-order solver to trade off compute vs. accuracy. Experiments on inverse imaging and molecule generation control illustrate the performance of new insight.", "questions": "1.Failure Analysis: The greedy method fails badly for some molecule generation. Please analyze why (e.g. non-convexity, learning-rate issues).\n\n2. Baseline Comparison: Explicitly compare to similar recent works(“greedy (midpoint)” v.s. midpoint guidance (Moufad et al., 2025)) to better position your hybrids.\n\nReferences\nMoufad, B., Janati, Y., Bedin, L., Durmus, A. O., Douc, R., Moulines, E., and Olsson, J. (2025). “Variational Diffusion Posterior Sampling with Midpoint Guidance.” Proceedings of the 13th International Conference on Learning Representations. URL: https://openreview.net/forum?id=6EUtjXAvmj (cit. on pp. 17, 18, 32, 34, 42)", "limitations": "Yes.", "rating": 2, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "Strengths\n1. First to cast posterior guidance as a greedy end-to-end approximation under certain assumptions.\n2. Rigorous theoretical analysis with clear theorems on greedy vs. full gradients and then derive error bounds (Theorems 5.4, 6.1).\n\nWeaknesses\n1. Results rely on affine paths and smoothness assumptions. Real-world flow/diffusion models often include non-affine terms or non-Gaussian noise, so it’s unclear whether the proofs can extend to more complex settings.\n2. Experiments are only proof of concept and do not demonstrate clear practical superiority.Limited empirical evidence of real-world benefit and underperform SOTA.For example, in Table 1 Greedy (2-step Euler) falls behind DAPS(B. Zhang et al. 2024) by only a narrow margin, whereas the basic Greedy algorithm performs significantly worse than D-flow(Ben-Hamu et al.2024).\n3. Experimental illustrations are limited,offers error bounds but no empirical runtime vs. quality trade-off Figures to guide method selection.The experimental evaluation lacks sufficient task diversity, omitting important scenarios such as editing and conditional generation.\n\nReferences\nBen-Hamu, H., Puny, O., Gat, I., Karrer, B., Singer, U., and Lipman, Y. (2024). “D-Flow: Differentiating through Flows for Controlled Generation.” Proceedings of the 41st International Conference on Machine Learning. URL: https://openreview.net/forum?id=SE20BFqj6J (cit. on pp. 1–3, 6, 8, 9, 17–19, 21, 22, 24, 33, 34)\nZhang, B., Chu, W., Berner, J., Meng, C., Anandkumar, A., and Song, Y. (2024). “Improving Diffusion Inverse Problem Solving with Decoupled Noise Annealing.” arXiv preprint arXiv:2407.01521 [cs.LG]. URL: https://arxiv.org/abs/2407.01521 (cit. on pp. 7, 8, 17, 32–35)", "was_revised": true, "final_justification": "After the rebuttal, I now better appreciate that the assumptions made are standard, and I acknowledge the additional analyses and experiments the authors have provided. However, I remain firm in my evaluation that the paper’s empirical support is too weak to fully endorse its significance for real-world guided generation tasks. My critique was always about empirical competitiveness and stability-and on that front, the responses, while informative, show that the proposed approach is still playing catch-up to existing methods (in some cases by a wide margin). At present, given the continued concerns about empirical robustness and competitiveness, I regret that I must maintain my original rating.", "rebuttal": "We thank reviewer Hyh7 for their detailed feedback and constructive suggestions.\nWe are glad that they find our contribution of casting \"posterior guidance as a greedy approximation of end-to-end guidance\" to be novel and that they appreciate our \"rigorous theoretical analysis with clear theorems.\"\nWe address the concerns and questions raised by the reviewer below.\n\n> Results rely on affine paths and smoothness assumptions. Real-world flow/diffusion models often include non-affine terms or non-Gaussian noise, so it’s unclear whether the proofs can extend to more complex settings.\n\nWe wish to clarify that real world flow/diffusion models **do** in fact use affine terms and a Gaussian source distribution.\nDiffusion models (in the ODE form) all fall within this formulation (remember that diffusion models diffuse the data distribution to a Gaussian distribution) and *most* flow models choose to use a Gaussian source distribution. We refer the reviewer to [1, Section 4.8] for more details.\nMoreover, many models are trained used the model of affine probability paths.\nThis analysis is standard and has been used in prior works [2].\n\nThus these assumptions in **no way** hinder the application or utility of our theoretical results in real world scenarios.\n\n> Experiments are only proof of concept and do not demonstrate clear practical superiority.Limited empirical evidence of real-world benefit and under perform SOTA.For example, in Table 1 Greedy (2-step Euler) falls behind DAPS(B. Zhang et al. 2024) by only a narrow margin, whereas the basic Greedy algorithm performs significantly worse than D-flow(Ben-Hamu et al.2024).\n\nWe wish to emphasis that the **core contribution of this paper is not to outperform the SOTA but to connect end-to-end and posterior guidance together.**\nThus our performance is bounded by these two approaches and we explore the connection between them.\nThus our core significant contribution is the rigorous mathematical analysis in unifying these two families together.\n\n> Experimental illustrations are limited,offers error bounds but no empirical runtime vs. quality trade-off Figures to guide method selection.The experimental evaluation lacks sufficient task diversity, omitting important scenarios such as editing and conditional generation.\n\nWhile we did not explore experiments applications such as editing, we do explore conditional generation in **Section 7.2** and in **Appendices H.2 and I.1**.\nFurthermore, we would like to reiterate that our experimental scope is in line with other works in the space [2-5].\n\nTo explore performance vs complexity tradeoff we would like to direct the reviewer's attention to **Theorem 5.1, Proposition 5.2, and Theorem 6.1** which explore several facets of the problem.\n\n**1.** In **Theorem 5.1** and **Proposition 5.2** we show that the greedy guidance strategy results in a covariance projection of the loss function; however, this projection is weaker and less significant as $t \\to 1$. Thus for later time steps the greedy gradient is more likely to push samples off the data manifold.\nWe posit this is why there was little stability concerns for the inverse image problems; whereas the the controlled molecule generation problem is less resilient to being pushed away from the data manifold. This makes intuitive sense as the very small changes could result in invalid molecules.\nThis corroborates with the analyses of prior work [2].\n\n**2.** Then in **Theorem 6.1** we show that the accuracy of the gradients is bound by the local truncation error of the underlying numerical scheme; however, for large step sizes higher-order solvers may be *less* suitable.\nThis however, depends on the the straightness of the underlying ODE, if the true underlying ODE is perfectly straight (highly idealized, but presented for illustrative purposes) then all the schemes would have no error.\n\nAdditionally, we have run some additional experiments to help clear up this demonstration empirically.\n\n| Property                | $\\alpha$          | $\\Delta \\varepsilon$ | $\\varepsilon_{\\mathrm{HOMO}}$ | $\\varepsilon_{\\mathrm{LUMO}}$ | $\\mu$ | $C_v$                                                |\n|:-------------------------|:-------------------:|:----------------------:|:-------:|:-------:|:-------:|:------------------------------------------------------:|\n| Greedy (Euler)          | 11.282            | 1265                 | 725                           | 1092                          | 1.559 | 6.469                                                |\n| Greedy (midpoint)       | 5.313             | 1196                 | 599                           | 1057                          | 1.417 | 2.967                                                |\n| Greedy (2-step Euler) | 5.377             | 1275                 | 560                           | 1204                          | 1.563 | 2.975                                                |\n| Greedy (3-step Euler)   | 5.098             | 1152                 | 600                           | 1152                          | 1.384 | 3.229                                                |\n| Greedy (5-step Euler)   | 4.177             | 1083                 | 571                           | 939                           | 1.328 | 2.332                                                |\n| DTO                     | 1.404             | 401                  | 176                           | 373                           | 0.372 | 0.866     |\n| EquiFM                  | 9.525             | 1494                 | 622                           | 1523                          | 1.628 | 6.689                                                |\n| Lower bound             | 0.10              | 64                   | 39                            | 46                            | 0.043 | 0.040                                                |\n\nWe note that generally, taking more discretization steps resulted in a more *useful* gradient. This corroborates with recent work which explored taking two Euler steps [1] (*N.B.*, they approach the problem from a different perspective and use different terminology, but the equivalence is valid).\n\n\n> Failure Analysis: The greedy method fails badly for some molecule generation. Please analyze why (e.g. non-convexity, learning-rate issues).\n\nWe discuss the instability more in Appendix H.2, but to summarize it relates to **Proposition 5.2** and the greedy gradient being pushed off the data manifold for $t \\gg 0$.\nWe provide a more detailed response below:\n\nIn **Theorem 5.1** and **Proposition 5.2** we show that the greedy guidance strategy results in a covariance projection of the loss function; however, this projection is weaker and less significant as $t \\to 1$. Thus for later time steps the greedy gradient is more likely to push samples off the data manifold.\nWe posit this is why there was little stability concerns for the inverse image problems; whereas the the controlled molecule generation problem is less resilient to being pushed away from the data manifold. This makes intuitive sense as the very small changes could result in invalid molecules.\nThis corroborates with the analyses of prior work [2].\n\n> Baseline Comparison: Explicitly compare to similar recent works(“greedy (midpoint)” v.s. midpoint guidance (Moufad et al., 2025)) to better position your hybrids.\n\nThe recent work of Moufad et al., 2025 can be viewed as analagous to the \"Greedy (2-step Euler)\" experimental runs and fits nicely within our framework.\nWe wish to note that this work was approached from a different perspective than ours; however, it ties in nicely to our analysis in the \"Beyond Euler\" section.\nWe discuss this connection in greater detail in **Appendix A.1**.\n\n[1] Lipman, Y., Havasi, M., Holderrieth, P., Shaul, N., Le, M., Karrer, B., Chen, R. T., Lopez-Paz,\nD., Ben-Hamu, H., and Gat, I. (2024). “Flow Matching Guide and Code”. In: arXiv preprint\narXiv:2412.06264\n\n[2] Ben-Hamu, H., Puny, O., Gat, I., Karrer, B., Singer, U., and Lipman, Y. (2024). “D-Flow: Differentiating through Flows for Controlled Generation.” Proceedings of the 41st International Conference on Machine Learning. URL: https://openreview.net/forum?id=SE20BFqj6J\n\n[3] Zhang, B., Chu, W., Berner, J., Meng, C., Anandkumar, A., and Song, Y. (2024). Improving Diffusion\nInverse Problem Solving with Decoupled Noise Annealing. arXiv: 2407.01521 [cs.LG]. URL:\nhttps://arxiv.org/abs/2407.01521\n\n[4] Blasingame and Liu, C. (2024). “AdjointDEIS: Efficient Gradients for Diffusion Models”. In:\nAdvances in Neural Information Processing Systems. Ed. by A. Globerson, L. Mackey, D. Bel-\ngrave, A. Fan, U. Paquet, J. Tomczak, and C. Zhang. Vol. 37. Curran Associates, Inc., pp. 2449–\n2483.\n\n[5] Wang, L., Cheng, C., Liao, Y., Qu, Y., and Liu, G. (2025). “Training Free Guided Flow-Matching\nwith Optimal Control”. In: The Thirteenth International Conference on Learning Representations.\nURL: https://openreview.net/forum?id=61ss5RA1MM\n\n## Closing remarks\nWe thank reviewer Hyh7 for their suggestions on how to improve our work.\nWe hope that our detailed responses helped clarify the strength of our contributions and alleviate any concerns that the reviewer had concerning our experimental work.\nWe are more than happy to consider any additional questions or further suggestions.\n\nWe kindly ask the reviewer to consider increasing their score if our responses address their concerns satisfactorily."}]

Author rebuttal:

Meta-review: The paper presents a unifying framework for posterior guidance and end-to-end guidance, two techniques used in training-free guided generation in flow and diffusion models. The main contribution is that posterior guidance can be viewed as a greedy first-order Euler approximation of end-to-end guidance. This sheds light on potential trade-offs between computational cost and gradient accuracy.
The theory is validated on inverse image problems and property-guided molecular generation.
During the rebuttal, the authors addressed the reviewers' questions by providing more experiments and clarifications. 
While there was some discussion around the fact that the results do not outperform SOTA, as pointed out by reviewer Hyh7 in particular, the authors have clarified that their point was not SOTA but to explain the connection between posterior and end-to-end guidance, which the experiments address.
Decision: Accept (poster)