Acquisition Function-Guided Generation for Bayesian Optimization over Discrete Spaces

# Acquisition Function-Guided Generation for Bayesian Optimization over Discrete Spaces
Source: [https://arxiv.org/html/2610.22763](https://arxiv.org/html/2610.22763)
Tommaso AncilliAffiliation:Institute of Machine Learning and Neural Computation, TU Graz, AustriaEmail:[name\.surname@tugraz\.at](mailto:[email protected])Thomas Wedenig††thanks:Joint first authorAffiliation:Institute of Machine Learning and Neural Computation, TU Graz, AustriaAffiliation:Graz Center for Machine Learning \(GraML\), AustriaEmail:[henry\.moss@lancaster\.ac\.uk](mailto:[email protected])Robert PeharzAffiliation:Institute of Machine Learning and Neural Computation, TU Graz, AustriaAffiliation:Graz Center for Machine Learning \(GraML\), Austria

###### Abstract

Bayesian optimization \(BO\) over discrete spaces is challenging due to missing gradients and the sparsity of valid solutions\. Latent\-space BO sidesteps both issues by fitting a surrogate in the latent space of a generative model, which however substantially complicates the direct incorporation of domain knowledge that is naturally expressed in the original discrete space\. In this work, we instead combine a pretrained generative model with a surrogate defined directly in data space, where inductive biases such as domain\-specific kernels can be readily incorporated\. Rather than maximizing the acquisition function over the discrete design space, our method samples candidates from the generative model under a distribution tilted by the acquisition function\. To recover useful gradients for sampling, we embed this tilted distribution into a diffusion bridge that yields a smooth, posterior\-averaged acquisition function\. This leads to*acquisition function guided generation*\(AFGG\), which we instantiate with a categorical diffusion model and an autoregressive normalizing flow\. Across a broad suite of molecular benchmarks, the diffusion model variant performs on par with leading BO methods, while the flow model variant outperforms state\-of\-the\-art methods in most cases and remains highly competitive otherwise\. Our implementation is available at[https://anonymous\.4open\.science/r/afgg](https://anonymous.4open.science/r/afgg)\.

## 1Introduction

Bayesian optimization \(BO\) offers a sample\-efficient approach to problems such as molecular and protein design, where objective evaluations are costly and the search space is discrete and structured\([Garnett, 2023](https://arxiv.org/html/2610.22763#bib.bib2);[Yang et al\., 2025](https://arxiv.org/html/2610.22763#bib.bib14)\)\. Unfortunately, direct optimization over such spaces is hard because valid designs are sparsely populated in a combinatorial space and standard continuous optimizers have no useful gradients to follow\([Moss et al\., 2020](https://arxiv.org/html/2610.22763#bib.bib16)\)\. Latent\-space Bayesian optimization \(LSBO\) addresses both challenges by training a generative model over the discrete space—thereby capturing the sparse structure of valid designs—and by performing BO in its continuous latent space, where gradients for acquisition optimization are readily available\([Gómez\-Bombarelli et al\., 2018](https://arxiv.org/html/2610.22763#bib.bib13)\)\.

However, domain knowledge is harder to incorporate in an abstract latent space, while in data space, such knowledge can often be encoded directly through inductive biases such as domain\-specific kernels\([Rogers and Hahn, 2010](https://arxiv.org/html/2610.22763#bib.bib8)\)\. Furthermore, the latent space must also “do two jobs at once:” support faithful generation and provide a geometry in which the objective is easy to model by a surrogate\. These goals need not align, which motivates many LSBO methods to adapt the latent representation during BO\([Maus et al\., 2022](https://arxiv.org/html/2610.22763#bib.bib31);[Lee et al\., 2023](https://arxiv.org/html/2610.22763#bib.bib36);[Chu et al\., 2024](https://arxiv.org/html/2610.22763#bib.bib32);[Tripp et al\., 2020](https://arxiv.org/html/2610.22763#bib.bib30);[Lee et al\., 2025](https://arxiv.org/html/2610.22763#bib.bib35);[Deshwal and Doppa, 2021](https://arxiv.org/html/2610.22763#bib.bib17)\), despite the small\-data regime in which such adaptation can overfit\([Ober et al\., 2021](https://arxiv.org/html/2610.22763#bib.bib15)\)\. Moreover, even a strong generative model only approximates the sparse set of valid designs, so acquisition optimization can move into poorly supported regions of latent space where decoding becomes unreliable\([Moss et al\., 2025](https://arxiv.org/html/2610.22763#bib.bib12);[Bodin et al\., 2025](https://arxiv.org/html/2610.22763#bib.bib40)\)\.

A natural strategy for avoiding these drawbacks is to decouple the two roles: use a strong pretrained generative model to capture the sparse structure of valid designs, while keeping the surrogate and acquisition function in data space, so that domain knowledge and inductive biases can be included\. To the best of our knowledge, COWBOYS\([Moss et al\., 2025](https://arxiv.org/html/2610.22763#bib.bib12)\)is the only prior BO method that follows this strategy, but it is tied to a VAE as the generative model and probability of improvement as the acquisition function\.

In this paper, we propose*acquisition function guided generation*\(AFGG\), which replaces acquisition function maximization with sampling from a distribution that tilts a frozen pretrained generative model toward designs favored by the acquisition function\. The construction is generic in both components: it requires only a generative model that supports guided sampling and any non\-negative acquisition function that can be evaluated pointwise; the latter need neither be differentiable nor define a probability\. Since the discrete, combinatorial design space makes direct sampling from the tilted distribution difficult, we embed it into a diffusion process over the generative model’s continuous space\. At each noise level, this yields a smooth posterior\-averaged acquisition function that provides the gradients needed for guided sampling\. The resulting gradients can then guide sampling in the same spirit as guidance in diffusion models\([Dhariwal and Nichol, 2021](https://arxiv.org/html/2610.22763#bib.bib21);[Ho and Salimans, 2022](https://arxiv.org/html/2610.22763#bib.bib22)\), which gives the method its name\.

Our contributions are threefold:

- •Acquisition function guided generation\.We formulate the BO inner loop as sampling from an acquisition\-tilted distribution that combines a frozen pretrained generative model with a surrogate defined directly in data space \(Section[3\.1](https://arxiv.org/html/2610.22763#S3.SS1)\)\. We show that this target can be recovered exactly in the generative model’s continuous space and that the resulting guidance term reduces to replacing the model’s posterior mean by its acquisition\-weighted counterpart \(Section[3\.3](https://arxiv.org/html/2610.22763#S3.SS3)\)\.
- •A unified construction for different generative models\.We derive AFGG from a common diffusion\-bridge construction and show how it applies to both categorical diffusion models and autoregressive normalizing flows \(Section[3\.4](https://arxiv.org/html/2610.22763#S3.SS4)\)\. The generator remains frozen throughout BO, while the surrogate and acquisition function stay in data space\.
- •State\-of\-the\-art empirical performance\.Across a broad suite of molecular design benchmarks under tight evaluation budgets, AFGG with the categorical diffusion model performs on par with leading BO methods\. More strikingly, AFGG with the flow model achieves state\-of\-the\-art performance on most benchmarks and remains highly competitive on the rest \(Section[4](https://arxiv.org/html/2610.22763#S4)\)\.

## 2Background

### 2\.1Bayesian Optimization over Discrete Spaces

We consider Bayesian optimization over discrete, structured design spaces\. In particular, we focus on sequence\-valued settings and denote the design space as𝒳:=𝒱L\{\\mathcal\{X\}\}:=\\mathcal\{V\}^\{L\}, the set of length\-LLsequences over a discrete vocabulary𝒱\\mathcal\{V\}of sizeVV\. We assume that valid or useful designs occupy a constrained and sparsely populated subset of the combinatorial space\. This setting covers, for example, molecules represented as SELFIES strings\([Krenn et al\., 2020](https://arxiv.org/html/2610.22763#bib.bib28)\)and proteins represented as amino\-acid sequences\. We seek the maximizerw∗w^\{\\ast\}of an expensive\-to\-evaluate and black\-box objectivef:𝒳→ℝf\\colon\{\\mathcal\{X\}\}\\to\\mathbb\{R\}, which provides no gradients and returns measurementsyythat may be contaminated by Gaussian noise:

w∗∈arg​maxw∈𝒳⁡f​\(w\),y⁡\(w\)=f⁡\(w\)\+εobs,εobs∼𝒩⁡\(0,σobs2\)\.w^\{\\ast\}\\in\\argmax\_\{w\\in\{\\mathcal\{X\}\}\}~f\(w\),\\qquad y\(w\)=f\(w\)\+\\varepsilon\_\{\\mathrm\{obs\}\},\\quad\\varepsilon\_\{\\mathrm\{obs\}\}\\sim\\mathcal\{N\}\(0,\\sigma\_\{\\mathrm\{obs\}\}^\{2\}\)\.\(1\)
Bayesian optimization \(BO\) fits a probabilistic surrogate to the observations collected so far and uses its predictive uncertainty to choose the next design to evaluate\([Shahriari et al\., 2016](https://arxiv.org/html/2610.22763#bib.bib3);[Garnett, 2023](https://arxiv.org/html/2610.22763#bib.bib2)\)\. Throughout this work, we use Gaussian processes \(GPs\)\([Rasmussen and Williams, 2006](https://arxiv.org/html/2610.22763#bib.bib1)\), whose posterior mean and variance are available in closed form\. However, our method is not tied to GPs and only requires a probabilistic surrogate from which an acquisition function can be computed or approximated\.

Defining the surrogate directly on the discrete space allows domain knowledge to be incorporated through suitable inductive biases, such as domain\-specific kernels\. For example, in our molecular experiments we use the Tanimoto kernel\([Ralaivola et al\., 2005](https://arxiv.org/html/2610.22763#bib.bib6);[Tripp et al\., 2023](https://arxiv.org/html/2610.22763#bib.bib7)\)on binary Morgan fingerprintsψ⁡\(w\)∈\{0,1\}dfp\\psi\(w\)\\in\\\{0,1\\\}^\{d\_\{\\text\{fp\}\}\}, where usuallydfp=2048d\_\{\\text\{fp\}\}=2048\([Rogers and Hahn, 2010](https://arxiv.org/html/2610.22763#bib.bib8)\),

k⁡\(w,w′\)=⟨ψ⁡\(w\),ψ⁡\(w′\)⟩∥ψ⁡\(w\)∥2\+∥ψ⁡\(w′\)∥2−⟨ψ⁡\(w\),ψ⁡\(w′\)⟩∈\[0,1\]\.k\(w,w^\{\\prime\}\)=\\frac\{\\langle\\psi\(w\),\\psi\(w^\{\\prime\}\)\\rangle\}\{\\lVert\\psi\(w\)\\rVert^\{2\}\+\\lVert\\psi\(w^\{\\prime\}\)\\rVert^\{2\}\-\\langle\\psi\(w\),\\psi\(w^\{\\prime\}\)\\rangle\}\\in\[0,1\]\.\(2\)
An acquisition functionα:𝒳→ℝ\\alpha\\colon\{\\mathcal\{X\}\}\\to\\mathbb\{R\}ranks candidate designs by balancing predicted performance against uncertainty\. We consider two standard choices, probability of improvement \(PI\) and expected improvement \(EI\), relative to the incumbenty⋆y^\{\\star\}:

αPI​\(w\)=Pr⁡\[f⁡\(w\)≥y⋆∣𝒟\]∈\[0,1\],αEI​\(w\)=𝔼⁡\[max⁡\(f⁡\(w\)−y⋆,0\)∣𝒟\]∈\[0,∞\),\\alpha^\{\\mathrm\{PI\}\}\(w\)=\\Pr\\big\[f\(w\)\\geq y^\{\\star\}\\mid\\mathcal\{D\}\\big\]\\in\[0,1\],\\qquad\\alpha^\{\\mathrm\{EI\}\}\(w\)=\\mathbb\{E\}\\big\[\\max\(f\(w\)\-y^\{\\star\},0\)\\mid\\mathcal\{D\}\\big\]\\in\[0,\\infty\),\(3\)where𝒟=\{\(wj,yj\)\}j=1n\\mathcal\{D\}=\\\{\(w\_\{j\},y\_\{j\}\)\\\}\_\{j=1\}^\{n\}denotes the observations collected so far\. Both are available in closed form under a GP\([Ament et al\., 2023](https://arxiv.org/html/2610.22763#bib.bib4)\)\. PI is a probability, whereas EI is non\-negative but potentially unbounded; this distinction will matter for our method\. Standard BO selects the next design by maximizing the acquisition functionwnext∈arg​maxw∈𝒳⁡α​\(w\)w\_\{\\mathrm\{next\}\}\\in\\argmax\_\{w\\in\{\\mathcal\{X\}\}\}~\\alpha\(w\)\. For discrete, combinatorial𝒳\{\\mathcal\{X\}\}, this inner optimization is generally hard, and gradient\-based methods for continuous BO do not apply directly\.

### 2\.2Generative Models over Sequences

Our method requires a pretrained generative model over𝒳\{\\mathcal\{X\}\}that supports guided sampling\. We consider two well\-suited model classes for discrete sequences: a categorical diffusion model and an autoregressive normalizing flow\. Both connect the discrete design space𝒳=𝒱L\{\\mathcal\{X\}\}=\\mathcal\{V\}^\{L\}to a continuous embedding spaceℝL×d\\mathbb\{R\}^\{L\\times d\}in the same way: an embedding tableE∈ℝ\|𝒱\|×dE\\in\\mathbb\{R\}^\{\|\\mathcal\{V\}\|\\times d\}\(learned by the model\) with rowseve\_\{v\},v∈𝒱v\\in\\mathcal\{V\}, maps a sequence to its embeddings, and a deterministic decode mapτ:ℝL×d→𝒳\\tau\\colon\\mathbb\{R\}^\{L\\times d\}\\to\{\\mathcal\{X\}\}clamps each position to the closest token in cosine similarity,

x=E\(w\)≜\(ew1,…,ewL\)∈ℝL×d,τ\(x\)i=arg​maxv∈𝒱cos\(xi,ev\),i=1,…,L\.x=E\(w\)\\triangleq\\big\(e\_\{w\_\{1\}\},\\dots,e\_\{w\_\{L\}\}\\big\)\\in\\mathbb\{R\}^\{L\\times d\},\\qquad\\tau\(x\)\_\{i\}=\\argmax\_\{v\\in\\mathcal\{V\}\}\\cos\\big\(x\_\{i\},e\_\{v\}\\big\),\\quad i=1,\\dots,L\.\(4\)
Continuous diffusion for categorical data \(CDCD\)\([Dieleman et al\., 2022](https://arxiv.org/html/2610.22763#bib.bib25)\)runs a variance\-exploding diffusion\([Karras et al\., 2022](https://arxiv.org/html/2610.22763#bib.bib20)\)on the embeddings,

zt=x\+t​ϵ,ϵ∼𝒩⁡\(0,I\),z\_\{t\}=x\+t\\,\\epsilon,\\qquad\\epsilon\\sim\\mathcal\{N\}\(0,I\),\(5\)so that the forward kernel isqt​\(zt∣x\)=𝒩⁡\(zt,x,t2​I\)q\_\{t\}\(z\_\{t\}\\mid x\)=\\mathcal\{N\}\(z\_\{t\};x,t^\{2\}I\)\. Samples are generated by integrating the reverse\-time SDE or probability\-flow ODE fromt=Tt=Ttot=0t=0\([Song et al\., 2021](https://arxiv.org/html/2610.22763#bib.bib19)\), which requires the score∇zt​log​pt​\(zt\)\\nabla\_\{z\_\{t\}\}\\log p\_\{t\}\(z\_\{t\}\)of the noised marginal\. Since∇zt​log​qt​\(zt∣x\)=\(x−zt\)/t2\\nabla\_\{z\_\{t\}\}\\log q\_\{t\}\(z\_\{t\}\\mid x\)=\(x\-z\_\{t\}\)/t^\{2\}is affine inxx, the score is the conditional score averaged over the denoising posterior, and in the discrete case, where each position ofxxis one of𝒱\\mathcal\{V\}’s embeddings, it has the form

∇zt​log​pt​\(zt\)=z^−ztt2,z^​\(zt,t\)i=∑v=1Vp⁡\(wi=v∣zt\)​ev\.\\nabla\_\{z\_\{t\}\}\\log p\_\{t\}\(z\_\{t\}\)=\\frac\{\\hat\{z\}\-z\_\{t\}\}\{t^\{2\}\},\\qquad\\hat\{z\}\(z\_\{t\},t\)\_\{i\}=\\sum\_\{v=1\}^\{V\}p\(w\_\{i\}=v\\mid z\_\{t\}\)\\,e\_\{v\}\.\(6\)The score therefore depends on the joint denoising posterior only through its per\-position marginals\([Dieleman et al\., 2022](https://arxiv.org/html/2610.22763#bib.bib25)\), and these are what must be learned: a transformer, trained with categorical cross\-entropy on noised embeddings, outputsLLmass functions over𝒱\\mathcal\{V\},

pθ\(wi=v∣zt,t\)=softmax\(logitsθ\(zt,t\)i\)v,i=1,…,L\.p\_\{\\theta\}\(w\_\{i\}=v\\mid z\_\{t\},t\)=\\mathrm\{softmax\}\\big\(\\mathrm\{logits\}\_\{\\theta\}\(z\_\{t\},t\)\_\{i\}\\big\)\_\{v\},\\qquad i=1,\\dots,L\.\(7\)Substitutingpθp\_\{\\theta\}forppin Eq\. \([6](https://arxiv.org/html/2610.22763#S2.E6)\) gives the learned score\.

Autoregressive normalizing flow \(SeqFlow\)\([Lee et al\., 2025](https://arxiv.org/html/2610.22763#bib.bib35)\)draws a Gaussian latent variablez∼𝒩⁡\(0,I\)z\\sim\\mathcal\{N\}\(0,I\),z∈ℝL×dz\\in\\mathbb\{R\}^\{L\\times d\}, and maps it through a diffeomorphismgθ:ℝL×d→ℝL×dg\_\{\\theta\}:\\mathbb\{R\}^\{L\\times d\}\\to\\mathbb\{R\}^\{L\\times d\}to a sequence of continuous token embeddings,x=gθ​\(z\)x=g\_\{\\theta\}\(z\), whichτ\\taudecodes by Eq\. \([4](https://arxiv.org/html/2610.22763#S2.E4)\)\. Becausegθg\_\{\\theta\}is a bijection andτ\\tauis deterministic, the resulting distribution over designs is exactly the pushforward of the latent Gaussian,

Pθ​\(w\)=∫Aw𝒩⁡\(z,0,I\)​𝑑z,Aw:=\(τ∘gθ\)−1​\(w\)⊂ℝL×d,P\_\{\\theta\}\(w\)=\\int\_\{A\_\{w\}\}\\mathcal\{N\}\(z;0,I\)\\,dz,\\qquad A\_\{w\}:=\(\\tau\\circ g\_\{\\theta\}\)^\{\-1\}\(w\)\\subset\\mathbb\{R\}^\{L\\times d\},\(8\)where the cells\{Aw\}w∈𝒳\\\{A\_\{w\}\\\}\_\{w\\in\{\\mathcal\{X\}\}\}partition the latent space\. We refer to[Lee et al\. \(2025\)](https://arxiv.org/html/2610.22763#bib.bib35)for details of the architecture and training procedure\.

## 3Acquisition Function\-Guided Generation

We now present AFGG, which steers a pretrained generative model toward designs with high acquisition values\. We first define the acquisition\-tilted target distribution and explain why it is hard to sample it directly in data space \(Section[3\.1](https://arxiv.org/html/2610.22763#S3.SS1)\)\. We then introduce a diffusion bridge that smooths this target and recovers informative gradients \(Section[3\.2](https://arxiv.org/html/2610.22763#S3.SS2)\)\. The resulting guidance term reduces to a difference of posterior means, which we estimate by sampling \(Section[3\.3](https://arxiv.org/html/2610.22763#S3.SS3)\)\. Finally, we show how the same construction applies to our two generative backbones, which differ only in how they provide the base score and posterior samples \(Section[3\.4](https://arxiv.org/html/2610.22763#S3.SS4)\)\.

### 3\.1The Sampling Problem

As described in Section[2](https://arxiv.org/html/2610.22763#S2), the generative model defines a densitypθ​\(x\)p\_\{\\theta\}\(x\)on the continuous embedding spaceℝL×d\\mathbb\{R\}^\{L\\times d\}together with a deterministic decoderτ:ℝL×d→𝒳\\tau\\colon\\mathbb\{R\}^\{L\\times d\}\\to\{\\mathcal\{X\}\}\. Each discrete design sequencew∈𝒳w\\in\{\\mathcal\{X\}\}corresponds to the region of embedding space that decodes to it,

τ−1​\(w\)≜\{x∈ℝL×d:τ⁡\(x\)=w\}\.\\tau^\{\-1\}\(w\)\\triangleq\\\{x\\in\\mathbb\{R\}^\{L\\times d\}\\colon\\tau\(x\)=w\\\}\.\(9\)These regions partition the embedding space, and the probability of a sequence is the mass thatpθp\_\{\\theta\}assigns to its corresponding region\. Hence, the decoder pushespθp\_\{\\theta\}forward to a probability mass functionPθP\_\{\\theta\}over the discrete sequence space,

Pθ​\(w\)=∫τ−1​\(w\)pθ​\(x\)​𝑑x\.P\_\{\\theta\}\(w\)=\\int\_\{\\tau^\{\-1\}\(w\)\}p\_\{\\theta\}\(x\)\\,\\mathrm\{d\}x\.\(10\)
###### Definition 3\.1\(Tilted target\)\.

Given the induced probability mass functionPθP\_\{\\theta\}over𝒳\{\\mathcal\{X\}\}and a non\-negative acquisition functionℓ:𝒳→ℝ\+\\ell\\colon\{\\mathcal\{X\}\}\\to\\mathbb\{R\}\_\{\+\}, we define the*tilted target distribution*

P⋆​\(w\)≜1Z​Pθ​\(w\)​ℓ​\(w\),Z≜∑w∈𝒳Pθ​\(w\)​ℓ​\(w\)\.P^\{\\star\}\(w\)\\triangleq\\frac\{1\}\{Z\}P\_\{\\theta\}\(w\)\\ell\(w\),\\qquad Z\\triangleq\\sum\_\{w\\in\{\\mathcal\{X\}\}\}P\_\{\\theta\}\(w\)\\ell\(w\)\.\(11\)

The tilted distribution combines the two components of our approach:PθP\_\{\\theta\}places mass on sequences supported by the pretrained generative model, whileℓ\\ellshifts this mass toward sequences favored by the surrogate\. The requirement thatℓ\\ellbe non\-negative is mild, as a real\-valued acquisition functionℓ~\\tilde\{\\ell\}can be turned into a positive tilt, for example throughℓ​\(w\)=exp⁡\(ℓ~​\(w\)\)\\ell\(w\)=\\exp\(\\tilde\{\\ell\}\(w\)\)\.

This product form also has a natural interpretation through entropy\-regularized control\([Uehara et al\., 2024](https://arxiv.org/html/2610.22763#bib.bib37)\)\. Indeed,P⋆P^\{\\star\}is the solution of

P⋆=arg​maxP\{𝔼P\[logℓ\(w\)\]−DKL\(P∥Pθ\)\}\.P^\{\\star\}=\\argmax\_\{P\}\\left\\\{\\mathbb\{E\}\_\{P\}\\big\[\\log\\ell\(w\)\\big\]\-D\_\{\\mathrm\{KL\}\}\\big\(P\\,\\\|\\,P\_\{\\theta\}\\big\)\\right\\\}\.\(12\)The first term rewards designs with high acquisition values, while the KL term keeps the resulting distribution close to the pretrained generative model\. Equivalently, entropy\-regularized control yields a target of the formPθ​\(w\)​exp⁡\(r⁡\(w\)\)P\_\{\\theta\}\(w\)\\exp\(r\(w\)\), and our tilted distribution follows by takingr⁡\(w\)=log⁡ℓ⁡\(w\)r\(w\)=\\log\\ell\(w\)\.

While sampling from the generative modelPθP\_\{\\theta\}is easy, the acquisition tilt changes this distribution such that directly sampling fromP⋆P^\{\\star\}would require handling the combinatorial space𝒳\{\\mathcal\{X\}\}, facing much the same difficulty as maximizing the acquisition function over𝒳\{\\mathcal\{X\}\}\. To sample fromP⋆P^\{\\star\}, we instead return to the continuous space of the generative model\. Using Eq\. \([10](https://arxiv.org/html/2610.22763#S3.E10)\), the corresponding tilted density is

p⋆​\(x\)≜1Z​pθ​\(x\)​ℓ​\(τ⁡\(x\)\)\.p^\{\\star\}\(x\)\\triangleq\\frac\{1\}\{Z\}p\_\{\\theta\}\(x\)\\ell\(\\tau\(x\)\)\.\(13\)Samplingx∼p⋆x\\sim p^\{\\star\}and decodingw=τ⁡\(x\)w=\\tau\(x\)yields exactlyw∼P⋆w\\sim P^\{\\star\}; we prove this in Appendix[A\.1](https://arxiv.org/html/2610.22763#A1.SS1)\.

The main technical challenge is thatℓ⁡\(τ⁡\(x\)\)\\ell\(\\tau\(x\)\)is constant within each cellτ−1​\(w\)\\tau^\{\-1\}\(w\), so∇xℓ​\(τ⁡\(x\)\)=0\\nabla\_\{x\}\\,\\ell\(\\tau\(x\)\)=0almost everywhere\. LSBO avoids this difficulty by moving the surrogate to latent space, but in doing so inherits the limitations discussed above\. The rest of this section shows how to recover informative gradients while keeping the surrogate in data space\.

### 3\.2Lifting the Target to a Bridge

We recover informative gradients by introducing a family of smoothed versions ofp⋆p^\{\\star\}, indexed byt∈\[0,T\]t\\in\[0,T\], witht=0t=0corresponding to the tilted target andt=Tt=Tto the noise distribution\. Note that, although this smoothing construction is borrowed from diffusion models\([Ho et al\., 2020](https://arxiv.org/html/2610.22763#bib.bib18);[Song et al\., 2021](https://arxiv.org/html/2610.22763#bib.bib19)\), the generative model itself need not be a diffusion model; it only needs to support the guided sampling procedure developed below\.

###### Definition 3\.2\(Bridge marginals\)\.

Starting from the tilted densityp⋆p^\{\\star\}and the original generative densitypθp\_\{\\theta\}, we define their*bridge marginals*at noise levelttas

pt⋆​\(zt\)=∫p⋆​\(x\)​qt​\(zt∣x\)​𝑑x,pt​\(zt\)=∫pθ​\(x\)​qt​\(zt∣x\)​𝑑x,p^\{\\star\}\_\{t\}\(z\_\{t\}\)=\\int p^\{\\star\}\(x\)\\,q\_\{t\}\(z\_\{t\}\\mid x\)\\,\\mathrm\{d\}x,\\qquad p\_\{t\}\(z\_\{t\}\)=\\int p\_\{\\theta\}\(x\)\\,q\_\{t\}\(z\_\{t\}\\mid x\)\\,\\mathrm\{d\}x,\(14\)

where

qt​\(zt∣x\)=𝒩⁡\(zt,αt​x,σt2​I\)q\_\{t\}\(z\_\{t\}\\mid x\)=\\mathcal\{N\}\\big\(z\_\{t\};\\alpha\_\{t\}x,\\sigma\_\{t\}^\{2\}I\\big\)\(15\)is a Gaussian noising kernel applied to the continuous variablexx\.

A key observation is that the acquisition tilt is preserved under this smoothing in a convenient form\. Let

p⁡\(x∣zt\)=pθ​\(x\)​qt​\(zt∣x\)pt​\(zt\)p\(x\\mid z\_\{t\}\)=\\frac\{p\_\{\\theta\}\(x\)q\_\{t\}\(z\_\{t\}\\mid x\)\}\{p\_\{t\}\(z\_\{t\}\)\}\(16\)denote the posterior over the clean variablexxunder the original generative model\.

###### Proposition 1\(Tilted marginal\)\.

The tilted marginal can be written as

pt⋆​\(zt\)∝pt​\(zt\)​ℓt​\(zt\),ℓt​\(zt\)≜𝔼p⁡\(x∣zt\)​\[ℓ⁡\(τ⁡\(x\)\)\],p^\{\\star\}\_\{t\}\(z\_\{t\}\)\\propto p\_\{t\}\(z\_\{t\}\)\\,\\ell\_\{t\}\(z\_\{t\}\),\\qquad\\ell\_\{t\}\(z\_\{t\}\)\\triangleq\\mathbb\{E\}\_\{p\(x\\mid z\_\{t\}\)\}\\big\[\\ell\(\\tau\(x\)\)\\big\],\(17\)whereℓt\\ell\_\{t\}is the acquisition function averaged over the posterior uncertainty about the clean design sequence\. Moreover, the corresponding tilted posterior satisfies

p⋆​\(x∣zt\)∝p⁡\(x∣zt\)​ℓ​\(τ⁡\(x\)\)\.p^\{\\star\}\(x\\mid z\_\{t\}\)\\propto p\(x\\mid z\_\{t\}\)\\,\\ell\(\\tau\(x\)\)\.\(18\)

###### Proof\.

The proof is in Appendix[A\.2](https://arxiv.org/html/2610.22763#A1.SS2)\. ∎

This factorization makes guidance convenient\. To sample fromp⋆p^\{\\star\}by reversing the bridge, we need the score of the tilted marginal,

∇zt​log​pt⋆​\(zt\)=∇zt​log​pt​\(zt\)\+∇zt​log​ℓt​\(zt\)\.\\nabla\_\{z\_\{t\}\}\\log p^\{\\star\}\_\{t\}\(z\_\{t\}\)=\\nabla\_\{z\_\{t\}\}\\log p\_\{t\}\(z\_\{t\}\)\+\\nabla\_\{z\_\{t\}\}\\log\\ell\_\{t\}\(z\_\{t\}\)\.\(19\)The first term is the score of the pretrained generative model at noise leveltt, while the second steers it toward regions favored by the acquisition function\. This has the same form as guidance in diffusion models\([Dhariwal and Nichol, 2021](https://arxiv.org/html/2610.22763#bib.bib21)\)\.

Importantly,ℓt\\ell\_\{t\}is no longer the piecewise\-constant functionℓ∘τ\\ell\\circ\\tau\. Instead, it averagesℓ⁡\(τ⁡\(x\)\)\\ell\(\\tau\(x\)\)over the posteriorp⁡\(x∣zt\)p\(x\\mid z\_\{t\}\), and can therefore vary smoothly with the continuous variableztz\_\{t\}\. Hence,∇zt​log​ℓt​\(zt\)\\nabla\_\{z\_\{t\}\}\\log\\ell\_\{t\}\(z\_\{t\}\)can provide an informative guidance signal even though∇xℓ​\(τ⁡\(x\)\)=0\\nabla\_\{x\}\\ell\(\\tau\(x\)\)=0almost everywhere\.

Evaluating Eq\. \([19](https://arxiv.org/html/2610.22763#S3.E19)\) therefore requires two ingredients: \(i\) the untilted score∇zt​log​pt​\(zt\)\\nabla\_\{z\_\{t\}\}\\log p\_\{t\}\(z\_\{t\}\)and \(ii\) samples from the posteriorp⁡\(x∣zt\)p\(x\\mid z\_\{t\}\)\. The next section shows how these ingredients yield the guidance term, while Section[3\.4](https://arxiv.org/html/2610.22763#S3.SS4)explains how they are obtained for two generative backbones\.

### 3\.3Guidance Is a Tilted Mean Difference

The central object in Eq\. \([19](https://arxiv.org/html/2610.22763#S3.E19)\) is the guidance term∇zt​log​ℓt​\(zt\)\\nabla\_\{z\_\{t\}\}\\log\\ell\_\{t\}\(z\_\{t\}\)\. For the Gaussian kernel Eq\. \([15](https://arxiv.org/html/2610.22763#S3.E15)\), this term has a particularly simple form: it is proportional to the difference between the ordinary posterior mean and its acquisition\-weighted counterpart\.

###### Definition 3\.3\(Tilted estimate\)\.

Withp⋆​\(x∣zt\)p^\{\\star\}\(x\\mid z\_\{t\}\)as in Eq\. \([18](https://arxiv.org/html/2610.22763#S3.E18)\), define

z^tilt≜𝔼p⋆​\(x∣zt\)​\[x\]=𝔼p⁡\(x∣zt\)​\[ℓ⁡\(τ⁡\(x\)\)​x\]𝔼p⁡\(x∣zt\)​\[ℓ​\(τ​\(x\)\)\],z^≜𝔼p⁡\(x∣zt\)​\[x\]\.\\hat\{z\}^\{\\mathrm\{tilt\}\}\\triangleq\\mathbb\{E\}\_\{p^\{\\star\}\(x\\mid z\_\{t\}\)\}\\big\[x\\big\]=\\frac\{\\mathbb\{E\}\_\{p\(x\\mid z\_\{t\}\)\}\\big\[\\ell\(\\tau\(x\)\)\\,x\\big\]\}\{\\mathbb\{E\}\_\{p\(x\\mid z\_\{t\}\)\}\\big\[\\ell\(\\tau\(x\)\)\\big\]\},\\qquad\\hat\{z\}\\triangleq\\mathbb\{E\}\_\{p\(x\\mid z\_\{t\}\)\}\\big\[x\\big\]\.\(20\)

Thus,z^\\hat\{z\}is the posterior mean under the pretrained generative model, whereasz^tilt\\hat\{z\}^\{\\mathrm\{tilt\}\}gives more weight to posterior samples whose decoded designs have high acquisition value\.

###### Proposition 2\(Guidance score as tilted mean difference\)\.

Let the kernel of Definition[3\.2](https://arxiv.org/html/2610.22763#S3.Thmdefinition2)be the Gaussian kernel \([15](https://arxiv.org/html/2610.22763#S3.E15)\), withαt,σt\>0\\alpha\_\{t\},\\sigma\_\{t\}\>0\. Then

∇zt​log​ℓt​\(zt\)=αt​\(z^tilt−z^\)σt2,∇zt​log​pt​\(zt\)=αt​z^−ztσt2\.\\nabla\_\{z\_\{t\}\}\\log\\ell\_\{t\}\(z\_\{t\}\)=\\frac\{\\alpha\_\{t\}\\big\(\\hat\{z\}^\{\\mathrm\{tilt\}\}\-\\hat\{z\}\\big\)\}\{\\sigma\_\{t\}^\{2\}\},\\qquad\\nabla\_\{z\_\{t\}\}\\log p\_\{t\}\(z\_\{t\}\)=\\frac\{\\alpha\_\{t\}\\,\\hat\{z\}\-z\_\{t\}\}\{\\sigma\_\{t\}^\{2\}\}\.\(21\)Substituting both expressions into Eq\. \([19](https://arxiv.org/html/2610.22763#S3.E19)\) gives

∇zt​log​pt⋆​\(zt\)=1σt2​\[\(αt​z^−zt\)\+αt​\(z^tilt−z^\)\]=αt​z^tilt−ztσt2\.\\nabla\_\{z\_\{t\}\}\\log p^\{\\star\}\_\{t\}\(z\_\{t\}\)=\\frac\{1\}\{\\sigma\_\{t\}^\{2\}\}\\Big\[\\big\(\\alpha\_\{t\}\\hat\{z\}\-z\_\{t\}\\big\)\+\\alpha\_\{t\}\\big\(\\hat\{z\}^\{\\mathrm\{tilt\}\}\-\\hat\{z\}\\big\)\\Big\]=\\frac\{\\alpha\_\{t\}\\hat\{z\}^\{\\mathrm\{tilt\}\}\-z\_\{t\}\}\{\\sigma\_\{t\}^\{2\}\}\.\(22\)

###### Proof\.

The proof is in Appendix[A\.4](https://arxiv.org/html/2610.22763#A1.SS4)\. ∎

The guided score therefore has exactly the same form as the original score, with the posterior meanz^\\hat\{z\}being replaced by its acquisition\-weighted counterpartz^tilt\\hat\{z\}^\{\\mathrm\{tilt\}\}\. This*tilted mean difference*is a key simplification, as guidance does not require differentiating the acquisition function, but only estimatingz^tilt\\hat\{z\}^\{\\mathrm\{tilt\}\}\.

In general,z^tilt\\hat\{z\}^\{\\mathrm\{tilt\}\}is not available in closed form becauseℓ⁡\(τ⁡\(x\)\)\\ell\(\\tau\(x\)\)need not factorize across sequence positions, so we estimate it from samplesx\(n\)∼p\(⋅∣zt\)x^\{\(n\)\}\\sim p\(\\cdot\\mid z\_\{t\}\)by self\-normalized importance sampling\. ForNNsamples, let

ℓn=ℓ⁡\(τ⁡\(x\(n\)\)\),hn=ℓn∑m=1Nℓm,z^tilt≈∑n=1Nhn​x\(n\)\.\\ell\_\{n\}=\\ell\\big\(\\tau\(x^\{\(n\)\}\)\\big\),\\qquad h\_\{n\}=\\frac\{\\ell\_\{n\}\}\{\\sum\_\{m=1\}^\{N\}\\ell\_\{m\}\},\\qquad\\hat\{z\}^\{\\mathrm\{tilt\}\}\\approx\\sum\_\{n=1\}^\{N\}h\_\{n\}x^\{\(n\)\}\.\(23\)This estimator is consistent \(Appendix[A\.5](https://arxiv.org/html/2610.22763#A1.SS5)\); guidance strength can further be controlled by a scaleλ\\lambda\(see Eq\. \([45](https://arxiv.org/html/2610.22763#A1.E45)\)\) or a temperatureγ\\gammaon the importance weights \(see Eq\. \([46](https://arxiv.org/html/2610.22763#A1.E46)\)\)\.

### 3\.4Two Backbones, One Bridge

#### CDCD\.

CDCD’s forward process in Eq\. \([5](https://arxiv.org/html/2610.22763#S2.E5)\) is the kernel in Eq\. \([15](https://arxiv.org/html/2610.22763#S3.E15)\) with the variance\-exploding scheduleαt≡1\\alpha\_\{t\}\\equiv 1,σt=t\\sigma\_\{t\}=t\([Karras et al\., 2022](https://arxiv.org/html/2610.22763#bib.bib20)\), sopt⋆=p⋆∗𝒩⁡\(0,t2​I\)p^\{\\star\}\_\{t\}=p^\{\\star\}\*\\mathcal\{N\}\(0,t^\{2\}I\)andztz\_\{t\}lives in the embedding space of Eq\. \([4](https://arxiv.org/html/2610.22763#S2.E4)\)\. The network provides both required quantities: the learned score Eq\. \([6](https://arxiv.org/html/2610.22763#S2.E6)\) and the per\-position posterior marginals Eq\. \([7](https://arxiv.org/html/2610.22763#S2.E7)\), from which posterior draws are taken under a mean\-field approximation \(Appendix[B](https://arxiv.org/html/2610.22763#A2)\)\. At sufficiently largeTT, the added noise dominates bothpθp\_\{\\theta\}and the tilt,

pT⋆≈pT≈𝒩⁡\(0,T2​I\),p^\{\\star\}\_\{T\}\\approx p\_\{T\}\\approx\\mathcal\{N\}\(0,T^\{2\}I\),\(24\)so reversing the bridge with the PF\-ODE or SDE driven by Eq\. \([19](https://arxiv.org/html/2610.22763#S3.E19)\) yieldsx∼p⋆x\\sim p^\{\\star\}, which decodes tow=τ⁡\(x\)∼P⋆w=\\tau\(x\)\\sim P^\{\\star\}\.

#### SeqFlow\.

SeqFlow has no built\-in noising process\. Althoughpθ​\(x\)p\_\{\\theta\}\(x\)is tractable, applying Eq\. \([15](https://arxiv.org/html/2610.22763#S3.E15)\) in embedding space makes both the noised marginalptp\_\{t\}and the posteriorp⁡\(x∣zt\)p\(x\\mid z\_\{t\}\)intractable\. We therefore move the bridge to the flow latentz=gθ−1​\(x\)z=g\_\{\\theta\}^\{\-1\}\(x\), whose untilted law is exactly𝒩⁡\(0,I\)\\mathcal\{N\}\(0,I\)by Eq\. \([8](https://arxiv.org/html/2610.22763#S2.E8)\)\. Sincegθg\_\{\\theta\}is a diffeomorphism, transporting the tilted density Eq\. \([13](https://arxiv.org/html/2610.22763#S3.E13)\) to the latent carries the same Jacobian factorJJas transportingpθp\_\{\\theta\}, so the factor cancels and only the tilt survives \(Appendix[A\.3](https://arxiv.org/html/2610.22763#A1.SS3)\),

π⋆​\(z\)≜p⋆​\(gθ​\(z\)\)​\|detJgθ​\(z\)\|=1Z​𝒩​\(z,0,I\)​ℓ​\(τ⁡\(gθ​\(z\)\)\),\\pi^\{\\star\}\(z\)\\triangleq p^\{\\star\}\\big\(g\_\{\\theta\}\(z\)\\big\)\\,\\big\\lvert\\det J\_\{g\_\{\\theta\}\}\(z\)\\big\\rvert=\\frac\{1\}\{Z\}\\,\\mathcal\{N\}\(z;0,I\)\\,\\ell\\big\(\\tau\(g\_\{\\theta\}\(z\)\)\\big\),\(25\)withτ⁡\(gθ​\(z\)\)∼P⋆\\tau\(g\_\{\\theta\}\(z\)\)\\sim P^\{\\star\}forz∼π⋆z\\sim\\pi^\{\\star\}\. As in Section[3\.1](https://arxiv.org/html/2610.22763#S3.SS1), the tilt remains piecewise constant on the cellsAwA\_\{w\}of Eq\. \([8](https://arxiv.org/html/2610.22763#S2.E8)\), so we smooth it using

qt​\(zt∣z\)=𝒩⁡\(zt,αt​z,σt2​I\),q\_\{t\}\(z\_\{t\}\\mid z\)=\\mathcal\{N\}\\big\(z\_\{t\};\\alpha\_\{t\}z,\\sigma\_\{t\}^\{2\}I\\big\),\(26\)with the variance\-preserving scheduleσt=1−αt2\\sigma\_\{t\}=\\sqrt\{1\-\\alpha\_\{t\}^\{2\}\},αt∈\(0,1\]\\alpha\_\{t\}\\in\(0,1\]\. Gaussian conjugacy then gives both required quantities in closed form,

\(i\)​∇zt​log⁡pt​\(zt\)=−zt,\(ii\)​p​\(z∣zt\)=𝒩⁡\(z,αt​zt,\(1−αt2\)​I\)\.\\text\{\(i\)\}\\ \\nabla\_\{z\_\{t\}\}\\log p\_\{t\}\(z\_\{t\}\)=\-z\_\{t\},\\qquad\\text\{\(ii\)\}\\ p\(z\\mid z\_\{t\}\)=\\mathcal\{N\}\\Big\(z;\\alpha\_\{t\}z\_\{t\},\(1\-\\alpha\_\{t\}^\{2\}\)I\\Big\)\.\(27\)The posterior is exact and network\-free; the flow enters only through the tilt, evaluated once per draw and never differentiated\. Thus, Section[3\.3](https://arxiv.org/html/2610.22763#S3.SS3)applies withxxreplaced byzz, while CDCD operates in embedding space\. Table[2](https://arxiv.org/html/2610.22763#A2.T2)summarizes the correspondence between the two backbones\.

### 3\.5Trust Region

Restricting search to a neighborhood of the incumbent, the best design observed so far, is a common strategy for improving the sample efficiency of high\-dimensional BO\([Eriksson et al\., 2019](https://arxiv.org/html/2610.22763#bib.bib5)\)\. In AFGG, we implement this locality through the initialization of the guided sampling trajectories\. Both backbones start sampling from a Gaussian,zT∼𝒩⁡\(0,σ2​I\)z\_\{T\}\\sim\\mathcal\{N\}\(0,\\sigma^\{2\}I\), withσ=T\\sigma=Tfor CDCD \(see Eq\. \([24](https://arxiv.org/html/2610.22763#S3.E24)\)\) andσ=1\\sigma=1for SeqFlow\. Instead, we wish to start closer to an anchorz⋆z^\{\\star\}that represents the incumbent in the same space\. However, sample quality deteriorates if the model must process an input that is unlikely under theDD\-dimensional Gaussian it expects, which concentrates its mass on a shell of radiusσ​D\\sigma\\sqrt\{D\}around the origin\([Blum et al\., 2020](https://arxiv.org/html/2610.22763#bib.bib41), Thm\. 2\.9\)\. We therefore sample

zT∼𝒩⁡\(zT,ρ​z⋆,\(1−ρ2\)​σ2​I\),z\_\{T\}\\sim\\mathcal\{N\}\\big\(z\_\{T\};\\,\\rho z^\{\\star\},\\,\(1\-\\rho^\{2\}\)\\,\\sigma^\{2\}I\\big\),\(28\)whereρ∈\[0,1\]\\rho\\in\[0,1\]controls how tightly the trust region concentrates aroundz⋆z^\{\\star\}:ρ=0\\rho=0recovers the unconditional initialization andρ=1\\rho=1starts every trajectory atz⋆z^\{\\star\}\. Since𝔼⁡\[∥zT∥2∣z⋆\]=ρ2​∥z⋆∥2\+\(1−ρ2\)​σ2​D\\mathbb\{E\}\[\\lVert z\_\{T\}\\rVert^\{2\}\\mid z^\{\\star\}\]=\\rho^\{2\}\\lVert z^\{\\star\}\\rVert^\{2\}\+\(1\-\\rho^\{2\}\)\\,\\sigma^\{2\}D, an anchor on the shell,∥z⋆∥2=σ2​D\\lVert z^\{\\star\}\\rVert^\{2\}=\\sigma^\{2\}D, gives𝔼⁡\[∥zT∥2∣z⋆\]=σ2​D\\mathbb\{E\}\[\\lVert z\_\{T\}\\rVert^\{2\}\\mid z^\{\\star\}\]=\\sigma^\{2\}Dfor everyρ\\rho, so the initializations stay near the shell\.

The complete AFGG optimization loop is described in Appendix[D](https://arxiv.org/html/2610.22763#A4)\.

## 4Experiments

We evaluate AFGG on the 25 molecular optimization objectives of[Gao et al\. \(2022\)](https://arxiv.org/html/2610.22763#bib.bib29)and[González\-Duque et al\. \(2024\)](https://arxiv.org/html/2610.22763#bib.bib9), following the small\-data protocol of[Moss et al\. \(2025\)](https://arxiv.org/html/2610.22763#bib.bib12)\. We consider both a*sequential setting*, with 10 initial observations followed by 300 oracle evaluations, one per BO iteration, and a*batch setting*, with 100 initial observations followed by 500 oracle evaluations in batches of five\. We instantiate AFGG with two different generative backbones, SeqFlow\([Lee et al\., 2025](https://arxiv.org/html/2610.22763#bib.bib35)\)and CDCD\([Dieleman et al\., 2022](https://arxiv.org/html/2610.22763#bib.bib25)\), as described in Section[3\.4](https://arxiv.org/html/2610.22763#S3.SS4), allowing us to test the same acquisition\-guided construction with two substantially different generative models\. For SeqFlow, we use the released GuacaMol\-trained checkpoint; since no pretrained CDCD checkpoint was available, we train one on the same dataset \(Appendix[B\.1](https://arxiv.org/html/2610.22763#A2.SS1)\)\.

We compare against five strong recent methods that are particularly relevant to our setting: CoBO\([Lee et al\., 2023](https://arxiv.org/html/2610.22763#bib.bib36)\), LOL\-BO\([Maus et al\., 2022](https://arxiv.org/html/2610.22763#bib.bib31)\), COWBOYS\([Moss et al\., 2025](https://arxiv.org/html/2610.22763#bib.bib12)\), InvBO\([Chu et al\., 2024](https://arxiv.org/html/2610.22763#bib.bib32)\), and NFBO\([Lee et al\., 2025](https://arxiv.org/html/2610.22763#bib.bib35)\)\. These include both the closest prior approach, COWBOYS, and NFBO, which uses the same SeqFlow generative backbone as one of our AFGG variants\. For the sequential setting, Appendix[E\.1](https://arxiv.org/html/2610.22763#A5.SS1)\(Table[6](https://arxiv.org/html/2610.22763#A5.T6)\) broadens the comparison to 11 baselines, additionally including CMA\-ES, differential evolution \(DE\), genetic algorithms \(GA\), hill climbing \(HC\), RLBO\([Kirschner et al\., 2019](https://arxiv.org/html/2610.22763#bib.bib46)\), and TuRBO\([Eriksson et al\., 2019](https://arxiv.org/html/2610.22763#bib.bib5)\)\. Unless stated otherwise, AFGG uses EI with the temperature tilt of Eq\. \([46](https://arxiv.org/html/2610.22763#A1.E46)\), which performs best in our ablations \(Appendix[E\.3](https://arxiv.org/html/2610.22763#A5.SS3)\)\.

Figure 1:Sequential setting\.Optimization trajectories on six representative objectives, with 10 initial observations followed by 300 oracle evaluations, one per BO iteration\. Curves show the median over 5 seeds\.Table 1:Sequential benchmark \(partial results\)\.Final best score on all 25 PMO objectives after 300 oracle calls \(10 initial points, 1 point per BO iteration\), mean±\\pmsample standard deviation over 5 seeds\. For readability the table shows five of the eleven baselines; Table[6](https://arxiv.org/html/2610.22763#A5.T6)gives the full comparison\. AFGG is shown in two instantiations:CDCDandSeqFlow\. Every cell is judged against the opposing group\.Bold: the cell’s mean exceeds every mean in the opposing group\.Green, shaded: lead exceeds the cell’s own sample standard deviation; bold without shading denotes a smaller lead\. Objective abbreviations are defined in Table[5](https://arxiv.org/html/2610.22763#A5.T5)\.Figure 2:Batch setting, uncoupled generation\.Optimization trajectories on the same six representative objectives, with 100 initial observations followed by 500 oracle evaluations in batches of five\. Curves show the median over 5 seeds\.Fig\.[1](https://arxiv.org/html/2610.22763#S4.F1)shows the optimization trajectories on six representative objectives in the sequential setting\. AFGG with SeqFlow makes particularly strong gains on several of these tasks, while the CDCD variant remains broadly competitive with the leading BO baselines\. To verify that this pattern is not specific to the selected objectives, Table[1](https://arxiv.org/html/2610.22763#S4.T1)reports the final performance on all 25 objectives against the five closest BO baselines\.

Fig\.[2](https://arxiv.org/html/2610.22763#S4.F2)repeats the comparison in the batch setting on the same six representative objectives\. The advantage of the SeqFlow variant persists whenq=5q=5candidates are selected per BO round\. Appendix[E\.1](https://arxiv.org/html/2610.22763#A5.SS1)reports results on all 25 objectives for both uncoupled and coupled batch constructions\.

Across both settings, the two AFGG backbones show a clear and consistent pattern\. AFGG with CDCD performs on par with the strongest BO baselines, whereas AFGG with SeqFlow achieves the best result on most objectives and remains competitive on the others\. The same ordering across the sequential and batch settings suggests that the stronger generative backbone translates into substantially stronger optimization performance\.Taken together, these results demonstrate state\-of\-the\-art performance for AFGG with SeqFlow across this benchmark suite: it achieves the best result on most objectives and remains close to the strongest baseline on the others\.

Additional analyses in Appendix[E\.2](https://arxiv.org/html/2610.22763#A5.SS2)isolate the contribution of the generative backbone and relate optimization performance to the quality of its unconditional samples\.

## 5Related Work

AFGG is related to*inference\-time alignment*of pretrained generative models, which likewise samples from a distribution obtained by tilting a generative model toward a reward\([Krishnamoorthy et al\., 2023](https://arxiv.org/html/2610.22763#bib.bib27);[Li et al\., 2025](https://arxiv.org/html/2610.22763#bib.bib39);[Uehara et al\., 2024](https://arxiv.org/html/2610.22763#bib.bib37);[Uehara et al\., 2025](https://arxiv.org/html/2610.22763#bib.bib38)\)\. The key difference is that inference\-time alignment typically uses a fixed reward, whereas in Bayesian optimization the acquisition function changes after every new observation\. AFGG therefore applies the same general idea of guided generation in a setting where the tilt is continually updated by the surrogate\.

Another line of related work is*active generation*, where the generative model is adapted repeatedly as new data arrive, for example by conditioning or variationally fitting it to an acquisition\-weighted target\([Brookes et al\., 2019](https://arxiv.org/html/2610.22763#bib.bib33);[Steinberg et al\., 2025](https://arxiv.org/html/2610.22763#bib.bib11);[Yun et al\., 2025](https://arxiv.org/html/2610.22763#bib.bib34);[Oliveira et al\., 2026](https://arxiv.org/html/2610.22763#bib.bib10)\); the same approach can also be applied to infilling problems, where the generated sequence has fixed length\([Stanton et al\., 2022](https://arxiv.org/html/2610.22763#bib.bib24);[Gruver et al\., 2023](https://arxiv.org/html/2610.22763#bib.bib23)\)\. AFGG takes a different route: the pretrained generator remains frozen, while only the surrogate and acquisition function are updated during BO\. This allows strong pretrained generative models to be reused throughout optimization without repeated retraining or fine\-tuning\.

Most generative BO methods place the surrogate in the learned latent space\([Tripp et al\., 2020](https://arxiv.org/html/2610.22763#bib.bib30);[Maus et al\., 2022](https://arxiv.org/html/2610.22763#bib.bib31);[Lee et al\., 2023](https://arxiv.org/html/2610.22763#bib.bib36);[Chu et al\., 2024](https://arxiv.org/html/2610.22763#bib.bib32);[Lee et al\., 2025](https://arxiv.org/html/2610.22763#bib.bib35)\), which makes inductive biases defined naturally on the original designs harder to use directly\. To the best of our knowledge, COWBOYS\([Moss et al\., 2025](https://arxiv.org/html/2610.22763#bib.bib12)\)is the only prior BO method that similarly combines a frozen generative model with a surrogate defined in data space\. It is therefore the closest precursor to AFGG, but its construction is tied to a VAE and to probability of improvement\. In contrast, AFGG only requires a generative model that supports guided sampling and a non\-negative acquisition function that can be evaluated pointwise\. The acquisition function need neither be differentiable nor define a probability, allowing, for example, EI to be used directly\.

## 6Conclusion

We introduced AFGG, which replaces acquisition\-function maximization with sampling from an acquisition\-tilted generative distribution\. A diffusion bridge makes this target amenable to guided sampling while leaving the surrogate and acquisition function in data space\. The generator remains frozen throughout BO, and the acquisition function only needs to be non\-negative and evaluable pointwise\. On the molecular benchmarks studied here, AFGG with CDCD is competitive with strong recent BO methods, while AFGG with SeqFlow achieves the best result on most objectives and remains close to the strongest baselines on the rest\. The gap between the two variants further shows that the quality of the generative backbone matters directly for optimization performance\. This makes stronger pretrained generators a natural next target for AFGG\. In particular, protein foundation models\([Abramson et al\., 2024](https://arxiv.org/html/2610.22763#bib.bib43);[Passaro et al\., 2025](https://arxiv.org/html/2610.22763#bib.bib44);[Watson et al\., 2023](https://arxiv.org/html/2610.22763#bib.bib42)\)could potentially serve as frozen backbones\. Applying AFGG to such models will mainly require efficient approximations to the posteriorp⁡\(x∣zt\)p\(x\\mid z\_\{t\}\)used for guidance\.

## 7AI Use Statement

Large language models were used as an assistive tool during the preparation of this work, under author supervision, in three roles:

- •Writing\.Copy\-editing of author\-written text for clarity and concision, and suggestions for reorganizing existing material\. No section was drafted from scratch by a model; all technical claims, derivations and experimental conclusions are the authors’\.
- •Related work\.Surfacing candidate references and drafting initial summaries of them\. Every cited work was subsequently read by the authors, and all attributions and comparisons were verified against the primary sources\.
- •Code\.Inspecting the released implementations of the baselines\. Standardizing the implementation and evaluation protocol across methods required modifications to the original code, and an AI assistant was used to help locate and implement them\. Every deviation from an original implementation is documented, with the corresponding line numbers\.

The authors take full responsibility for the entire content of this paper, including any text, code or references produced with AI assistance\.

## 8Reproducibility Statement

An anonymized implementation is available at[https://anonymous\.4open\.science/r/afgg](https://anonymous.4open.science/r/afgg)\. It covers the guided samplers of Algorithm[1](https://arxiv.org/html/2610.22763#alg1)and Algorithm[2](https://arxiv.org/html/2610.22763#alg2), the BO loop, and our modified baseline code, together with the configuration files and random seeds needed to reproduce every table and figure reported here\. The guidance construction is stated in full in Section[3](https://arxiv.org/html/2610.22763#S3)and proved in Appendix[A](https://arxiv.org/html/2610.22763#A1)\. Hyperparameters and training details for the CDCD generator are given in Table[4](https://arxiv.org/html/2610.22763#A2.T4)and Appendix[B\.1](https://arxiv.org/html/2610.22763#A2.SS1); the SeqFlow backbone is the pretrained model of[Lee et al\. \(2025\)](https://arxiv.org/html/2610.22763#bib.bib35), used without modification\.

## 9Acknowledgments

This work was supported by the Austrian Science Fund \(FWF\) through the doc\.funds “BiotechPredict” \(10\.55776/DOC2686925\) and the Cluster of Excellence “Bilateral AI” \(10\.55776/COE12\)\. Additional support was provided by the TU Graz Leadprojekt “DigiBioTech” and Lancaster University’s Mathematics for AI in Real\-world Systems E3 grant\.

## References

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## Appendix AProofs

### A\.1Tilting Commutes with Decoding

###### Proposition 3\(Lifting\)\.

Letp⋆p^\{\\star\}be the tilted density Eq\. \([13](https://arxiv.org/html/2610.22763#S3.E13)\) andP⋆P^\{\\star\}the tilted target of Definition[3\.1](https://arxiv.org/html/2610.22763#S3.Thmdefinition1)\. Then a drawx∼p⋆x\\sim p^\{\\star\}returnsτ⁡\(x\)∼P⋆\\tau\(x\)\\sim P^\{\\star\}\.

###### Proof\.

WriteZ=∫ℝL×dpθ​\(x\)​ℓ​\(τ⁡\(x\)\)​𝑑xZ=\\int\_\{\\mathbb\{R\}^\{L\\times d\}\}p\_\{\\theta\}\(x\)\\,\\ell\(\\tau\(x\)\)\\,\\mathrm\{d\}xfor the normalizing constant ofp⋆p^\{\\star\}\. On the cellτ−1​\(w\)\\tau^\{\-1\}\(w\)the decode map is constant by definition,τ⁡\(x\)=w\\tau\(x\)=w, soℓ⁡\(τ⁡\(x\)\)=ℓ⁡\(w\)\\ell\(\\tau\(x\)\)=\\ell\(w\)factors out of the integral:

Prx∼p⋆\[τ\(x\)=w\]=∫τ−1​\(w\)p⋆\(x\)dx=1Z∫τ−1​\(w\)pθ\(x\)ℓ\(τ\(x\)\)dx\\displaystyle\\Pr\_\{x\\sim p^\{\\star\}\}\\big\[\\tau\(x\)=w\\big\]=\\int\_\{\\tau^\{\-1\}\(w\)\}p^\{\\star\}\(x\)\\,\\mathrm\{d\}x=\\frac\{1\}\{Z\}\\int\_\{\\tau^\{\-1\}\(w\)\}p\_\{\\theta\}\(x\)\\,\\ell\(\\tau\(x\)\)\\,\\mathrm\{d\}x\(29\)=ℓ⁡\(w\)Z​∫τ−1​\(w\)pθ​\(x\)​dx=Pθ​\(w\)​ℓ​\(w\)Z\.\\displaystyle=\\frac\{\\ell\(w\)\}\{Z\}\\int\_\{\\tau^\{\-1\}\(w\)\}p\_\{\\theta\}\(x\)\\,\\mathrm\{d\}x=\\frac\{P\_\{\\theta\}\(w\)\\,\\ell\(w\)\}\{Z\}\.The same decomposition identifies the constant\. Since the cells\{τ−1​\(w\)\}w∈𝒳\\\{\\tau^\{\-1\}\(w\)\\\}\_\{w\\in\{\\mathcal\{X\}\}\}partitionℝL×d\\mathbb\{R\}^\{L\\times d\},

Z=∑w∈𝒳∫τ−1​\(w\)pθ​\(x\)​ℓ​\(τ⁡\(x\)\)​𝑑x=∑w∈𝒳Pθ​\(w\)​ℓ​\(w\),Z=\\sum\_\{w\\in\{\\mathcal\{X\}\}\}\\int\_\{\\tau^\{\-1\}\(w\)\}p\_\{\\theta\}\(x\)\\,\\ell\(\\tau\(x\)\)\\,\\mathrm\{d\}x=\\sum\_\{w\\in\{\\mathcal\{X\}\}\}P\_\{\\theta\}\(w\)\\,\\ell\(w\),\(30\)which is exactly the normalizing constant ofP⋆P^\{\\star\}in Eq\. \([11](https://arxiv.org/html/2610.22763#S3.E11)\)\. Henceτ⁡\(x\)∼P⋆\\tau\(x\)\\sim P^\{\\star\}forx∼p⋆x\\sim p^\{\\star\}\. ∎

Note that only the constancy ofℓ∘τ\\ell\\circ\\tauon cells was used; no property ofpθp\_\{\\theta\}and no regularity ofτ\\taubeyond measurability is required\. This is what licenses tilting in the continuous space and decoding afterwards, and for SeqFlow it is applied a second time withτ\\taureplaced byτ∘gθ\\tau\\circ g\_\{\\theta\}\(Appendix[A\.3](https://arxiv.org/html/2610.22763#A1.SS3)\)\.

### A\.2Proof of Proposition[1](https://arxiv.org/html/2610.22763#Thmproposition1)

#### Statement\.

Withℓt​\(zt\)≜𝔼p⁡\(x∣zt\)​\[ℓ⁡\(τ⁡\(x\)\)\]\\ell\_\{t\}\(z\_\{t\}\)\\triangleq\\mathbb\{E\}\_\{p\(x\\mid z\_\{t\}\)\}\\big\[\\ell\(\\tau\(x\)\)\\big\], Proposition[1](https://arxiv.org/html/2610.22763#Thmproposition1)asserts that the tilted marginal and the tilted posterior factor as

pt⋆​\(zt\)∝pt​\(zt\)​ℓt​\(zt\),p⋆​\(x∣zt\)∝p⁡\(x∣zt\)​ℓ​\(τ⁡\(x\)\),p^\{\\star\}\_\{t\}\(z\_\{t\}\)\\;\\propto\\;p\_\{t\}\(z\_\{t\}\)\\,\\ell\_\{t\}\(z\_\{t\}\),\\qquad p^\{\\star\}\(x\\mid z\_\{t\}\)\\;\\propto\\;p\(x\\mid z\_\{t\}\)\\,\\ell\(\\tau\(x\)\),the first being Eq\. \([17](https://arxiv.org/html/2610.22763#S3.E17)\)\.

###### Proof\.

Expandpt⋆p^\{\\star\}\_\{t\}using Definition[3\.1](https://arxiv.org/html/2610.22763#S3.Thmdefinition1)and Definition[3\.2](https://arxiv.org/html/2610.22763#S3.Thmdefinition2):

pt⋆​\(zt\)=∫qt​\(zt∣x\)​p⋆​\(x\)​𝑑x∝∫qt​\(zt∣x\)​pθ​\(x\)​ℓ​\(τ⁡\(x\)\)​𝑑x\.p^\{\\star\}\_\{t\}\(z\_\{t\}\)=\\int q\_\{t\}\(z\_\{t\}\\mid x\)\\,p^\{\\star\}\(x\)\\,\\mathrm\{d\}x\\propto\\int q\_\{t\}\(z\_\{t\}\\mid x\)\\,p\_\{\\theta\}\(x\)\\,\\ell\(\\tau\(x\)\)\\,\\mathrm\{d\}x\.\(31\)By Bayes’ rule,qt​\(zt∣x\)​pθ​\(x\)=p⁡\(x∣zt\)​pt​\(zt\)q\_\{t\}\(z\_\{t\}\\mid x\)\\,p\_\{\\theta\}\(x\)=p\(x\\mid z\_\{t\}\)\\,p\_\{t\}\(z\_\{t\}\), so

pt⋆​\(zt\)∝pt​\(zt\)​∫p⁡\(x∣zt\)​ℓ​\(τ⁡\(x\)\)​𝑑x=pt​\(zt\)​ℓt​\(zt\),p^\{\\star\}\_\{t\}\(z\_\{t\}\)\\propto p\_\{t\}\(z\_\{t\}\)\\int p\(x\\mid z\_\{t\}\)\\,\\ell\(\\tau\(x\)\)\\,\\mathrm\{d\}x=p\_\{t\}\(z\_\{t\}\)\\,\\ell\_\{t\}\(z\_\{t\}\),\(32\)withℓt​\(zt\)=𝔼p⁡\(x∣zt\)​\[ℓ⁡\(τ⁡\(x\)\)\]\\ell\_\{t\}\(z\_\{t\}\)=\\mathbb\{E\}\_\{p\(x\\mid z\_\{t\}\)\}\[\\ell\(\\tau\(x\)\)\], which is Eq\. \([17](https://arxiv.org/html/2610.22763#S3.E17)\)\.

For the second claim, apply Bayes’ rule to the tilted pair\(p⋆,qt\)\\big\(p^\{\\star\},q\_\{t\}\\big\)and substitute both expansions above:

p⋆​\(x∣zt\)=p⋆​\(x\)​qt​\(zt∣x\)pt⋆​\(zt\)∝pθ​\(x\)​ℓ​\(τ⁡\(x\)\)​qt​\(zt∣x\)pt​\(zt\)​ℓt​\(zt\)=pθ​\(x\)​qt​\(zt∣x\)pt​\(zt\)⏟=p⁡\(x∣zt\)⋅ℓ⁡\(τ⁡\(x\)\)ℓt​\(zt\)\.p^\{\\star\}\(x\\mid z\_\{t\}\)=\\frac\{p^\{\\star\}\(x\)\\,q\_\{t\}\(z\_\{t\}\\mid x\)\}\{p^\{\\star\}\_\{t\}\(z\_\{t\}\)\}\\propto\\frac\{p\_\{\\theta\}\(x\)\\,\\ell\(\\tau\(x\)\)\\,q\_\{t\}\(z\_\{t\}\\mid x\)\}\{p\_\{t\}\(z\_\{t\}\)\\,\\ell\_\{t\}\(z\_\{t\}\)\}=\\underbrace\{\\frac\{p\_\{\\theta\}\(x\)\\,q\_\{t\}\(z\_\{t\}\\mid x\)\}\{p\_\{t\}\(z\_\{t\}\)\}\}\_\{=\\;p\(x\\mid z\_\{t\}\)\}\\cdot\\frac\{\\ell\(\\tau\(x\)\)\}\{\\ell\_\{t\}\(z\_\{t\}\)\}\.\(33\)Sinceℓt​\(zt\)\\ell\_\{t\}\(z\_\{t\}\)does not depend onxx, it is absorbed into the normalizing constant, givingp⋆​\(x∣zt\)∝p⁡\(x∣zt\)​ℓ​\(τ⁡\(x\)\)p^\{\\star\}\(x\\mid z\_\{t\}\)\\propto p\(x\\mid z\_\{t\}\)\\,\\ell\(\\tau\(x\)\), which is Eq\. \([18](https://arxiv.org/html/2610.22763#S3.E18)\)\. ∎

### A\.3Change of Variables to the Flow Latent

Section[3\.4](https://arxiv.org/html/2610.22763#S3.SS4)runs SeqFlow in its latentz=gθ−1​\(x\)z=g\_\{\\theta\}^\{\-1\}\(x\)rather than inxx\. This requires two facts: the tilted target Eq\. \([13](https://arxiv.org/html/2610.22763#S3.E13)\), pushed throughgθ−1g\_\{\\theta\}^\{\-1\}, is again a Gaussian tilted by the acquisition, now composed with the decoder, Eq\. \([25](https://arxiv.org/html/2610.22763#S3.E25)\); and decoding a draw of it recoversP⋆P^\{\\star\}\. Both follow from the change\-of\-variables formula, because the Jacobian of the flow enters the tilted and the untilted density in the same way\.

###### Proposition 4\(Flow\-latent tilt\)\.

Letgθg\_\{\\theta\}be a diffeomorphism ofℝL×d\\mathbb\{R\}^\{L\\times d\}such thatz∼𝒩⁡\(0,I\)z\\sim\\mathcal\{N\}\(0,I\)impliesgθ​\(z\)∼pθg\_\{\\theta\}\(z\)\\sim p\_\{\\theta\}, as in Eq\. \([8](https://arxiv.org/html/2610.22763#S2.E8)\), and letπ⋆​\(z\)≜p⋆​\(gθ​\(z\)\)​\|J⁡\(z\)\|\\pi^\{\\star\}\(z\)\\triangleq p^\{\\star\}\\big\(g\_\{\\theta\}\(z\)\\big\)\\,\\lvert J\(z\)\\rvert, withJ⁡\(z\)J\(z\)the Jacobian determinant ofgθg\_\{\\theta\}atzz\. Then

π⋆​\(z\)=1Z​𝒩​\(z,0,I\)​ℓ​\(τ⁡\(gθ​\(z\)\)\),\\pi^\{\\star\}\(z\)=\\frac\{1\}\{Z\}\\,\\mathcal\{N\}\(z;0,I\)\\;\\ell\\big\(\\tau\(g\_\{\\theta\}\(z\)\)\\big\),which is Eq\. \([25](https://arxiv.org/html/2610.22763#S3.E25)\), andz∼π⋆z\\sim\\pi^\{\\star\}impliesτ⁡\(gθ​\(z\)\)∼P⋆\\tau\(g\_\{\\theta\}\(z\)\)\\sim P^\{\\star\}\.

###### Proof\.

Sincegθg\_\{\\theta\}is a diffeomorphism, the change\-of\-variables formula applies top⋆p^\{\\star\}and topθp\_\{\\theta\}with the*same*Jacobian factor\. WritingJ⁡\(z\)J\(z\)for the Jacobian determinant ofgθg\_\{\\theta\}atzz,

π⋆​\(z\)=p⋆​\(gθ​\(z\)\)​\|J⁡\(z\)\|,𝒩⁡\(z,0,I\)=pθ​\(gθ​\(z\)\)​\|J⁡\(z\)\|,\\pi^\{\\star\}\(z\)=p^\{\\star\}\\big\(g\_\{\\theta\}\(z\)\\big\)\\,\\lvert J\(z\)\\rvert,\\qquad\\mathcal\{N\}\(z;0,I\)=p\_\{\\theta\}\\big\(g\_\{\\theta\}\(z\)\\big\)\\,\\lvert J\(z\)\\rvert,\(34\)the second becausegθg\_\{\\theta\}transports𝒩⁡\(0,I\)\\mathcal\{N\}\(0,I\)topθp\_\{\\theta\}, as in Eq\. \([8](https://arxiv.org/html/2610.22763#S2.E8)\)\. Substitutingp⋆=1Z​pθ⋅\(ℓ∘τ\)p^\{\\star\}=\\tfrac\{1\}\{Z\}\\,p\_\{\\theta\}\\cdot\(\\ell\\circ\\tau\)from Eq\. \([13](https://arxiv.org/html/2610.22763#S3.E13)\) into the first and recognizing the second,

π⋆​\(z\)=1Z​pθ​\(gθ​\(z\)\)​\|J⁡\(z\)\|⏟=𝒩⁡\(z,0,I\)​ℓ​\(τ⁡\(gθ​\(z\)\)\),\\pi^\{\\star\}\(z\)=\\frac\{1\}\{Z\}\\,\\underbrace\{p\_\{\\theta\}\\big\(g\_\{\\theta\}\(z\)\\big\)\\,\\lvert J\(z\)\\rvert\}\_\{=\\;\\mathcal\{N\}\(z;0,I\)\}\\;\\ell\\big\(\\tau\(g\_\{\\theta\}\(z\)\)\\big\),\(35\)which is Eq\. \([25](https://arxiv.org/html/2610.22763#S3.E25)\): the Jacobian is common to both densities and cancels in the ratioπ⋆/𝒩⁡\(0,I\)=\(p⋆/pθ\)∘gθ\\pi^\{\\star\}/\\mathcal\{N\}\(0,I\)=\(p^\{\\star\}/p\_\{\\theta\}\)\\circ g\_\{\\theta\}\. Finally,z∼π⋆z\\sim\\pi^\{\\star\}givesgθ​\(z\)∼p⋆g\_\{\\theta\}\(z\)\\sim p^\{\\star\}by the first identity of Eq\. \([34](https://arxiv.org/html/2610.22763#A1.E34)\), soτ⁡\(gθ​\(z\)\)∼P⋆\\tau\(g\_\{\\theta\}\(z\)\)\\sim P^\{\\star\}by Appendix[A\.1](https://arxiv.org/html/2610.22763#A1.SS1)\. ∎

### A\.4Proof of Proposition[2](https://arxiv.org/html/2610.22763#Thmproposition2)

#### Statement\.

For the Gaussian kernel Eq\. \([15](https://arxiv.org/html/2610.22763#S3.E15)\) withαt,σt\>0\\alpha\_\{t\},\\sigma\_\{t\}\>0, and withz^,z^tilt\\hat\{z\},\\hat\{z\}^\{\\mathrm\{tilt\}\}as in Definition[3\.3](https://arxiv.org/html/2610.22763#S3.Thmdefinition3), Proposition[2](https://arxiv.org/html/2610.22763#Thmproposition2)asserts

∇zt​log​ℓt​\(zt\)=αt​\(z^tilt−z^\)σt2,∇zt​log​pt​\(zt\)=αt​z^−ztσt2,\\nabla\_\{z\_\{t\}\}\\log\\ell\_\{t\}\(z\_\{t\}\)=\\frac\{\\alpha\_\{t\}\\big\(\\hat\{z\}^\{\\mathrm\{tilt\}\}\-\\hat\{z\}\\big\)\}\{\\sigma\_\{t\}^\{2\}\},\\qquad\\nabla\_\{z\_\{t\}\}\\log p\_\{t\}\(z\_\{t\}\)=\\frac\{\\alpha\_\{t\}\\,\\hat\{z\}\-z\_\{t\}\}\{\\sigma\_\{t\}^\{2\}\},so that the guided score Eq\. \([19](https://arxiv.org/html/2610.22763#S3.E19)\) is\(αt​z^tilt−zt\)/σt2\\big\(\\alpha\_\{t\}\\hat\{z\}^\{\\mathrm\{tilt\}\}\-z\_\{t\}\\big\)/\\sigma\_\{t\}^\{2\}, i\.e\. Eq\. \([21](https://arxiv.org/html/2610.22763#S3.E21)\) and Eq\. \([22](https://arxiv.org/html/2610.22763#S3.E22)\)\.

###### Proof\.

We differentiatelog⁡ℓt\\log\\ell\_\{t\}directly and exhibit the two means whoseztz\_\{t\}cancels\. Write

I⁡\(zt\)≜∫ℓ⁡\(τ⁡\(x\)\)​qt​\(zt∣x\)​pθ​\(x\)​𝑑x\.I\(z\_\{t\}\)\\triangleq\\int\\ell\(\\tau\(x\)\)\\,q\_\{t\}\(z\_\{t\}\\mid x\)\\,p\_\{\\theta\}\(x\)\\,\\mathrm\{d\}x\.\(36\)By Bayes’ rule,p⁡\(x∣zt\)=qt​\(zt∣x\)​pθ​\(x\)/pt​\(zt\)p\(x\\mid z\_\{t\}\)=q\_\{t\}\(z\_\{t\}\\mid x\)\\,p\_\{\\theta\}\(x\)/p\_\{t\}\(z\_\{t\}\), so the definition ofℓt\\ell\_\{t\}in Proposition[1](https://arxiv.org/html/2610.22763#Thmproposition1)reads

ℓt​\(zt\)=∫ℓ⁡\(τ⁡\(x\)\)​p​\(x∣zt\)​𝑑x=1pt​\(zt\)​∫ℓ⁡\(τ⁡\(x\)\)​qt​\(zt∣x\)​pθ​\(x\)​𝑑x=I⁡\(zt\)pt​\(zt\),\\ell\_\{t\}\(z\_\{t\}\)=\\int\\ell\(\\tau\(x\)\)\\,p\(x\\mid z\_\{t\}\)\\,\\mathrm\{d\}x=\\frac\{1\}\{p\_\{t\}\(z\_\{t\}\)\}\\int\\ell\(\\tau\(x\)\)\\,q\_\{t\}\(z\_\{t\}\\mid x\)\\,p\_\{\\theta\}\(x\)\\,\\mathrm\{d\}x=\\frac\{I\(z\_\{t\}\)\}\{p\_\{t\}\(z\_\{t\}\)\},\(37\)and therefore

∇zt​log​ℓt​\(zt\)=∇zt​log​I​\(zt\)−∇zt​log​pt​\(zt\)\.\\nabla\_\{z\_\{t\}\}\\log\\ell\_\{t\}\(z\_\{t\}\)=\\nabla\_\{z\_\{t\}\}\\log I\(z\_\{t\}\)\-\\nabla\_\{z\_\{t\}\}\\log p\_\{t\}\(z\_\{t\}\)\.\(38\)Sinceqtq\_\{t\}is the Gaussian kernel Eq\. \([15](https://arxiv.org/html/2610.22763#S3.E15)\),∇zt​log​qt​\(zt∣x\)=\(αt​x−zt\)/σt2\\nabla\_\{z\_\{t\}\}\\log q\_\{t\}\(z\_\{t\}\\mid x\)=\(\\alpha\_\{t\}\\,x\-z\_\{t\}\)/\\sigma\_\{t\}^\{2\}, so by the log\-derivative rule∇ztqt=qt​\(αt​x−zt\)/σt2\\nabla\_\{z\_\{t\}\}q\_\{t\}=q\_\{t\}\\,\(\\alpha\_\{t\}\\,x\-z\_\{t\}\)/\\sigma\_\{t\}^\{2\}\. Consequently

∇zt​log​I​\(zt\)\\displaystyle\\nabla\_\{z\_\{t\}\}\\log I\(z\_\{t\}\)=∇ztI​\(zt\)I⁡\(zt\)=∫ℓ⁡\(τ⁡\(x\)\)​αt​x−ztσt2​qt​\(zt∣x\)​pθ​\(x\)​𝑑x∫ℓ⁡\(τ⁡\(x\)\)​qt​\(zt∣x\)​pθ​\(x\)​𝑑x\\displaystyle=\\frac\{\\nabla\_\{z\_\{t\}\}I\(z\_\{t\}\)\}\{I\(z\_\{t\}\)\}=\\frac\{\\displaystyle\\int\\ell\(\\tau\(x\)\)\\,\\frac\{\\alpha\_\{t\}\\,x\-z\_\{t\}\}\{\\sigma\_\{t\}^\{2\}\}\\,q\_\{t\}\(z\_\{t\}\\mid x\)\\,p\_\{\\theta\}\(x\)\\,\\mathrm\{d\}x\}\{\\displaystyle\\int\\ell\(\\tau\(x\)\)\\,q\_\{t\}\(z\_\{t\}\\mid x\)\\,p\_\{\\theta\}\(x\)\\,\\mathrm\{d\}x\}\(39\)=∫ℓ⁡\(τ⁡\(x\)\)​αt​x−ztσt2​p​\(x∣zt\)​𝑑x∫ℓ⁡\(τ⁡\(x\)\)​p​\(x∣zt\)​𝑑x\\displaystyle=\\frac\{\\displaystyle\\int\\ell\(\\tau\(x\)\)\\,\\frac\{\\alpha\_\{t\}\\,x\-z\_\{t\}\}\{\\sigma\_\{t\}^\{2\}\}\\,p\(x\\mid z\_\{t\}\)\\,\\mathrm\{d\}x\}\{\\displaystyle\\int\\ell\(\\tau\(x\)\)\\,p\(x\\mid z\_\{t\}\)\\,\\mathrm\{d\}x\}=1σt2⋅αt​∫x​ℓ​\(τ⁡\(x\)\)​p​\(x∣zt\)​𝑑x−zt​ℓt​\(zt\)ℓt​\(zt\)\\displaystyle=\\frac\{1\}\{\\sigma\_\{t\}^\{2\}\}\\cdot\\frac\{\\alpha\_\{t\}\\displaystyle\\int x\\,\\ell\(\\tau\(x\)\)\\,p\(x\\mid z\_\{t\}\)\\,\\mathrm\{d\}x\\;\-\\;z\_\{t\}\\,\\ell\_\{t\}\(z\_\{t\}\)\}\{\\ell\_\{t\}\(z\_\{t\}\)\}=1σt2​\(αt​∫x​ℓ⁡\(τ⁡\(x\)\)​p​\(x∣zt\)ℓt​\(zt\)​dx−zt\)\\displaystyle=\\frac\{1\}\{\\sigma\_\{t\}^\{2\}\}\\left\(\\alpha\_\{t\}\\int x\\,\\frac\{\\ell\(\\tau\(x\)\)\\,p\(x\\mid z\_\{t\}\)\}\{\\ell\_\{t\}\(z\_\{t\}\)\}\\,\\mathrm\{d\}x\\;\-\\;z\_\{t\}\\right\)=1σt2​\(αt​∫x​p⋆​\(x∣zt\)​dx−zt\)=αt​z^tilt−ztσt2\.\\displaystyle=\\frac\{1\}\{\\sigma\_\{t\}^\{2\}\}\\left\(\\alpha\_\{t\}\\int x\\,p^\{\\star\}\(x\\mid z\_\{t\}\)\\,\\mathrm\{d\}x\\;\-\\;z\_\{t\}\\right\)=\\frac\{\\alpha\_\{t\}\\,\\hat\{z\}^\{\\mathrm\{tilt\}\}\-z\_\{t\}\}\{\\sigma\_\{t\}^\{2\}\}\.
The second term of Eq\. \([38](https://arxiv.org/html/2610.22763#A1.E38)\) follows from the same steps applied topt​\(zt\)=∫qt​\(zt∣x\)​pθ​\(x\)​𝑑xp\_\{t\}\(z\_\{t\}\)=\\int q\_\{t\}\(z\_\{t\}\\mid x\)\\,p\_\{\\theta\}\(x\)\\,\\mathrm\{d\}x\. Now the denominator ispt​\(zt\)p\_\{t\}\(z\_\{t\}\)itself, so dividing by it givesp⁡\(x∣zt\)p\(x\\mid z\_\{t\}\)at once, and the second member carries∫p⁡\(x∣zt\)​𝑑x=1\\int p\(x\\mid z\_\{t\}\)\\,\\mathrm\{d\}x=1in place ofℓt​\(zt\)\\ell\_\{t\}\(z\_\{t\}\):

∇zt​log​pt​\(zt\)\\displaystyle\\nabla\_\{z\_\{t\}\}\\log p\_\{t\}\(z\_\{t\}\)=∇ztpt​\(zt\)pt​\(zt\)=∫αt​x−ztσt2​qt​\(zt∣x\)​pθ​\(x\)pt​\(zt\)​𝑑x\\displaystyle=\\frac\{\\nabla\_\{z\_\{t\}\}p\_\{t\}\(z\_\{t\}\)\}\{p\_\{t\}\(z\_\{t\}\)\}=\\int\\frac\{\\alpha\_\{t\}\\,x\-z\_\{t\}\}\{\\sigma\_\{t\}^\{2\}\}\\,\\frac\{q\_\{t\}\(z\_\{t\}\\mid x\)\\,p\_\{\\theta\}\(x\)\}\{p\_\{t\}\(z\_\{t\}\)\}\\,\\mathrm\{d\}x\(40\)=1σt2​\(αt​∫x​p​\(x∣zt\)​dx−zt​∫p⁡\(x∣zt\)​dx\)=αt​z^−ztσt2,\\displaystyle=\\frac\{1\}\{\\sigma\_\{t\}^\{2\}\}\\left\(\\alpha\_\{t\}\\int x\\,p\(x\\mid z\_\{t\}\)\\,\\mathrm\{d\}x\\;\-\\;z\_\{t\}\\int p\(x\\mid z\_\{t\}\)\\,\\mathrm\{d\}x\\right\)=\\frac\{\\alpha\_\{t\}\\,\\hat\{z\}\-z\_\{t\}\}\{\\sigma\_\{t\}^\{2\}\},wherez^=𝔼p\(⋅∣zt\)\[x\]\\hat\{z\}=\\mathbb\{E\}\_\{p\(\\cdot\\mid z\_\{t\}\)\}\[x\]; this is Tweedie’s formula for this kernel and is the untilted score quoted in Proposition[2](https://arxiv.org/html/2610.22763#Thmproposition2)\.

Substituting Eq\. \([39](https://arxiv.org/html/2610.22763#A1.E39)\) and Eq\. \([40](https://arxiv.org/html/2610.22763#A1.E40)\) into Eq\. \([38](https://arxiv.org/html/2610.22763#A1.E38)\), theztz\_\{t\}terms — identical in both — cancel:

∇zt​log​ℓt​\(zt\)=αt​z^tilt−ztσt2−αt​z^−ztσt2=αt​\(z^tilt−z^\)σt2,\\nabla\_\{z\_\{t\}\}\\log\\ell\_\{t\}\(z\_\{t\}\)=\\frac\{\\alpha\_\{t\}\\,\\hat\{z\}^\{\\mathrm\{tilt\}\}\-z\_\{t\}\}\{\\sigma\_\{t\}^\{2\}\}\-\\frac\{\\alpha\_\{t\}\\,\\hat\{z\}\-z\_\{t\}\}\{\\sigma\_\{t\}^\{2\}\}=\\frac\{\\alpha\_\{t\}\\big\(\\hat\{z\}^\{\\mathrm\{tilt\}\}\-\\hat\{z\}\\big\)\}\{\\sigma\_\{t\}^\{2\}\},\(41\)which is Eq\. \([21](https://arxiv.org/html/2610.22763#S3.E21)\)\. Adding Eq\. \([40](https://arxiv.org/html/2610.22763#A1.E40)\) back recovers Eq\. \([22](https://arxiv.org/html/2610.22763#S3.E22)\)\. ∎

### A\.5Estimating the Tilted Mean

By Proposition[2](https://arxiv.org/html/2610.22763#Thmproposition2), the guidance term depends on the frozen model only through the tilted posterior meanz^tilt\\hat\{z\}^\{\\mathrm\{tilt\}\}of Definition[3\.3](https://arxiv.org/html/2610.22763#S3.Thmdefinition3)\. Its untilted counterpartz^\\hat\{z\}comes with the score \(i\), butz^tilt\\hat\{z\}^\{\\mathrm\{tilt\}\}is a ratio of two expectations underp\(⋅∣zt\)p\(\\cdot\\mid z\_\{t\}\)withℓ\\ella black box, and has no closed form\. This section gives the estimator used in Section[3\.3](https://arxiv.org/html/2610.22763#S3.SS3), how its draws are obtained for each backbone, and the two parameters that set the strength of the tilt\.

###### Proposition 5\(Self\-normalized importance\-sampling \(SNIS\) estimator\)\.

DrawNNi\.i\.d\. samplesx\(n\)∼p⁡\(x∣zt\)x^\{\(n\)\}\\sim p\(x\\mid z\_\{t\}\)and writeℓn≜ℓ⁡\(τ⁡\(x\(n\)\)\)\\ell\_\{n\}\\triangleq\\ell\(\\tau\(x^\{\(n\)\}\)\)\. Then

z^tilt≈∑n=1Nhn​x\(n\),hn=ℓn∑mℓm,∑nhn=1\.\\hat\{z\}^\{\\mathrm\{tilt\}\}\\approx\\sum\_\{n=1\}^\{N\}h\_\{n\}\\,x^\{\(n\)\},\\qquad h\_\{n\}=\\frac\{\\ell\_\{n\}\}\{\\sum\_\{m\}\\ell\_\{m\}\},\\qquad\\sum\_\{n\}h\_\{n\}=1\.\(42\)

###### Proof\.

We start by rewriting Definition[3\.3](https://arxiv.org/html/2610.22763#S3.Thmdefinition3)\. By Eq\. \([18](https://arxiv.org/html/2610.22763#S3.E18)\),p⋆​\(x∣zt\)=p⁡\(x∣zt\)​ℓ​\(τ⁡\(x\)\)/ℓt​\(zt\)p^\{\\star\}\(x\\mid z\_\{t\}\)=p\(x\\mid z\_\{t\}\)\\,\\ell\(\\tau\(x\)\)/\\ell\_\{t\}\(z\_\{t\}\), hence

z^tilt=1ℓt​\(zt\)​∫x​p​\(x∣zt\)​ℓ​\(τ⁡\(x\)\)​𝑑x=𝔼p\(⋅∣zt\)\[ℓ\(τ\(x\)\)x\]𝔼p\(⋅∣zt\)\[ℓ\(τ\(x\)\)\],\\hat\{z\}^\{\\mathrm\{tilt\}\}=\\frac\{1\}\{\\ell\_\{t\}\(z\_\{t\}\)\}\\int x\\,p\(x\\mid z\_\{t\}\)\\,\\ell\(\\tau\(x\)\)\\,\\mathrm\{d\}x=\\frac\{\\mathbb\{E\}\_\{p\(\\cdot\\mid z\_\{t\}\)\}\\big\[\\ell\(\\tau\(x\)\)\\,x\\big\]\}\{\\mathbb\{E\}\_\{p\(\\cdot\\mid z\_\{t\}\)\}\\big\[\\ell\(\\tau\(x\)\)\\big\]\},\(43\)the last step by the definition ofℓt\\ell\_\{t\}in Proposition[1](https://arxiv.org/html/2610.22763#Thmproposition1)\. Neither expectation admits a closed form:ℓ\\ellis a scalar function of the whole decoded object and does not factorize over the coordinates ofxx, so numerator and denominator both couple allLLpositions\. Estimating the two expectations of Eq\. \([43](https://arxiv.org/html/2610.22763#A1.E43)\) by Monte Carlo, using the sameNNdrawsx\(n\)∼p\(⋅∣zt\)x^\{\(n\)\}\\sim p\(\\cdot\\mid z\_\{t\}\)in both, gives

z^tilt≈1N​∑n=1Nℓn​x\(n\)1N​∑m=1Nℓm=∑n=1Nℓn∑mℓm⏟hn​x\(n\),∑nhn=1\.\\hat\{z\}^\{\\mathrm\{tilt\}\}\\approx\\frac\{\\tfrac\{1\}\{N\}\\sum\_\{n=1\}^\{N\}\\ell\_\{n\}\\,x^\{\(n\)\}\}\{\\tfrac\{1\}\{N\}\\sum\_\{m=1\}^\{N\}\\ell\_\{m\}\}=\\sum\_\{n=1\}^\{N\}\\underbrace\{\\frac\{\\ell\_\{n\}\}\{\\sum\_\{m\}\\ell\_\{m\}\}\}\_\{h\_\{n\}\}x^\{\(n\)\},\\qquad\\sum\_\{n\}h\_\{n\}=1\.\(44\)The estimator is a ratio of two unbiased estimators sharing the same draws, hence consistent but biased at finiteNN, with biasO⁡\(1/N\)O\(1/N\); the self\-normalization inhnh\_\{n\}is what removes the unknownℓt​\(zt\)\\ell\_\{t\}\(z\_\{t\}\)\. ∎

#### Two knobs, and they are not the same operation\.

In Eq\. \([11](https://arxiv.org/html/2610.22763#S3.E11)\) the acquisition enters with unit exponent; in practice the strength of the tilt is a tuning parameter\. It can be adjusted at two points of the estimator: after the tilted mean has been formed, or inside the weights\. The first,λ\\lambda, scales the guidance term of Eq\. \([19](https://arxiv.org/html/2610.22763#S3.E19)\) after the tilted mean has been formed,λ​∇zt​log​ℓt=∇zt​log​ℓtλ\\lambda\\nabla\_\{z\_\{t\}\}\\log\\ell\_\{t\}=\\nabla\_\{z\_\{t\}\}\\log\\ell\_\{t\}^\{\\lambda\}:

∇zt​log​pt\+λ​∇zt​log​ℓt=αt​\(z^\+λ⁡\(z^tilt−z^\)\)−ztσt2=αt​z^tilt,γ−ztσt2\.\\nabla\_\{z\_\{t\}\}\\log p\_\{t\}\+\\lambda\\,\\nabla\_\{z\_\{t\}\}\\log\\ell\_\{t\}=\\frac\{\\alpha\_\{t\}\\big\(\\hat\{z\}\+\\lambda\(\\hat\{z\}^\{\\mathrm\{tilt\}\}\-\\hat\{z\}\)\\big\)\-z\_\{t\}\}\{\\sigma\_\{t\}^\{2\}\}=\\frac\{\\alpha\_\{t\}\\,\\hat\{z\}^\{\\mathrm\{tilt\},\\gamma\}\-z\_\{t\}\}\{\\sigma\_\{t\}^\{2\}\}\.\(45\)This leaves the draws and the weights of Eq\. \([42](https://arxiv.org/html/2610.22763#A1.E42)\) untouched and moves only the target, along the segment fromz^\\hat\{z\}toz^tilt\\hat\{z\}^\{\\mathrm\{tilt\}\}\. Poweringℓ\\ellfirst and marginalizing second changes only the weights,

hn\(γ\)=softmaxn\(1γlogℓn\)=ℓn1/γ∑mℓm1/γ,ℓ↦ℓ1/γ,ℓt\(γ\)=𝔼p\(⋅∣zt\)\[ℓ1/γ\]\.h\_\{n\}^\{\(\\gamma\)\}=\\mathrm\{softmax\}\_\{n\}\\Big\(\\tfrac\{1\}\{\\gamma\}\\log\\ell\_\{n\}\\Big\)=\\frac\{\\ell\_\{n\}^\{1/\\gamma\}\}\{\\sum\_\{m\}\\ell\_\{m\}^\{1/\\gamma\}\},\\qquad\\ell\\mapsto\\ell^\{1/\\gamma\},\\qquad\\ell\_\{t\}^\{\(\\gamma\)\}=\\mathbb\{E\}\_\{p\(\\cdot\\mid z\_\{t\}\)\}\\big\[\\ell^\{1/\\gamma\}\\big\]\.\(46\)The target keeps the form of Eq\. \([22](https://arxiv.org/html/2610.22763#S3.E22)\) and the weight vector is deformed instead:γ→0\\gamma\\to 0collapseshhonto the best sampled candidate, whileγ→∞\\gamma\\to\\inftyflattens it to uniform and recovers the untiltedz^\\hat\{z\}\.

## Appendix BCDCD

Table 2:What each backbone supplies to the guided score Eq\. \([19](https://arxiv.org/html/2610.22763#S3.E19)\)\. Quantities \(i\) and \(ii\) are backbone\-specific; the guidance term∇zt​log​ℓt\\nabla\_\{z\_\{t\}\}\\log\\ell\_\{t\}is estimated from the draws of \(ii\) \(Section[3\.3](https://arxiv.org/html/2610.22763#S3.SS3)\)\.Algorithm[1](https://arxiv.org/html/2610.22763#alg1)covers the guided\-sampling step when the backbone is a diffusion model\.

Algorithm 1GuidedSample— CDCD\. Instantiates Section[3\.4](https://arxiv.org/html/2610.22763#S3.SS4)at\(αt,σt\)=\(1,t\)\(\\alpha\_\{t\},\\sigma\_\{t\}\)=\(1,t\)\. CDCD’s code was never released, so we adapted a public reimplementation\([Sacco, 2024](https://arxiv.org/html/2610.22763#bib.bib26)\)\.0:

zinitz\_\{\\text\{init\}\}; frozen

netθ\\mathrm\{net\}\_\{\\theta\}and embedding table

EE;

α\\alpha,

γ\\gamma,

λ\\lambda,

NN,

𝒟\\mathcal\{D\}
1:

zT←zinitz\_\{T\}\\leftarrow z\_\{\\text\{init\}\}
2:for

t∈\[T,…,0\]t\\in\[T,\\dots,0\],

δ=tcurr−tnext\\delta=t\_\{\\text\{curr\}\}\-t\_\{\\text\{next\}\}do

3:

logits,z^←netθ​\(zt,t\)\\mathrm\{logits\},\\ \\hat\{z\}\\leftarrow\\mathrm\{net\}\_\{\\theta\}\(z\_\{t\},t\)⊳\\trianglerightEq\. \([6](https://arxiv.org/html/2610.22763#S2.E6)\)

4:

w\(n\)∼pθ\(⋅∣zt,t\)w^\{\(n\)\}\\sim p\_\{\\theta\}\(\\cdot\\mid z\_\{t\},t\),

n=1\.\.Nn=1\.\.N⊳\\trianglerightEq\. \([7](https://arxiv.org/html/2610.22763#S2.E7)\)

5:

hn←softmaxn​\(1γ​log⁡α⁡\(w\(n\),𝒟\)\)h\_\{n\}\\leftarrow\\mathrm\{softmax\}\_\{n\}\\big\(\\tfrac\{1\}\{\\gamma\}\\log\\alpha\(w^\{\(n\)\};\\mathcal\{D\}\)\\big\)⊳\\trianglerightEq\. \([46](https://arxiv.org/html/2610.22763#A1.E46)\)

6:

z^tilt←∑nhn​E​\(w\(n\)\)\\hat\{z\}^\{\\mathrm\{tilt\}\}\\leftarrow\\sum\_\{n\}h\_\{n\}\\,E\(w^\{\(n\)\}\)⊳\\trianglerightProposition[5](https://arxiv.org/html/2610.22763#Thmproposition5)or Proposition[6](https://arxiv.org/html/2610.22763#Thmproposition6)

7:

zt−δ←zt\+δ​t​z^\+λ⁡\(z^tilt−z^\)−ztt2z\_\{t\-\\delta\}\\leftarrow z\_\{t\}\+\\delta\\,t\\,\\dfrac\{\\hat\{z\}\+\\lambda\(\\hat\{z\}^\{\\mathrm\{tilt\}\}\-\\hat\{z\}\)\-z\_\{t\}\}\{t^\{2\}\}⊳\\trianglerightEq\. \([45](https://arxiv.org/html/2610.22763#A1.E45)\)

8:endfor

9:return

z0z\_\{0\}

#### Instantiation\.

CDCD’s forward process Eq\. \([5](https://arxiv.org/html/2610.22763#S2.E5)\) is the kernel Eq\. \([15](https://arxiv.org/html/2610.22763#S3.E15)\) of Definition[3\.2](https://arxiv.org/html/2610.22763#S3.Thmdefinition2)at\(αt,σt\)=\(1,t\)\(\\alpha\_\{t\},\\sigma\_\{t\}\)=\(1,t\), soqt​\(zt∣x\)=𝒩⁡\(zt,x,t2​I\)q\_\{t\}\(z\_\{t\}\\mid x\)=\\mathcal\{N\}\(z\_\{t\};x,t^\{2\}I\)withx=E⁡\(w\)x=E\(w\)\. Proposition[2](https://arxiv.org/html/2610.22763#Thmproposition2)then reads

∇zt​log​pt⋆​\(zt\)=1t2​\[\(z^−zt\)\+\(z^tilt−z^\)\]=z^tilt−ztt2,\\nabla\_\{z\_\{t\}\}\\log p^\{\\star\}\_\{t\}\(z\_\{t\}\)=\\frac\{1\}\{t^\{2\}\}\\big\[\(\\hat\{z\}\-z\_\{t\}\)\+\(\\hat\{z\}^\{\\mathrm\{tilt\}\}\-\\hat\{z\}\)\\big\]=\\frac\{\\hat\{z\}^\{\\mathrm\{tilt\}\}\-z\_\{t\}\}\{t^\{2\}\},\(47\)whose first term is CDCD’s own score Eq\. \([6](https://arxiv.org/html/2610.22763#S2.E6)\): guidance replacesz^\\hat\{z\}byz^tilt\\hat\{z\}^\{\\mathrm\{tilt\}\}and leaves the sampler otherwise untouched\. The sampler integrates Eq\. \([47](https://arxiv.org/html/2610.22763#A2.E47)\) by an Euler step \(Algorithm[1](https://arxiv.org/html/2610.22763#alg1), line 7\), which carries a discretization error per step — a cost SeqFlow does not pay \(Appendix[C](https://arxiv.org/html/2610.22763#A3)\)\.

Withx=E⁡\(w\)x=E\(w\), Definition[3\.3](https://arxiv.org/html/2610.22763#S3.Thmdefinition3)reads

z^tilt=∑w∈𝒱Lpθ​\(w∣zt\)​ℓ​\(w\)​E​\(w\)∑w∈𝒱Lpθ​\(w∣zt\)​ℓ​\(w\)\.\\hat\{z\}^\{\\mathrm\{tilt\}\}=\\frac\{\\sum\_\{w\\in\\mathcal\{V\}^\{L\}\}p\_\{\\theta\}\(w\\mid z\_\{t\}\)\\,\\ell\(w\)\\,E\(w\)\}\{\\sum\_\{w\\in\\mathcal\{V\}^\{L\}\}p\_\{\\theta\}\(w\\mid z\_\{t\}\)\\,\\ell\(w\)\}\.\(48\)Both sums range over𝒱L\\mathcal\{V\}^\{L\}, and neither decomposes:ℓ\\ellis a scalar function of the whole decoded molecule,ℓ⁡\(\(,,,,,\)\)\\ell\\big\(\(w\_\{1\},\\dots,w\_\{L\}\)\\big\), so it couples allLLpositions\. Note the contrast withz^\\hat\{z\}, which is an expectation of a quantity that is additive across positions and is therefore determined by the marginals Eq\. \([7](https://arxiv.org/html/2610.22763#S2.E7)\) alone;z^tilt\\hat\{z\}^\{\\mathrm\{tilt\}\}is not, and must be estimated by Proposition[5](https://arxiv.org/html/2610.22763#Thmproposition5)\.

#### Sampling the denoising posterior\.

This is where CDCD incurs its one structural approximation, and it is worth being precise about whose approximation it is\. Proposition[5](https://arxiv.org/html/2610.22763#Thmproposition5)needs draws fromp\(⋅∣zt\)p\(\\cdot\\mid z\_\{t\}\)\. For a diffusion model, drawing from the true denoising posterior means running the reverse process fromztz\_\{t\}all the way to the data end, once per sample: atNNsamples per guidance step andKKsteps per trajectory, that isN​KNKrollouts, which is prohibitive\. What is available for free is Eq\. \([7](https://arxiv.org/html/2610.22763#S2.E7)\), the per\-position marginals, since they come from the same forward pass that already producedz^\\hat\{z\}and drawing from a factorized categorical costs nothing\. We therefore adopt the mean\-field approximation

p⁡\(x∣zt\)≈p~​\(x∣zt\)≜∏i=1Lpθ​\(wi∣zt,t\),x=E⁡\(w\),p\(x\\mid z\_\{t\}\)\\ \\approx\\ \\tilde\{p\}\(x\\mid z\_\{t\}\)\\triangleq\\textstyle\\prod\_\{i=1\}^\{L\}p\_\{\\theta\}\(w\_\{i\}\\mid z\_\{t\},t\),\\qquad x=E\(w\),\(49\)and sample positions independently\.

Two things should be said about it\. First, it is*our*approximation, introduced in the estimator, not an assumption CDCD makes: the model never represents the joint because the score does not require it,z^\\hat\{z\}being determined by the marginals alone\. Second, it is empirically benign — we compared Eq\. \([49](https://arxiv.org/html/2610.22763#A2.E49)\) against exact rollouts of the reverse process and found no degradation in optimization performance, at a small fraction of the cost \(Table[3](https://arxiv.org/html/2610.22763#A2.T3)\)\. The SeqFlow version needs neither: its posterior Eq\. \([53](https://arxiv.org/html/2610.22763#A3.E53)\) is exactly Gaussian and fully correlated, so a draw from it is a draw from the true posterior\.

Table 3:The mean\-field approximation Eq\. \([49](https://arxiv.org/html/2610.22763#A2.E49)\) against rollouts of the reverse process\.KKis the number of inner denoising steps used to draw each of theNNSNIS samples:K=0K=0is the factorized draw from the per\-position marginals Eq\. \([7](https://arxiv.org/html/2610.22763#S2.E7)\), i\.e\. Eq\. \([49](https://arxiv.org/html/2610.22763#A2.E49)\), and is what Algorithm[1](https://arxiv.org/html/2610.22763#alg1)uses;K\>0K\>0replaces it by aKK\-step ancestral rollout, which couples the positions through the network and resolves the joint\. Final objective value after300300BO epochs on six GuacaMol/TDC objectives, mean±\\pmsd over33seeds; higher is better\.Δ\\DeltaisK=32K\{=\}32minusK=0K\{=\}0, and the last column expresses it in units of the pooled seed\-to\-seed sd\. AtK=32K=32— beyond the point where proposal fidelity saturates, and at≈24×\\approx 24\\timesthe wall clock \(8\.778\.77against0\.360\.36h per arm\) — no objective improves beyond the seed spread and one degrades\.
#### REINFORCE estimator\.

Proposition[5](https://arxiv.org/html/2610.22763#Thmproposition5)estimatesz^tilt\\hat\{z\}^\{\\mathrm\{tilt\}\}from draws of a posterior that is already available in closed form, so the conditional score is analytic and Monte Carlo is needed only for the sequence\-level expectation ofℓ\\ell\. That estimator works directly on Definition[3\.3](https://arxiv.org/html/2610.22763#S3.Thmdefinition3)\. A second, more direct route starts from the observation that Proposition[2](https://arxiv.org/html/2610.22763#Thmproposition2)can be rearranged asz^tilt=z^\+t2​∇zt​log⁡ℓt​\(zt\)\\hat\{z\}^\{\\mathrm\{tilt\}\}=\\hat\{z\}\+t^\{2\}\\,\\nabla\_\{z\_\{t\}\}\\log\\ell\_\{t\}\(z\_\{t\}\)\. Sincez^\\hat\{z\}is available from the forward pass of the model, the only remaining obstacle is∇zt​log​ℓt​\(zt\)\\nabla\_\{z\_\{t\}\}\\log\\ell\_\{t\}\(z\_\{t\}\)\.

###### Proposition 6\(REINFORCE estimator\)\.

Givenz^tilt=z^\+t2​∇zt​log⁡ℓt​\(zt\)\\hat\{z\}^\{\\mathrm\{tilt\}\}=\\hat\{z\}\+t^\{2\}\\,\\nabla\_\{z\_\{t\}\}\\log\\ell\_\{t\}\(z\_\{t\}\), its estimate may be written as

z^tilt≈z^\+t2​∑n=1Nhn​∇zt​log⁡pθ​\(w\(n\)∣zt\),hn=ℓ⁡\(w\(n\)\)∑mℓ⁡\(w\(m\)\)\.\\hat\{z\}^\{\\mathrm\{tilt\}\}\\approx\\hat\{z\}\+t^\{2\}\\sum\_\{n=1\}^\{N\}h\_\{n\}\\,\\nabla\_\{z\_\{t\}\}\\log p\_\{\\theta\}\(w^\{\(n\)\}\\mid z\_\{t\}\),\\qquad h\_\{n\}=\\frac\{\\ell\(w^\{\(n\)\}\)\}\{\\sum\_\{m\}\\ell\(w^\{\(m\)\}\)\}\.\(50\)

###### Proof\.

By Eq\. \([17](https://arxiv.org/html/2610.22763#S3.E17)\),ℓt​\(zt\)=∫ℓ⁡\(τ⁡\(x\)\)​p​\(x∣zt\)​𝑑x=∑w∈𝒱Lℓ⁡\(w\)​pθ​\(w∣zt\)\\ell\_\{t\}\(z\_\{t\}\)=\\int\\ell\(\\tau\(x\)\)\\,p\(x\\mid z\_\{t\}\)\\,\\mathrm\{d\}x=\\sum\_\{w\\in\\mathcal\{V\}^\{L\}\}\\ell\(w\)\\,p\_\{\\theta\}\(w\\mid z\_\{t\}\)\. Applying the log\-derivative trick, i\.e\. multiplying and dividing bypθp\_\{\\theta\},

∇ztℓt=∑w∈𝒱Lℓ⁡\(w\)​pθ​∇ztpθpθ=𝔼pθ​\[ℓ​∇zt​log⁡pθ\]\\nabla\_\{z\_\{t\}\}\\ell\_\{t\}=\\sum\_\{w\\in\\mathcal\{V\}^\{L\}\}\\ell\(w\)\\,p\_\{\\theta\}\\frac\{\\nabla\_\{z\_\{t\}\}p\_\{\\theta\}\}\{p\_\{\\theta\}\}=\\mathbb\{E\}\_\{p\_\{\\theta\}\}\\big\[\\ell\\,\\nabla\_\{z\_\{t\}\}\\log p\_\{\\theta\}\\big\]∇zt​log​ℓt=𝔼pθ​\[ℓ​∇zt​log⁡pθ\]𝔼pθ​\[ℓ\]=∑w∈𝒱L∇zt​log​pθ​\(w∣zt\)​ℓ​\(w\)​pθ​\(w∣zt\)∑w∈𝒱Lpθ​\(w∣zt\)​ℓ​\(w\)\.\\nabla\_\{z\_\{t\}\}\\log\\ell\_\{t\}=\\frac\{\\mathbb\{E\}\_\{p\_\{\\theta\}\}\\big\[\\ell\\,\\nabla\_\{z\_\{t\}\}\\log p\_\{\\theta\}\\big\]\}\{\\mathbb\{E\}\_\{p\_\{\\theta\}\}\\big\[\\ell\\big\]\}=\\frac\{\\displaystyle\\sum\_\{w\\in\\mathcal\{V\}^\{L\}\}\\nabla\_\{z\_\{t\}\}\\log p\_\{\\theta\}\(w\\mid z\_\{t\}\)\\,\\ell\(w\)\\,p\_\{\\theta\}\(w\\mid z\_\{t\}\)\}\{\\displaystyle\\sum\_\{w\\in\\mathcal\{V\}^\{L\}\}p\_\{\\theta\}\(w\\mid z\_\{t\}\)\\,\\ell\(w\)\}\.This is again a ratio of two expectations under the samepθp\_\{\\theta\}, with the same intractable denominatorℓt\\ell\_\{t\}, so Monte Carlo applies as in Proposition[5](https://arxiv.org/html/2610.22763#Thmproposition5):

∇zt​log​ℓt≈1N​∑n=1Nℓn​∇zt​log⁡pθ​\(w\(n\)∣zt\)1N​∑m=1Nℓm=∑n=1Nℓn∑mℓm⏟hn​∇zt​log​pθ​\(w\(n\)∣zt\),∑nhn=1\.\\nabla\_\{z\_\{t\}\}\\log\\ell\_\{t\}\\approx\\frac\{\\frac\{1\}\{N\}\\sum\_\{n=1\}^\{N\}\\ell\_\{n\}\\,\\nabla\_\{z\_\{t\}\}\\log p\_\{\\theta\}\(w^\{\(n\)\}\\mid z\_\{t\}\)\}\{\\frac\{1\}\{N\}\\sum\_\{m=1\}^\{N\}\\ell\_\{m\}\}=\\sum\_\{n=1\}^\{N\}\\underbrace\{\\frac\{\\ell\_\{n\}\}\{\\sum\_\{m\}\\ell\_\{m\}\}\}\_\{h\_\{n\}\}\\nabla\_\{z\_\{t\}\}\\log p\_\{\\theta\}\(w^\{\(n\)\}\\mid z\_\{t\}\),\\qquad\\sum\_\{n\}h\_\{n\}=1\.Substituting intoz^tilt=z^\+t2​∇zt​log⁡ℓt​\(zt\)\\hat\{z\}^\{\\mathrm\{tilt\}\}=\\hat\{z\}\+t^\{2\}\\,\\nabla\_\{z\_\{t\}\}\\log\\ell\_\{t\}\(z\_\{t\}\)gives Eq\. \([50](https://arxiv.org/html/2610.22763#A2.E50)\)\. ∎

The per\-sample score required by Eq\. \([50](https://arxiv.org/html/2610.22763#A2.E50)\) is itself available in closed form\.

#### Cost\.

Bothz^\\hat\{z\}and the categoricalpθ​\(w∣zt,t\)p\_\{\\theta\}\(w\\mid z\_\{t\},t\)come from a single forward pass ofnetθ\\mathrm\{net\}\_\{\\theta\}, so theNNdraws of Proposition[5](https://arxiv.org/html/2610.22763#Thmproposition5)are sampled from an object already in hand and cost no additional network evaluations\. Each step is therefore one network evaluation,NNdecodes, and one GP prediction overNNmolecules\.

### B\.1Training CDCD

No pretrained diffusion model was available off the shelf, so we trained one from scratch\. Table[4](https://arxiv.org/html/2610.22763#A2.T4)lists the hyperparameter configuration we used\.[Dieleman et al\. \(2022\)](https://arxiv.org/html/2610.22763#bib.bib25)never released the original code, so we started from a public reimplementation\([Sacco, 2024](https://arxiv.org/html/2610.22763#bib.bib26)\)and adapted it to our needs\.

Table 4:Hyperparameters of the CDCD generator\.

## Appendix CSeqFlow

Algorithm[2](https://arxiv.org/html/2610.22763#alg2)covers the guided\-sampling step when the backbone is a normalizing flow\.

Algorithm 2GuidedSample— SeqFlow\. Instantiates Section[3\.4](https://arxiv.org/html/2610.22763#S3.SS4)in the flow latent with the variance\-preserving schedule Eq\. \([51](https://arxiv.org/html/2610.22763#A3.E51)\)\.0:

zinitz\_\{\\text\{init\}\}; frozen flow

gθg\_\{\\theta\}and readout

τ\\tau; schedule

ρT<⋯<ρ0\\rho\_\{T\}<\\dots<\\rho\_\{0\};

α\\alpha,

γ\\gamma,

λ\\lambda,

NN,

𝒟\\mathcal\{D\}
1:

zT←zinitz\_\{T\}\\leftarrow z\_\{\\text\{init\}\}
2:for

t=T,…,0t=T,\\dots,0,

t′t^\{\\prime\}the next leveldo

3:

z^←ρt​zt\\hat\{z\}\\leftarrow\\rho\_\{t\}\\,z\_\{t\}⊳\\trianglerightclosed form; no network eval, Eq\. \([53](https://arxiv.org/html/2610.22763#A3.E53)\)

4:

z\(n\)∼𝒩⁡\(z^,\(1−ρt2\)​I\)z^\{\(n\)\}\\sim\\mathcal\{N\}\\big\(\\hat\{z\},\(1\-\\rho\_\{t\}^\{2\}\)I\\big\),

n=1\.\.Nn=1\.\.N⊳\\trianglerightexact posterior

5:

hn←softmaxn​\(1γ​log⁡α⁡\(τ⁡\(gθ​\(z\(n\)\)\),𝒟\)\)h\_\{n\}\\leftarrow\\mathrm\{softmax\}\_\{n\}\\big\(\\tfrac\{1\}\{\\gamma\}\\log\\alpha\(\\tau\(g\_\{\\theta\}\(z^\{\(n\)\}\)\);\\mathcal\{D\}\)\\big\)⊳\\trianglerightEq\. \([46](https://arxiv.org/html/2610.22763#A1.E46)\)

6:

z^tilt←∑nhn​z\(n\)\\hat\{z\}^\{\\mathrm\{tilt\}\}\\leftarrow\\sum\_\{n\}h\_\{n\}\\,z^\{\(n\)\}⊳\\trianglerightProposition[5](https://arxiv.org/html/2610.22763#Thmproposition5)

7:

zt′←ρt′​\(z^\+λ⁡\(z^tilt−z^\)\)\+1−ρt′2​ξz\_\{t^\{\\prime\}\}\\leftarrow\\rho\_\{t^\{\\prime\}\}\\big\(\\hat\{z\}\+\\lambda\(\\hat\{z\}^\{\\mathrm\{tilt\}\}\-\\hat\{z\}\)\\big\)\+\\sqrt\{1\-\\rho\_\{t^\{\\prime\}\}^\{2\}\}\\,\\xi,

ξ∼𝒩⁡\(0,I\)\\xi\\sim\\mathcal\{N\}\(0,I\)
8:endfor

9:return

z0z\_\{0\}, decoded as

τ⁡\(gθ​\(z0\)\)\\tau\(g\_\{\\theta\}\(z\_\{0\}\)\)

#### Instantiation\.

SeqFlow supplies no noising process of its own, so the kernel of Definition[3\.2](https://arxiv.org/html/2610.22763#S3.Thmdefinition2)is imposed rather than inherited\. We work in the flow latentz=gθ−1​\(x\)z=g\_\{\\theta\}^\{\-1\}\(x\), whose law is exactly𝒩⁡\(0,I\)\\mathcal\{N\}\(0,I\)while the tilt is the pullbackℓ​\(τ​\(gθ​\(z\)\)\)\\ell\\big\(\\tau\(g\_\{\\theta\}\(z\)\)\\big\), Eq\. \([25](https://arxiv.org/html/2610.22763#S3.E25)\)\. Set

qt​\(zt∣z\)=𝒩⁡\(zt,ρt​z,\(1−ρt2\)​I\),ρt∈\(0,1\],q\_\{t\}\(z\_\{t\}\\mid z\)=\\mathcal\{N\}\\big\(z\_\{t\};\\,\\rho\_\{t\}\\,z,\\,\(1\-\\rho\_\{t\}^\{2\}\)I\\big\),\\qquad\\rho\_\{t\}\\in\(0,1\],\(51\)i\.e\.\(αt,σt\)=\(ρt,1−ρt2\)\(\\alpha\_\{t\},\\sigma\_\{t\}\)=\\big\(\\rho\_\{t\},\\sqrt\{1\-\\rho\_\{t\}^\{2\}\}\\big\)in Eq\. \([26](https://arxiv.org/html/2610.22763#S3.E26)\)\. Two closed forms follow\. Since the latent law is𝒩⁡\(0,I\)\\mathcal\{N\}\(0,I\), the untilted marginal is

pt​\(zt\)=∫𝒩⁡\(zt,ρt​z,\(1−ρt2\)​I\)​𝒩​\(z,0,I\)​𝑑z=𝒩⁡\(zt,0,I\),p\_\{t\}\(z\_\{t\}\)=\\int\\mathcal\{N\}\\big\(z\_\{t\};\\rho\_\{t\}z,\(1\-\\rho\_\{t\}^\{2\}\)I\\big\)\\,\\mathcal\{N\}\(z;0,I\)\\,dz=\\mathcal\{N\}\(z\_\{t\};0,I\),\(52\)so the bridge is variance preserving and every level carries the same law as the flow’s own base; and by Gaussian conjugacy the posterior of Proposition[1](https://arxiv.org/html/2610.22763#Thmproposition1)is exact,

p⁡\(z∣zt\)=𝒩⁡\(z,ρt​zt,\(1−ρt2\)​I\),z^=ρt​zt,p\(z\\mid z\_\{t\}\)=\\mathcal\{N\}\\big\(z;\\,\\rho\_\{t\}\\,z\_\{t\},\\,\(1\-\\rho\_\{t\}^\{2\}\)I\\big\),\\qquad\\hat\{z\}=\\rho\_\{t\}\\,z\_\{t\},\(53\)which is Eq\. \([27](https://arxiv.org/html/2610.22763#S3.E27)\) withαt=ρt\\alpha\_\{t\}=\\rho\_\{t\}\. Neither requires a network evaluation\. Proposition[2](https://arxiv.org/html/2610.22763#Thmproposition2)then reads

∇zt​log​pt⋆​\(zt\)=ρt​z^tilt−zt1−ρt2\.\\nabla\_\{z\_\{t\}\}\\log p^\{\\star\}\_\{t\}\(z\_\{t\}\)=\\frac\{\\rho\_\{t\}\\,\\hat\{z\}^\{\\mathrm\{tilt\}\}\-z\_\{t\}\}\{1\-\\rho\_\{t\}^\{2\}\}\.\(54\)Nothing aboutgθg\_\{\\theta\}enters any of Eq\. \([52](https://arxiv.org/html/2610.22763#A3.E52)\)–Eq\. \([54](https://arxiv.org/html/2610.22763#A3.E54)\): the learned content of the flow is confined to the tilt, where it is evaluated pointwise by Proposition[5](https://arxiv.org/html/2610.22763#Thmproposition5)and never differentiated\. This is why no analogue of Proposition[6](https://arxiv.org/html/2610.22763#Thmproposition6)is needed here — there is no learned posterior whose score would have to be estimated\.

#### Why tilting the latent is legitimate\.

Definition[3\.1](https://arxiv.org/html/2610.22763#S3.Thmdefinition1)is stated over the object thatℓ\\ellscores, whereas Eq\. \([51](https://arxiv.org/html/2610.22763#A3.E51)\) tiltszz\. The two agree by the argument of Appendix[A\.1](https://arxiv.org/html/2610.22763#A1.SS1), applied withτ\\taureplaced by the composite readoutτ∘gθ\\tau\\circ g\_\{\\theta\}: writingΠ\\Pifor the distribution over molecules induced byπ⋆​\(z\)∝𝒩⁡\(z,0,I\)​ℓ​\(τ⁡\(gθ​\(z\)\)\)\\pi^\{\\star\}\(z\)\\propto\\mathcal\{N\}\(z;0,I\)\\,\\ell\(\\tau\(g\_\{\\theta\}\(z\)\)\)of Eq\. \([25](https://arxiv.org/html/2610.22763#S3.E25)\), and using thatℓ\\ellis constant on each cellAwA\_\{w\}of Eq\. \([8](https://arxiv.org/html/2610.22763#S2.E8)\),

Π⁡\(w\)=1Z​∫Aw𝒩⁡\(z,0,I\)​ℓ​\(τ⁡\(gθ​\(z\)\)\)​𝑑z=ℓ⁡\(w\)Z​∫Aw𝒩⁡\(z,0,I\)​𝑑z=Pθ​\(w\)​ℓ​\(w\)Z,\\Pi\(w\)=\\frac\{1\}\{Z\}\\int\_\{A\_\{w\}\}\\mathcal\{N\}\(z;0,I\)\\,\\ell\\big\(\\tau\(g\_\{\\theta\}\(z\)\)\\big\)\\,dz=\\frac\{\\ell\(w\)\}\{Z\}\\int\_\{A\_\{w\}\}\\mathcal\{N\}\(z;0,I\)\\,dz=\\frac\{P\_\{\\theta\}\(w\)\\,\\ell\(w\)\}\{Z\},\(55\)the intended target\. Tilting the latent and decoding is therefore the same distribution over molecules as tilting the molecules directly; the constancy ofℓ\\ellon cells is what licenses pulling it out of the integral\.

#### What remains approximate\.

Three things, none of them the posterior\. First, theNN\-sample estimate ofz^tilt\\hat\{z\}^\{\\mathrm\{tilt\}\}in Proposition[5](https://arxiv.org/html/2610.22763#Thmproposition5), which is common to both backbones\. Second, Algorithm[2](https://arxiv.org/html/2610.22763#alg2)substitutes the tilted posterior*mean*z^tilt\\hat\{z\}^\{\\mathrm\{tilt\}\}for a*draw*fromp⋆\(⋅∣zt\)p^\{\\star\}\(\\cdot\\mid z\_\{t\}\), which is under\-dispersed and mode\-seeking; resampling a singlez\(n\)z^\{\(n\)\}with probabilityhnh\_\{n\}instead recovers a transition that is asymptotically exact inNN, at the cost of variance\. Third, the rollout is initialized atzT∼pT=𝒩⁡\(0,I\)z\_\{T\}\\sim p\_\{T\}=\\mathcal\{N\}\(0,I\)rather than atπT⋆\\pi^\{\\star\}\_\{T\}, which by Proposition[1](https://arxiv.org/html/2610.22763#Thmproposition1)differs frompTp\_\{T\}by the tiltℓT\\ell\_\{T\}; the discrepancy vanishes asρT→0\\rho\_\{T\}\\to 0and is the price ofπT⋆\\pi^\{\\star\}\_\{T\}not being samplable\.

#### Cost\.

Each step drawsNNlatents from Eq\. \([53](https://arxiv.org/html/2610.22763#A3.E53)\) and decodes each throughgθg\_\{\\theta\}andτ\\tau, so the flow is evaluatedNNtimes per step\. CDCD amortizes a single forward pass across allNNdraws, since Eq\. \([7](https://arxiv.org/html/2610.22763#S2.E7)\) is already available from it\. This is the price of the exact posterior: the flow has no amortized categorical to sample from, and every draw must be pushed through the network to be scored\. The decode is latency\-bound rather than throughput\-bound, however, soNNis nearly free and the step count is the binding constraint — the opposite of CDCD’s economics\.

### C\.1Training SeqFlow

We used the pretrained model available in the authors’ repository\. For details of how it was trained and which hyperparameter configuration was used, we refer to[Lee et al\. \(2025\)](https://arxiv.org/html/2610.22763#bib.bib35)\.

## Appendix DAFGG Algorithm

Algorithm[3](https://arxiv.org/html/2610.22763#alg3)summarizes the complete AFGG optimization loop\. At each BO iteration, the surrogate is updated in data space, the guided sampler is initialized around the current incumbent using the trust region above, and the resulting candidate is decoded and evaluated\. The overall procedure is shared across both generative backbones; they differ only in the continuous variable in which the bridge and guidance are applied\. For CDCD, this variable is the token\-embedding representation, whereas for SeqFlow it is the Gaussian flow latent\. The complete guided samplers are given in Appendix[B](https://arxiv.org/html/2610.22763#A2)and Appendix[C](https://arxiv.org/html/2610.22763#A3)\(Algorithm[1](https://arxiv.org/html/2610.22763#alg1)and Algorithm[2](https://arxiv.org/html/2610.22763#alg2)\), and Table[2](https://arxiv.org/html/2610.22763#A2.T2)summarizes the correspondence between the two instantiations\.

Algorithm 3AFGG: full BO loop0:frozen generator; acquisition

α\\alpha; temperature

γ\\gamma; guidance scale

λ\\lambda; inner samples

NN; batch size

qq; oracle budget

BB; initial dataset

𝒟0\\mathcal\{D\}\_\{0\}; trust\-region coefficient

ρ\\rho; anchor scale

σ\\sigma
1:

𝒟←𝒟0\\mathcal\{D\}\\leftarrow\\mathcal\{D\}\_\{0\}
2:for

r=1,2,…,⌊B/q⌋r=1,2,\\dots,\\lfloor B/q\\rfloordo

3:fit GP on

𝒟\\mathcal\{D\}⊳\\trianglerightin design space, Section[2\.1](https://arxiv.org/html/2610.22763#S2.SS1)

4:

z⋆←z^\{\\star\}\\leftarrowanchor from the incumbent

5:

z\(1:q\)init←ρz⋆\+1−ρ2εz^\{\(1:q\)\}\_\{\\text\{init\}\}\\leftarrow\\rho\\,z^\{\\star\}\+\\sqrt\{1\-\\rho^\{2\}\}\\,\\varepsilon,

ε∼𝒩⁡\(0,σ2​I\)\\varepsilon\\sim\\mathcal\{N\}\(0,\\sigma^\{2\}I\)⊳\\trianglerightEq\. \([28](https://arxiv.org/html/2610.22763#S3.E28)\)

6:

z\(1:q\)←GuidedSample\(zinit\(1:q\)\)z^\{\(1:q\)\}\\leftarrow\\textsc\{GuidedSample\}\\big\(z^\{\(1:q\)\}\_\{\\text\{init\}\}\\big\)⊳\\trianglerightAlgorithm[1](https://arxiv.org/html/2610.22763#alg1)\(CDCD\) or Algorithm[2](https://arxiv.org/html/2610.22763#alg2)\(SeqFlow\)

7:

w\(i\)←τ⁡\(z\(i\)\)w^\{\(i\)\}\\leftarrow\\tau\(z^\{\(i\)\}\)\(CDCD\) or

τ⁡\(gθ​\(z\(i\)\)\)\\tau\\big\(g\_\{\\theta\}\(z^\{\(i\)\}\)\\big\)\(SeqFlow\),

i=1\.\.qi=1\.\.q⊳\\trianglerightdecode to design space

8:

y\(i\)←y^\{\(i\)\}\\leftarrowevaluate oracle on

w\(i\)w^\{\(i\)\},

i=1\.\.qi=1\.\.q
9:

𝒟←𝒟∪\{\(w\(i\),y\(i\)\)\}i=1q\\mathcal\{D\}\\leftarrow\\mathcal\{D\}\\cup\\\{\(w^\{\(i\)\},y^\{\(i\)\}\)\\\}\_\{i=1\}^\{q\}
10:endfor

11:return

arg​max\(w,y\)∈𝒟⁡y\\argmax\_\{\(w,y\)\\in\\mathcal\{D\}\}y

## Appendix EAdditional Material

### E\.1Additional Results on GuacaMol

This section reports the full benchmark behind Section[4](https://arxiv.org/html/2610.22763#S4): all 25 PMO objectives, in the sequential regime and in both batch regimes, as final scores and as complete trajectories\. Objective names are abbreviated as in Table[5](https://arxiv.org/html/2610.22763#A5.T5)\.

Table 5:Abbreviations of the 25 PMO objectives, used in all tables\.All available generative\-model checkpoints of the baselines were trained on GuacaMol; COWBOYS provides no checkpoint, so we retrain it on GuacaMol from the released code\.

One accounting point applies throughout\. CoBO\([Lee et al\., 2023](https://arxiv.org/html/2610.22763#bib.bib36)\)and LOL\-BO\([Maus et al\., 2022](https://arxiv.org/html/2610.22763#bib.bib31)\)spend oracle calls realigning a latent with the molecule it decodes to, and their released code does not charge those calls to the budget\. Following[Chu et al\. \(2024\)](https://arxiv.org/html/2610.22763#bib.bib32)we modified both to count them, so every method below is compared at the same number of oracle evaluations\.

Fig\.[3](https://arxiv.org/html/2610.22763#A5.F3)and Table[6](https://arxiv.org/html/2610.22763#A5.T6)give the sequential regime \(q=1q=1, 10 initial points, 300 oracle calls\) against all eleven baselines, of which Table[1](https://arxiv.org/html/2610.22763#S4.T1)in the main text shows five\. Each cell is judged against the opposing group, so a bold AFGG cell beats every baseline on that objective and a bold baseline cell beats both AFGG columns; green marks a lead exceeding the cell’s own standard deviation\. Two objectives are worth reading separately from the rest\.logpandsaare not scaled to be in\[0,1\]\[0,1\], and reward molecules that are large or synthetically implausible, so a high score there measures how far a method will leave the training distribution rather than how well it optimizes; the LSBO baselines lead on both\.

Figure 3:Sequential setting: full trajectories on all 25 PMO objectives \(10 initial points, 300 oracle calls, 1 point per BO iteration\)\.Table 6:Final best score on all 25 PMO objectives after 300 oracle calls \(10 initial points, 1 point per BO iteration\), mean±\\pmsample standard deviation over 5 seeds\. AFGG is shown in two instantiations:CDCDandSeqFlow\. Every cell is judged against the opposing group: an AFGG cell against the 11 baselines, a baseline cell against the two AFGG columns\.Bold: the cell’s mean exceeds every mean in the opposing group\.Green: that lead is significant, exceeding the cell’s own standard deviation; bold without green therefore means ahead but within noise\.The batch regime \(100 initial points, then 100 epochs ofq=5q=5\) admits two ways of turning a per\-candidate tilt into a batch, differing in where the interaction between members enters\.*Uncoupled*generation guides each trajectory by its ownℓ=αEI\\ell=\\alpha^\{\\mathrm\{EI\}\}and lets the points “meet” only at selection, where theqqpoints are chosen from the generated pool by joint qLogEI in design space\.*Coupled*generation uses one joint tilt per batch,ℓ\\ellbeing qLogEI evaluated on the partial batch, so each member is generated aware of those already fixed\. Coupled generation is the more faithful instantiation of batch BO — it is the one that targets the joint acquisition the batch is eventually scored by — but it is not the one that performs better here, so we report both\. Fig\.[4](https://arxiv.org/html/2610.22763#A5.F4)and Table[7](https://arxiv.org/html/2610.22763#A5.T7)give the uncoupled variant, the one used in Section[4](https://arxiv.org/html/2610.22763#S4); Fig\.[5](https://arxiv.org/html/2610.22763#A5.F5)and Table[8](https://arxiv.org/html/2610.22763#A5.T8)the coupled one\. The baseline columns are the same runs in both tables, since only the AFGG variants differ\.

![Refer to caption](https://arxiv.org/html/2610.22763v1/grid25_batch_uncoupled.png)Figure 4:Batch regime, uncoupled generation: 500 acquisitions \(100 initial points, then 100 epochs of 5\), all 25 objectives\.Table 7:Batch regime, uncoupled generation\.Final best score on all 25 PMO objectives after 500 acquisitions \(100 initial points, then 100 epochs of 5\), mean±\\pmsample standard deviation over 5 seeds\. AFGG generates each batch*uncoupled*here: every molecule is guided by its own acquisition value, and the batch only forms at selection\. Every cell is judged against the opposing group: an AFGG cell against the five baselines, a baseline cell against the two AFGG columns\.Bold: the cell’s mean exceeds every mean in the opposing group\.Green: that lead is significant, exceeding the cell’s own standard deviation; bold without green therefore means ahead but within noise\.![Refer to caption](https://arxiv.org/html/2610.22763v1/figures/batch-coupled/grid25_batch_coupled.png)Figure 5:Batch regime, coupled generation: 500 acquisitions \(100 initial points, then 100 epochs of 5\), all 25 objectives\.Table 8:Batch regime, coupled generation\.Final best score on all 25 PMO objectives after 500 acquisitions \(100 initial points, then 100 epochs of 5\), mean±\\pmsample standard deviation over 5 seeds\. AFGG generates each batch*coupled*here: one joint qLogEI tilt per candidate batch, so its members are generated aware of each other\. Every cell is judged against the opposing group: an AFGG cell against the five baselines, a baseline cell against the two AFGG columns\.Bold: the cell’s mean exceeds every mean in the opposing group\.Green: that lead is significant, exceeding the cell’s own standard deviation; bold without green therefore means ahead but within noise\.
### E\.2Effect of the Generative Prior

AFGG \(CDCD\) and AFGG \(SeqFlow\) share the same acquisition\-guided machinery, but differ in the generative model being steered: Table[9](https://arxiv.org/html/2610.22763#A5.T9)isolates that component: we draw50005000unconditional samples from each model and score them against a3030k\-molecule subset of GuacaMol\. Validity, uniqueness and novelty do not separate the two models: all six values exceed0\.960\.96and the residual differences, below two points, are in CDCD’s favor\. FCD does separate them, by a factor of roughly3\.53\.5: SeqFlow reaches6\.026\.02against CDCD’s20\.9920\.99, i\.e\. its samples sit far closer to the reference distribution in ChemNet feature space\. Guidance tilts the prior rather than extending its support, so what the acquisition function can propose is confined to the region the prior already covers, and a prior whose mass is better aligned with the data distribution gives the search a better\-conditioned region to act on\.

Table 9:Unconditional sample quality\.50005000molecules drawn from each generative model and scored against a3030k\-molecule subset of GuacaMol\.
### E\.3Hyperparameter Tuning for CDCD

Every column reports the final best score after 300 oracle calls, mean±\\pmsample standard deviation over 5 seeds\. All ablations use the CDCD backbone in the sequential setting \(10 initial points, 300 oracle calls, 1 point per epoch\)\. Every objective is scored on\[0,1\]\[0,1\]; bold marks the best mean in each row \(ties at the printed precision are all bolded\), and*Wins*counts only rows with a unique winner\.

#### Guidance on vs\. off\.

By settingλ=0\\lambda=0, it has been proven that the generative model is not the main driver for optimizing the function\. Having an additional force, represented by the acquisition function, has a dramatic effect on performance \(see Table[10](https://arxiv.org/html/2610.22763#A5.T10)for full results\)\.

Table 10:Guidance on vs\. off;*off*setsλ=0\\lambda=0, flattening the SNIS weights to uniform so the tilted mean equals the untilted one\.
#### Guidance estimator\.

SNIS and REINFORCE target the same quantity,z^tilt\\hat\{z\}^\{\\mathrm\{tilt\}\}\(Proposition[6](https://arxiv.org/html/2610.22763#Thmproposition6)and Proposition[5](https://arxiv.org/html/2610.22763#Thmproposition5)\)\. SNIS forms it directly as a weighted average of the sampled embeddings,∑nhn​E​\(w\(n\)\)\\sum\_\{n\}h\_\{n\}E\(w^\{\(n\)\}\): a closed\-form combination with no gradient involved Eq\. \([42](https://arxiv.org/html/2610.22763#A1.E42)\)\. REINFORCE instead routes through Eq\. \([50](https://arxiv.org/html/2610.22763#A2.E50)\), a score\-function estimator — these are well known to carry substantially higher variance than a direct, closed\-form average for a fixed sample budget, and atN=10N=10trajectories per epoch that variance shows: SNIS wins all seven objectives, several by a wide margin \(zale\_mpo: 0\.420 vs\. 0\.250\), Table[11](https://arxiv.org/html/2610.22763#A5.T11)for full results\.

Table 11:Guidance estimator: closed\-form SNIS vs\. REINFORCE \(score\-function estimator with a leave\-one\-out baseline\) in place of the mean\-swap identity\.
#### Acquisition function\.

Table[12](https://arxiv.org/html/2610.22763#A5.T12)shows AFGG’s versatility: it can be applied with any acquisition function\. As expected, EI yields higher performance than PI or UCB\.

Table 12:Acquisition function driving both the guidance tilt and candidate selection; UCB usesβ=0\.1\\beta=0\.1\.
#### Candidate pool size\.

Moving from 1 to 10 trajectories captures essentially all the achievable gain \(mean0\.470→0\.4800\.470\\to 0\.480\); doubling again to 20 does not clearly help \(mean0\.4770\.477, within a single row’s typical standard deviation of the 10\-trajectory result\), while doubling the per\-epoch generation cost\. We fix 10 as the point where that curve has flattened, Table[13](https://arxiv.org/html/2610.22763#A5.T13)for full results\.

Table 13:Candidate pool size: guided trajectories rolled out per epoch, of which the best by acquisition value is evaluated\.
#### Where the temperature acts\.

Power tilt reshapes the SNIS weights,hn∝ℓn1/γh\_\{n\}\\propto\\ell\_\{n\}^\{1/\\gamma\}\(see Eq\. \([46](https://arxiv.org/html/2610.22763#A1.E46)\)\) — always a convex combination of the sampled embeddings, so the guided target cannot leave their span\. Scale drift instead rescales the drift after the tilted mean is formed \(see Eq\. \([45](https://arxiv.org/html/2610.22763#A1.E45)\)\): atλ\>1\\lambda\>1this extrapolates pastz^tilt\\hat\{z\}^\{\\mathrm\{tilt\}\}, pushing the implied target outside the convex hull of what was actually sampled and decoded — a region the frozen denoiser never saw in training\. On this seven\-objective slice the two mechanisms land close together \(means0\.4740\.474–0\.4860\.486, all within a row’s typical standard deviation\), so we are not claiming a large empirical gap here; the convex\-hull argument, not this table, is why we default to power tilt at higher temperatures; Table[14](https://arxiv.org/html/2610.22763#A5.T14)for full results\.

Table 14:Different temperature parameters and how they affect the final generation process\.