Implicit Generative Property Enhancer

We thank the reviewer for their positive assessment, particularly acknowledging GPE's simplicity, data efficiency, and architecture-agnostic nature.

Limited real-world validation. In silico initial validation is standard for novel computational design methods, given the significant cost and time of in vitro studies. As additional empirical evidence we provide a summary of single-round wet-lab experiments comparing GPE (mPropEn variant) against LAMBO [Gruver et al 23] for improving therapeutic protein binding affinity across eight initial seeds over three distinct targets. Due to proprietary constraints, we do not disclose the targets. Our model demonstrates significantly higher and more consistent binding rates (94.6% vs. 44.2% overall) and design improvements (34.4% vs. 14.0% overall).

[Gruver et al 23] Protein design with guided discrete diffusion.

Multi-objective design challenges and experiments. This reviewer raised a valid concern, as simultaneously improving multiple criteria can result in sparse matched pairs. To address this, we compute a multivariate rank across all properties of interest and optimize for that single scalar score. This multivariate score can be derived using either the Hypervolume [Zitzler et al 03] or the CDF indicator, as proposed in [Park et al 24]. Both methods effectively mitigate the sparsity issue by reducing the problem to a single optimization target.

We demonstrate this approach on a multi-property antibody optimization task, selecting two orthogonal developability properties: positive charge and negative charge. We compute a multivariate ranking over these two properties to obtain a single scalar score used for matching. We then train individual single-property GPE (mPropEn variant, as described in the manuscript) models to optimize each property separately, as well as a multi-property GPE (multiGPE) model that directly optimizes the combined multivariate ranking score.

We observe that multiGPE consistently suggests more designs that lie on the Pareto front, whereas the single-objective models tend to generate designs that occupy the extremes (tails) of the front. Furthermore, when considering both objectives, the most optimal designs on the Pareto front are predominantly proposed by multiGPE variants. This supports the effectiveness of combining our matching strategy with multi-objective indicators. Further datasets and experiments will be explored in subsequent analyses.

[Zitzler et al 03] Performance assessment of multiobjective optimizers: An analysis and review

[Park et al 24] BOtied: Multi-objective Bayesian optimization with tied multivariate ranks

Applying GPE within a diffusion framework. Diffusion models and flow matching models (when source distribution is Gaussian—as in our case) are equivalent under some reparametrization (eg, see the nice explanation in [Gao et al 25]). Therefore, we anticipate that results with “matched diffusion models” would be similar to those of our mFM (matched flow matching). Moreover, mWJS (matched walk-jump sampling) instantiation is a score-based model, closely related to diffusion models. Implementing GPE with a full diffusion framework would involve defining a conditional diffusion process and learning its conditional score function, similar to how $p^+(x'|x)$ is defined.

[Gao et al 25] Diffusion Meets Flow Matching: Two Sides of the Same Coin

Sensitivity to number of iterations. Our empirical results in Figures 3 (a and b) demonstrate that GPE's performance, in terms of median fitness, rapidly increases and then plateaus after a certain number of iterations (e.g., around 10-12 for AAV on protein fitness, and 4 rounds for GFP). This suggests a (dataset-dependent) saturation point where further iterations yield diminishing returns, providing practical guidance for iteration choice. We noted that the models’ "hyperparameters are transferable between the protein tasks studied, suggesting some degree of robustness" (L324).

We thank the reviewer for the comprehensive and insightful review. We appreciate the positive comments on the simplicity, intuition, empirical effectiveness, modularity of our framework and that our work is "fascinating, and a very promising direction". We also thank the extra references, which will be added to the manuscript. Next, we address each of the reviewer's points and suggestions.

Clarifying problem setting. We will clarify in the introduction and problem setting that our work focuses on offline (one-round) design optimization, ie generating improved designs from seeds without iterative experimental feedback. As pointed by the reviewer, our iterative sampling and self-training are distinct from online/Bayesian optimization. This was indeed a source of confusion to Reviewer VQoe. We will also explicitly state that we directly learn new conditional generative models, rather than guiding pre-trained models. We briefly discussed this on Appendix A, but we will make it clearer on the main manuscript.

How important is using C(x,x') to prevent “out-of-domain” generation? The constraint $C(x,x')$ is important for preventing "out-of-domain" generation in GPE. This term, particularly the distance component like $\text{dist}(x,x')<\Delta x$​, ensures that the generated design $x'$ remain within a meaningful vicinity of its seed $x$. Without such constraint, the generative model could produce samples that are technically "improved" in property but are entirely unrealistic, or irrelevant to the original design domain. By construction, training on matched datasets provides theoretical guarantees for generating designs in distribution (as dx goes to 0), please see Theorem 2 in [Tagasovska et al 2024].

This is also relevant to the self-training strategy, as one of the primary concerns in such setups is model collapse—specifically, overfitting on synthetic data and forgetting the original training distribution. Current methods to mitigate model collapse typically involve either controlling the ratio of original to synthetic data or pruning synthetic examples that deviate significantly from the training data. Training on matched datasets, by design, inherently addresses both of these concerns.

Ablation on $\Delta_x$: Below we show the performance of mWJS trained on different matched data from both AAV-medium and AAV-hard, with varying $\Delta_x$ threshold.

For the AAV-medium dataset, we observe that fitness is relatively stable with lower thresholds (0.50-0.53 for $\Delta_x$ in 1-7), peaking at 0.53 for $\Delta_x$ values of 4, 5, and 6. As $\Delta_x$ increases beyond 7, fitness notably declines, reaching 0.45 at $\Delta_x$=10.

On the AAV-hard dataset, fitness displays a more pronounced sensitivity to $\Delta_x$. It starts lower, increases, and reaches its peak at 0.54 for Δx​ values of 5 and 6. Performance significantly underperforms for very small $\Delta_x$ (0.26 at $\Delta_x$=1​) and also deteriorates for large $\Delta_x$ (>7), dropping to 0.31 at $\Delta_x$=10.

Do you think the empirical performance of this method is from the increased training set size and (discriminator free) implicit improvement objective? We believe GPE's performance stems from both the increased effective training data (via matched pairs) and the discriminator-free implicit objective. While a hypothetical discriminator-based model could use similar paired data, GPE's key advantage is its simplicity and avoidance of a separate predictor training loop. GPE, which is formulated as a conditional density estimation problem, implicitly learns the distribution of improved samples rather than relying on a (potentially unreliable) point-wise prediction.

Moreover, introducing an auxiliary discriminative model adds non-trivial complexity to the overall pipeline (training two models, positive/negative sample reweighting, managing their interaction, potential for mode collapse or training instabilities if not carefully balanced). GPE's end-to-end, single-stage training bypasses these issues somehow.

Do you think there would be less need for iterative self training using a trained predictor on matched data? The iterative self-training helps for extrapolation, a benefit for which a reliable explicit discriminator might also reduce the need. But, obtaining such a discriminator (specially in low-data) is often the bottleneck itself. That being said, we note that our method is orthogonal to predictor-based methods, and therefore could use any discriminator to further guide generation, potentially combining the benefit of both approaches.

Limitations of ML Oracles. We agree with the reviewer's point regarding ML oracle limitations and acknowledge this in the manuscript (footnote 6, page 7). We use them for fair comparison with well-established benchmarks [Kirjner et al.]. We would also like to also emphasize the antibody experiment (Section 4.2) as an example of GPE extrapolating on a physicochemical property with an exact evaluation metric, serving as a form of "oracle-free" validation.

How does this work relate to DPO/preference learning. GPE shares the spirit of preference learning, aligning models with desired outcomes based on comparisons. Similar to DPO, we optimize directly without an explicit reward model or property oracle. Both frameworks theoretically show their paired objectives implicitly optimize the same quantity as explicit reward-maximizing procedures: DPO with a binary-cross-entropy loss on preference pairs, and GPE by learning a conditional density estimator on matched data. This "bakes" the optimization signal into the generator, yielding simpler, more stable training loops. Conceptually there are similarities between the two frameworks, but, we would like to highlight the following differences where using the matched reconstruction is beneficial:

• Automatic data densification: Matching inflates an n-point dataset to O(n2) ordered pairs, providing richer, automatically generated supervision than human-annotated sets.
• On-manifold generation: Nearest-neighbor matching ensures plausible, realistic generations within the data manifold, unlike DPO which can drift with thin preference coverage.
• Domain agnosticism & small-data robustness: GPE requires no language-specific likelihoods or large pre-trained policies, making it practical for scientific design where DPO's ratios are unavailable or poorly calibrated.

In short, both bypass discriminators/reward models,, but GPE's matched-pair strategy converts readily-available scalar property data into a dense, on-manifold, gradient-approximating signal---often cheaper and more informative than explicit preference labels.

Notation $y$ in the Lagrangian. The functional derivative acts by $\frac{\delta}{\delta q(x'|x)} q(y'|y) = \delta(x-y) \delta(x'-y')$ where the rhs denotes Dirac delta distributions. Marginalization over the dirac delta distributions then changes the arguments of the densities from $y$, $y'$ to $x$, $x'$ as required for the result stated in the theorem. All four variables $x$,$x'$, $y$, $y'$ take value in the same space. We will make this explicit in the proof and clarify the role of $y$.

R^d (continuous) vs. V^D (discrete). We will clarify that the Rd formulation is for generality, and its application to discrete sequences is direct via appropriate architectural choices.

Intractability of Z(x). We will add a clarifying sentence that Z(x) (the normalization constant in Equation 1) is generally intractable due to high-dimensional integration/combinatorial summation, which justify our likelihood approximation approach.

“Tangential question”: GPE’s implicit exploration/exploitation tradeoff. We 100% agree with the reviewer that “for offline optimisation tasks, we are more interested in exploitation (modulo diverse batch-design concerns)”. Therefore, GPE's primary mechanism is exploitation: it learns to generate designs ($x'$) that have higher property values ($g(x')>g(x)$) than given seeds ($x$), based on observed improvements in the training data. This is a strong drive towards known high-value regions. The iterative sampling further refines this exploitation.

However, GPE also exhibits some level of implicit exploration: it generates a distribution of improved designs, allowing exploration within high-value regions. The distance constraint ($\text{dist}(x,x')<\Delta_x$​) encourages localized exploration, ensuring generated designs remain near their seeds. Following [Kirjner et al. 24]'s suggestion, the Fitness metric can be interpreted as a proxy for exploitation, while the Novelty and Diversity can provide insights about exploration/exploitation trade-offs. Finally, iterative self-training can potentially enable GPE to extrapolate property values and explore regions beyond the original training data manifold by generating synthetic data and retraining, representing a more global, model-driven form of exploration.

GPE does not use explicit acquisition functions but inherently balances strong exploitation with localized, and eventually broader, exploration through its generative mechanism, constraints, and self-training.

We thank the reviewer for their feedback on Bayesian optimization (BO) and discriminative models. We believe there may be a misunderstanding of GPE's offline (one-round) optimization problem setting (see [Kim et al 25] for a survey on this topic), which fundamentally differs from iterative online/Bayesian optimization. Reviewer DwTD acknowledges this difference. We hope the following paragraphs can make this distinction clearer. We will incorporate these clarifications in the paper and better contextualize the problem setup.

[Kim et al, 25] Offline Model-Based Optimization: Comprehensive Review.

Problem formulation and "strawman" argument. We thank the reviewer for a thoughtful critique and for pointing us to [Tripp & Hernández‑Lobato 2024]. This work relies on the PMO benchmark (from [Gao et al 22]), a simplified setting consisting of surrogate properties of relatively simple properties on small molecules only. As the authors state in the discussion themselves,"what this paper presents should best be thought of as a very limited pilot study" and "it is unclear whether results from single-task, noiseless, and unconstrained optimization will translate to real-world problems which tend to be multitask, noisy, and highly constrained."

Below we clarify the scope of our claim, provide empirical evidence that reliable discriminative surrogate models remain an unsolved bottleneck for many real‑world molecular and protein design tasks, and explain why BO, while indispensable, is not always a universal remedy in these settings.

• We agree BO is valuable when the surrogate is trustworthy. [Tripp & Hernández‑Lobato 2024] show that careful priors, and optimizer settings let BO top the PMO small‑molecule benchmark.

• But, for most real objectives the surrogate remains the weak link, eg:

  • Protein stability (ΔΔG): state‑of‑the‑art models still hover around RMSE ≈ 1–2 kcal/mol and r ≈ 0.5 on fully independent test sets—above the ≤0.5 kcal/mol engineering threshold [Benevuta et al 23].
  • Protein‑ligand affinity: removing train–test leakage in PDBBind drops Pearson r to ≤ 0.30. [Li et al 24]
  • Small‑molecule phys‑chem (e.g. log S): OOD splits double the error reported on random splits. [Sorkun et al 19]

• When surrogates are unreliable, BO inherits three practical failure modes.

  • Long‑tail bias: the acquisition function is steered away from the true optimum by systematic surrogate error at the distribution edges (documented in the affinity study above).
  • High‑dimensional/discrete spaces: recent survey finds no GP‑ or neural‑BO variant that scales gracefully beyond ~100 categorical dimensions. [González-Duque et al 24]
  • Fragility and mis‑calibration: tiny changes in prior width or kernel smoothness erase BO’s edge, and uncertainty estimates in GNNs and protein‑fitness models remain poorly calibrated. [Greenman et al 25]

• Our contribution targets precisely this early regime. We learn local generative distributions that do not need a global oracle. Once higher‑quality experimental labels accumulate, the BO frameworks championed by the reviewer can be layered on top of our generator as a natural next step.

We will revise the introduction and discussion to reflect this balanced view: BO excels with reliable surrogates; our method tackles the many cases where such surrogates are not yet available.

[Benevenuta et al 23] Challenges in predicting stabilizing variations: An exploration

[Li et al 24] Leak Proof PDBBind: A Reorganized Dataset of Protein-Ligand Complexes for More Generalizable Binding Affinity Prediction

[Sorkun et al 19] AqSolDB, a curated reference set of aqueous solubility and 2D descriptors for a diverse set of compounds

[González-Duque et al 24] A survey and benchmark of high-dimensional Bayesian optimization of discrete sequences

[Greenman et al 25] Benchmarking uncertainty quantification for protein engineering

GGS and "smoothed fitness landscape". We agree that GGS's "smoothed fitness landscape" aims to mitigate real-world challenges related to noisy/rugged fitness landscapes. Our "predictor-free" approach is not presented as the only necessary paradigm shift, but rather as a potential alternative that sidesteps the complexities and potential pitfalls (eg, overfitting or unreliability at tails) associated with training and relying on an explicit discriminative predictor for design generation in an offline context. GPE provides a conceptual simplicity and potential robustness by implicitly learning property improvement, instead of explicitly modeling or smoothing the fitness landscape.

Our method is orthogonal to GGS (or any other model that leverages a predictor) and could be combined for enhanced offline optimization. As mentioned above, these methods could also be integrated into BO loops. We will clarify this in the related work section.

Modern Bayesian optimization approaches with deep generative models. Recent BO models integrate deeply with generative models, using them to navigate complex, structured design spaces. However, we reiterate that these approaches are still embedded within the online, iterative framework of BO, where the generative model often aids in proposing candidates for subsequent evaluation guided by an acquisition function derived from a surrogate model. On the other hand, GPE was made for one-shot batch generation for offline applications. We will integrate these references into related works to provide a more comprehensive overview of how generative models are used in both online and offline optimization contexts.

Ablation study of C(x,x’). We conducted ablation studies on the $\Delta_x$ (Levenshtein distance) and $\Delta_g$ (property improvement) thresholds, as shown in the tables below for mWJS on the AAV-medium and AAV-hard datasets.

Ablation study of $\Delta_x$ thresholds:

For the AAV-medium (Table 1), we observe that fitness remains relatively stable with lower $\Delta_x$ thresholds (0.50-0.53 for $\Delta_x$ in 1-7), peaking at 0.53 for $\Delta_x$ values of 4, 5, and 6. As $\Delta_x$ increases beyond 7, fitness notably declines, reaching 0.45 at $\Delta_x$=10. Diversity generally increases with $\Delta_x$, while novelty shows minor fluctuations.

On AAV-hard (Table 2), fitness displays a more pronounced sensitivity to $\Delta_x$. It starts lower, increases, and reaches its peak at 0.54 for $\Delta_x$ values of 5 and 6. Performance significantly underperforms for very small $\Delta_x$ (0.26 at $\Delta_x$=1) and also deteriorates for large $\Delta_x$ (>7), dropping to 0.31 at $\Delta_x$=10.

These results indicate that a moderate range for $\Delta_x$ (e.g., 4-7 for AAV-medium, 5-6 for AAV-hard) yields optimal or near-optimal fitness. Very small thresholds likely overly restrict generation, while very large thresholds may dilute meaningful local relationships.

Ablation study of $\Delta_g$ thresholds ($\Delta_x$ fixed at 5):

Tables 3 and 4 report an ablation study where we vary the $\Delta_g$ threshold (while keeping $\Delta_x$ fixed at 5). Here, we observe that performance is stable for smaller $\Delta_g$ values. However, fitness starts to decline notably when $\Delta_g$ exceeds 0.8, particularly evident in the AAV-hard dataset.

We will include a detailed discussion of this comprehensive ablation study in the revised manuscript.

“Why is GPE’s approach better than the principled exploration/exploitation tradeoff we get from model uncertainty based optimization of Bayesian optimization?” GPE is not better than Bayesian Optimization (BO), but it's a different approach. While both BO and GPE aim at design optimization in small sample sizes, the two frameworks solve different problems. The goal of GPE is to generate designs, whereas in BO the goal is to choose the most promising designs that should be labeled in order to improve predictors’ performance or find the best candidate—i.e. the focus is selection. In the context of optimizing designs with deep generative models, one would have a suite of (1) generative models, (2) property predictors and (3) BO/active learning modules that will do the final selection across a pool of candidates. GPE falls in (1), the category of generative models section that will contribute to the library of potential candidates. Thus, GPE complements BO; it's ideal for producing promising design batches for costly offline validation, or can be integrated into BO for warm-starting or batch acquisition.

We thank the reviewer for their reviews and for highlighting the key strength of our work: the ability to perform design optimization without relying on unreliable surrogate estimates, particularly crucial in low-data and underexplored regions of the design space. Below we address the reviewer's concerns.

Lack of theoretical novelty. We respectfully disagree with the assessment that our theoretical contribution lacks novelty. While Theorem 1 adapts a known maximum likelihood result, its application within our framework is novel (and foundational) for GPE. In particular, we demonstrate that by formulating design optimization as a conditional density estimation problem and leveraging the implicitly defined "improved distribution" $p^+(x'∣x)$ through data matching, we can theoretically recover the optimal conditional density without any explicit property predictor. This theoretical grounding provides the basis for training diverse off-the-shelf generative models to directly sample improved designs, which is a significant departure from conventional predictor-guided optimization paradigms. The novelty lies in how this theoretical result enables a simple, end-to-end, and predictor-free approach to a challenging problem.

"Match variants are simply conditional generative models". The underlying generative models (VAE, flow matching, walk-jump sampling, diffusion, autoregressive, etc.) are indeed "off-the-shelf" in their core formulation. However, the novelty is not in inventing new generative model architectures, but in the GPE framework itself and how it trains and applies these existing models to tackle the design optimization problem. The key innovations are:

• The "matched dataset" construction: This novel data pairing ($(x,x’)$ where $x'$ has higher property than $x$) implicitly encodes the desired property improvement.

• The conditional density estimation problem formulation: We cast the problem as learning to sample from $p^+(x'∣x)$, the distribution of improved designs conditioned on a seed, which is implicitly defined by the matched data.

• Predictor-free optimization: By training these conditional generative models directly on the matched dataset using objectives aligned with conditional density estimation (e.g., maximum likelihood or its approximations), GPE bypasses the need for a separate, often unreliable, discriminative property predictor. Therefore, while the generative models themselves are known, their instantiation within the GPE framework for implicit, predictor-free design optimization represent the core novelty.

"Fix the reference to 'the least squares loss 3.2' in line 172. What is it referring to?" The reference "least-squares loss (3.2)" should refer to the mWJS loss defined on L166 ($\mathcal{L}_{mWJS}$), which is a least-squares loss. We will assign a specific equation number to the mWJS loss and refer to it in the revised manuscript.

"In each training round, how many samples are generated per conditioning seed sequences? Have you performed any ablation on the number of samples generated per seed? Does it have a significant impact?" This is an interesting question and we thank the reviewer for raising it. Indeed, not all seeds are equal. For some which have more neighbors, i.e. lie in a more dense region in the training data, the model has better chances of learning the implicit direction of improvement and therefore more opportunity for generating diverse candidate designs. Hence, the number of pairs around each seed at round 0, is indicative of the opportunities of improvement.

Unfortunately, we cannot include figures in the rebuttal but we will add more intuition and ablation plot on this point in the updated version of the manuscript. For the moment, we hope the following statistics from an ablation will answer your question. We notice the expected trend, that is, at each round the number of designs (diversity) per seed goes down both in terms of mean and standard deviation.

M_k mismatch between Section 3.4 and Algorithm 2. We apologize for this inconsistency in notation—thank you for pointing out. The correct version is the one described in Section 3.4 (L197). We will fix Algorithm 2 in the revised manuscript, ensuring consistency between the main text and the appendix. Please find below the essential changes that correct the error:

P^(k) ← { (x_i, x'_i) : dist( x'_i, x_i ) ≤ Δ_x }
M^(k+1) ← M^(k) ∪ P^(k)  // Augment training set
// 4. Retrain on the full (real + pseudo) matches
θ^(k+1) ← argmin_{θ}  L( θ; M^(k+1) )

Fixed initial seeds vs random initial seed during benchmarking. For benchmarking, we used fixed initial seeds after observing no considerable performance difference with random seeds in preliminary tests. This allowed us to isolate and systematically compare optimization algorithms from a standardized starting point.

For the AAV-medium dataset (Table 2), the performance across all metrics (fitness, diversity, and novelty) remains highly consistent, regardless of whether fixed or random seeds are used. The slight variations observed are well within the reported standard deviations, confirming the stability of the results.

Similarly, for the more challenging AAV-hard dataset (Table 3), the mWJS model demonstrates robust performance. While minor fluctuations are present, the fitness, diversity, and novelty scores for both fixed and random seeds remain comparable, reinforcing our preliminary findings that the choice of initial seed does not considerably impact the algorithm's performance.

This corroborates our initial observations and confirms that the use of fixed seeds provides a reliable basis for comparison without sacrificing the generalizability of our findings. We will include this ablation experiment in the final version of the manuscript.

Missed citation: The citation of Ghaffari et al. (2024) will be explicitly added to the related work section. Thanks for pointing it out.