Posterior Inference with Diffusion Models for High-dimensional Black-box Optimization

Thank you for your positive assessment of our paper's extensive experiment results. We've attempted to answer your questions below.

Claims And Evidence) Therefore, whether simple reweighted training achieve weighted likelihood such as Eq. 11 remains questionable.

Thank you for pointing out the question regarding Eq.11. As you mentioned, we try to maximize the ELBO instead of the marginal likelihood. We will fix the Eq.11 in the final manuscript.

Other Strengths And Weaknesses) One part that I do not like about this paper is that it contains to many sub-parts and tricks. The paper is driven by a clear performance target, but there is no clear technical clue that lead this paper.

Thank you for your constructive feedback. While there are several sub-parts in our method, please note that we systemically analyze the effect of each component through extensive ablation studies to verify that each component is crucial for improving performance.

Furthermore, please note that several ideas we imported in this paper are already considered as a reasonable choice for training diffusion models as an amortized sampler effectively. For example, off-policy training is suggested in various GFlowNets literature [1, 2, 3].

[1] Venkatraman, Siddarth, et al. "Amortizing intractable inference in diffusion models for vision, language, and control."

[2] Akhound-Sadegh, Tara, et al. "Iterated denoising energy matching for sampling from boltzmann densities."

[3] Rector-Brooks, Jarrid, et al. "Steering masked discrete diffusion models via discrete denoising posterior prediction."

Thank you again for your comments. We hope we have addressed them satisfactorily above, but do not hesitate to let us know if you have further questions. We are always ready to engage in further discussion!

Thank you for your positive comment and for considering our key idea, incorporating the diffusion model as prior and casting sampling as posterior inference for solving high-dimensional black-box optimization, as novel. We answer your questions below.

Other Comments Or Suggestions) Given the expressive power of diffusion model, I think some tasks including structured input space may further enhance the paper (e.g. chemical/protein design).

As you mentioned, tasks including structured input space further enhance the usefulness of our method. To this end, we conduct experiments on molecular optimization following [1]. As shown in the table, our method achieves not only higher performance but also high sample efficiency compared to recent BO baselines for structured inputs. We promise to add these results in our final manuscript.

Experiment results on structured inputs. Experiments are conducted with four random seeds.

[1] Maus, Natalie, et al. "Local latent space bayesian optimization over structured inputs."

[2] Lee, Seunghun, et al. "Advancing bayesian optimization via learning correlated latent space."

Questions For Authors) How to set the number of ensembles to guarantee the uncertainty quantification is reasonable?

Thank you for your interest in our work. The number of ensembles is crucial to reasonably quantify the uncertainty of the surrogate model. To this end, we conduct experiments by varying number of ensembles, $K$. As shown in the table, there is no big difference in performance when we increase $K$ more than $K=5$. However, without uncertainty quantification or too small number of ensembles leads to poor performance, which indicates that uncertainty quantification is crucial for high-dimensional black-box optimization problems.

Ablation studies on the number of ensembles ($K$). Experiments are conducted with four random seeds.

Thank you for your concrete review. Below we answer the questions and concerns you raised.

Claims and Evidence) There is no measurement regarding the uncertainty of the generative model or discussion about the relationships between two terms of uncertainty.

While utilizing the uncertainty of the diffusion model can be an interesting future work, measuring the uncertainty of the diffusion models is mostly complex [1]. We conducted a brief literature search on this topic [2, 3], but most of them focus on detecting poor-quality images, which lies outside of our research scope.

[1] Wu, Dongxia, et al. "Diff-BBO: Diffusion-Based Inverse Modeling for Black-Box Optimization."

[2] Kou, Siqi, et al. "Bayesdiff: Estimating pixel-wise uncertainty in diffusion via bayesian inference."

[3] Jazbec, Metod, et al. "Generative Uncertainty in Diffusion Models."

Methods and Evaluation Criteria) It is helpful to verify that by including the black-box optimization benchmarks with structured data in the design space, such as molecular optimization tasks.

Our method can be directly applied to benchmarks with structured data, such as molecular optimization tasks. To this end, we conducted additional experiments on benchmarks with structured inputs. We follow the standard evaluation pipeline with [4].

As shown in the table, our method achieves both higher performance and better sample efficiency compared to recent BO baselines for structured inputs. We promise to add these results in our final manuscript.

Experiment results on structured inputs. Experiments are conducted with 4 random seeds.

[4] Maus, Natalie, et al. "Local latent space bayesian optimization over structured inputs."

[5] Lee, Seunghun, et al. "Advancing bayesian optimization via learning correlated latent space."

Theoretical Claims) There is no theoretical guarantee provided for the sampling approach.

While we acknowledge that the theoretical guarantee of our algorithm could further enhance its reliability, the guarantee of finding optimal solutions using deep learning models is almost impossible. Our paper makes a methodological and empirical contribution to solving high-dimensional black-box optimization problems effectively. We believe that our method represents a new departure for solving high-dimensional black-box optimization by importing ideas from diffusion models and amortized posterior inference.

Experimental Designs or Analyses) My concern is that many baselines for high-dimensional black-box optimization are missing. For the ablation study in the appendix, it will be helpful if the authors can include other baselines in the analysis of batch size and initial dataset size.

We apologize for missing some crucial baselines in high-dimensional BO. However, we would like to emphasize that we present 4 strong BO-based baselines (We also already include LA-MCTS, which you mentioned in [5]). In particular, MCMC-BO and CMA-BO, which published last year, outperform most of the baselines listed above in various benchmarks.

Nevertheless, we conducted experiments with additional baselines, logEI and MCTS-VS. As shown in the table, we outperform those baselines in terms of performance. We promise to conduct experiments on all benchmarks and update the results in our final manuscript.

Experiment results of DiBO and additional baselines. Experiments are conducted with four random seeds.

Regarding ablation studies, we also include other baselines in the analysis of batch size and initial dataset size. As shown in the table, even with different experiment settings, our method consistently outperforms other baselines by a large margin. Furthermore, as depicted in Figure 9 in the Appendix, our method does not degrade in terms of performance even with a small initial dataset size.

Ablation studies on other baselines in terms of initial experiment settings. Experiments are conducted with four random seeds.