TFG-Flow: Training-free Guidance in Multimodal Generative Flow

Thank you for your detailed response. I address the points below.

• In practice, people typically find that taking smaller steps leads to higher quality samples [a,b]

I disagree that referencing flow matching here is relevant. In the flow matching model taking smaller steps is justified, because the flow-matching model is a continuous normalizing flow (CNF), i.e. sampling is given by solving an ODE. Therefore it is well understood how taking smaller steps in solving an ODE leads to a smaller error. There is no $p_{1|t}$ in flow matching.

• $p_{1|t}$ is a sub-moudule implemented as a neural network. It takes both the continous and discrete parts of as input, and outputs a sample $X_t$ for the continous part

I am afraid this does not answer my question. I asked how the distribution $p_{1|t}$ is defined. "Submodule", nor "neural network" does not define a distribution. The discrete part is clear, you output a probability distribution over an entire discrete space. How about the continuous part? The notations and derivations assume that $p_{1|t}$ is a distribution. How is this distribution defined? Saying that "it is a submodule implemented as a neural network that outputs a sample" does not define a distribution. Where does the randomness come from if it is a distribution? For example, if the neural also took noise as input, then it would define an implicit distribution (like GANs). However, Eq (8) in [1] suggests that $p_{1|t}$ is trained with maximum likelihood. That rules out GANs. I presume that the authors use $p_{1|t}(x_1|x_t) = \mathcal{N}(x_1 | \mu(x_t), \sigma(x_t)I)$, where $\mu$ is the neural network that the authors refer to. However, I do not see this defined anywhere in your paper, this reponse, nor in [1]. An example of how one might define a distribution is e.g. the definition of $p_{t|1}$ (reverse conditioning) in Eq (11) in [1]. Not every neural network that takes some input and produces some output defines a probability distribution (unless the authors mean a Dirac delta distribution, but I would need to check if the derivations hold in that case; although this is also not stated in the text)

• In [1], the authors use a single sample for the Monte Carlo estimation

In your paper do you also use a single sample? I also asked about the variance.

Your suggestion seems to align with a “generate-then-filter” approach, where multiple samples are generated, and the top-1 sample with the highest target value is selected

This is not what I am suggesting. I do not mean sampling $G_1 \sim p_1$. I mean in TGF-Flow, instead of generating any random sample $G_1 \sim p_{1|t}$, sampling multiple $G_1 \sim p_{1|t}$ and choosing the best one before moving $t \to t + dt$.

the merit of training-free guidance is not “efficiency” in terms of speed or runtime.

If runtime is not the issue, why not train additional classifiers and use them for guidance? In that case, I am not sure I understand the motivation of the study. What is the added value of the method being "training free" if it's not efficiency?

Table 1 runtime and parameter count

Do I understand correctly that all the baselines and the proposed method share the exact same architecture? Regarding training times, what does it mean "e.g., 1h10min for training the target predictor and 2h for training the generative models)". Does every method require training a target predictor? Let me clarify what I meant. It would be helpful if you provided a table that summarizes the total training time, total parameter count, and total sampling time for each method, so that every component of every model is included in all three categories. E.g. is none of the baseline methods using regular (classifier-free or classifier-based) guidance? If not, why not? This would be a reasonable baseline. If yes, then I would be surprised if they require exactly the same training time as those that do not require this.

• The performance of training-based guidance methods (e.g., EEGSDE) typically serves as an upper bound for training-free methods [...] the merit of training-free methods lies not in their efficiency or speed but in their flexibility and independence from time-dependent classifiers

This is what I think I'm missing. To reiterate the point above: If speed is not a concern then I don't see the gain in pursuing training-free guidance methods. I can just spend X amount of additional time to train the time-dependent classifier and use regular guidance. Especially since the performance gap to the training-based remains significant.

I would also reiterate the point I made in the original review. I would suggest changing the presentation in Table 2. In the current revision it still implies that the proposed method is the best overall. It is not the case. It is significantly worse than EEGSDE. The only gain is that TGF-Flow is "training-free" (but not necessarily faster).

To demonstrate the benefits of the training-free property to leverage flexible target predictors, we included results using an open-sourced foundation model UniMol [2] as the target predictor for experiments in Table 1. UniMol, a strong off-the-shelf model, significantly enhances the performance of TFG-Flow, narrowing the gap with EEGSDE

But the performance is still worse than EEGSDE. Respectfully, I do not see how this demonstrates the benefits of the training-free property. Just to clarify, does EEGSDE also use UniMol as guidance in this experiment? (What I mean is an additional, perhaps time-dependent, classifier trained to guide EEGSDE towards the signal from UniMol?)

• we incorporated a comparison with optimization-based methods

Could you please clarify two things for me? In this table, is "time" measuring the time to generate a single molecule? I am curious about the runtime, because you mention in the generate-then-filter you first generate 10 candidate molecules and then choose the best one. In [2] it is reported that estimating Vina score for a single protein-ligand complex takes 20 seconds, so this method should at least take 200 seconds to produce a single molecule. Does Vina2 run faster on your machine?

• The claim on the distributional property follows from existing works [4,5,6].

Could the authors point me to the specific derivation/equation/claim in e.g. [4] where I can find that derivation?

• To clarify, we train the drug quality prediction network to guide the flow model during training

I am afraid this confused me more. So what does training-free mean? Since the guidance is needed during training?

• The subtlety here is that $z_t$ is a function of $t$ .

Again, I must respectfully disagree. I went over the derivation in [3] and I do not think what you are referring to is an error, but rather a slight abuse of notation. Oftentimes, when one writes $\nabla_{z_t}f(z_t, t)$ it actually means $\nabla_z f(z, t)|_{z=z_t}$, which is clearly what the authors meant given the preceding derivation. In that case $z$ and $t$ are independent variables and $t$ is treated as constant when taking the gradient w.r.t. $z$.

However, funnily enough, I do believe that [3] makes a critical error in their reasoning. It is however in Equation (3), not in Equation (9). The authors of [3] assume that a "well-trained" flow is linear. This is clearly not true. I will reach out to the authors of [3] for an explanation.

• Table 5

Please fix the citations in Table 5. Neither RGA, nor 3D-MCTS were developed by [2].

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[1] Campbell et al. "Generative Flows on Discrete State-Spaces: Enabling Multimodal Flows with Applications to Protein Co-Design" (ICML 2024)

[2] Karczewski et al. "What Ails Generative Structure-Based Drug Design Too Little or Too Much Expressivity?" (Arxiv)

[3] Sun et al. "RectifID: Personalizing Rectified Flow with Anchored Classifier Guidance" (NeurIPS 2024)