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Optimization · Optimization and Learning under Uncertainty

Rafael Oliveira, Dan Steinberg, Edwin Bonilla

We present a general strategy for turning generative models into candidate solution samplers for batch Bayesian optimization (BO). The use of generative models for BO enables large batch scaling as generative sampling, optimization of non-continuous design spaces, and high-dimensional and combinatorial design. Inspired by the success of direct preference optimization (DPO), we show that one can train a generative model with noisy, simple utility values directly computed from observations to then form proposal distributions whose densities are proportional to the expected utility, i.e., BO's acquisition function values. Furthermore, this approach is generalizable beyond preference-based feedback to general types of reward signals and loss functions. This perspective avoids the construction of surrogate (regression or classification) models, common in previous methods that have used generative models for black-box optimization. Theoretically, we show that the generative models within the BO process follow a sequence of distributions which asymptotically approximate an optimal target under certain conditions. We also evaluate the performance through experiments on challenging optimization problems involving large batches in high dimensions.

Chongqi Qu, Meiqin Liu, Jian Lan, Shanling Dong, Zhunga Liu

Bayesian optimization (BO) is a powerful approach for black-box optimization, and in many real-world problems, domain experts possess valuable prior knowledge about promising regions of the search space. However, existing prior-informed BO methods are often overly complex, tied to specific acquisition functions, or highly sensitive to inaccurate priors. We propose DynMeanBO, a simple and general framework that incorporates expert priors into the Gaussian process mean function with a dynamic decay mechanism. This design allows BO to exploit expert knowledge in the early stages while gradually reverting to standard BO behavior, ensuring robustness against misleading priors while retaining the exploratory behavior of standard BO. DynMeanBO is broadly compatible with acquisition functions, introduces negligible computational cost, and comes with convergence guarantees under Expected Improvement and Upper Confidence Bound. Experiments on synthetic benchmarks and hyperparameter optimization tasks show that DynMeanBO accelerates convergence with informative priors and remains robust under biased ones.

Optimization · Zero-order and Black-box Optimization

Yimeng Zeng, Natalie Maus, Haydn Jones, Jeffrey Tao, Fangping Wan, Marcelo Der Torossian Torres, Cesar de la Fuente, Ryan Marcus, Osbert Bastani, Jacob Gardner

In multi-task Bayesian optimization, the goal is to leverage experience from optimizing existing tasks to improve the efficiency of optimizing new ones. While approaches using multi-task Gaussian processes or deep kernel transfer exist, the performance improvement is marginal when scaling beyond a moderate number of tasks. We introduce **BOLT**, an initialization-only transfer strategy that distills prior BO runs into an LLM which proposes candidates for new tasks, while the surrogate at test time remains single-task. The LLM is periodically fine-tuned on top solutions from completed runs, creating a closed loop where better BO outputs yield better initializations over time. This decoupled design scales to roughly 1500 tasks without the saturation observed for shared-surrogate MTBO and adds only a small, amortized overhead relative to the BO inner loops. We evaluate on two domains: database query optimization and antimicrobial peptide design. We demonstrate that LLM-generated initializations steadily improve and accelerate BO, and with sufficient fine-tuning, a few LLM samples often match or surpass full ''from-scratch'' BO with far fewer oracle calls.

Optimization · Zero-order and Black-box Optimization

David Stenger, Armin Lindicke, Alexander von Rohr, Sebastian Trimpe

Searching large and complex design spaces for a global optimum can be infeasible and unnecessary. A practical alternative is to iteratively refine the neighborhood of an initial design using local optimization methods such as gradient descent. We propose local entropy search (LES), a Bayesian optimization paradigm that explicitly targets the solutions reachable by the descent sequences of iterative optimizers. The algorithm propagates the posterior belief over the objective through the optimizer, resulting in a probability distribution over descent sequences. It then selects the next evaluation by maximizing mutual information with that distribution, using a combination of analytic entropy calculations and Monte-Carlo sampling of descent sequences. Empirical results on high-complexity synthetic objectives and benchmark problems show that LES achieves strong sample efficiency compared to existing local and global Bayesian optimization methods.

Optimization · Zero-order and Black-box Optimization

Anthony Bardou, Antoine Gonon, Aryan Ahadinia, Patrick Thiran

Bayesian Optimization (BO) is a powerful framework for optimizing noisy, expensive-to-evaluate black-box functions. When the objective exhibits invariances under a group action, exploiting these symmetries can substantially improve BO efficiency. While using maximum similarity across group orbits has long been considered in other domains, the fact that the max kernel is not positive semidefinite (PSD) has prevented its use in BO. In this work, we revisit this idea by considering a PSD projection of the max kernel. Compared to existing invariant (and non-invariant) kernels, we show it achieves significantly lower regret on both synthetic and real-world BO benchmarks, without increasing computational complexity.

Optimization · Global Optimization

Yilin Xie, Shiqiang Zhang, Jixiang Qing, Ruth Misener, Calvin Tsay

Graph-structured data are central to many scientific and industrial applications where the goal is to optimize expensive black-box objectives defined over graph structures or node configurations---as seen in molecular design, supply chains, and sensor placement. Bayesian optimization offers a principled approach for such settings, but existing methods largely focus on functions defined over nodes of a fixed graph. Moreover, graph optimization is often approached heuristically, and it remains unclear how to systematically incorporate structural constraints into BO. To address these gaps, we build on shortest-path graph kernels to develop a principled framework for acquisition optimization over unseen graph structures and associated node attributes. Through a novel formulation based on mixed-integer programming, we enable global exploration of the combinatorial domain over graph structures and explicit embedding of problem-specific constraints. We demonstrate that our method, BoGrape, is competitive both on general synthetic benchmarks and representative molecular design case studies with application-specific constraints.

Theory · Probabilistic Methods

Xinyu Zhang, Conor Hassan, Julien Martinelli, Daolang Huang, Samuel Kaski

Balancing competing objectives is omnipresent across disciplines, from drug design to autonomous systems. Multi-objective Bayesian optimization is a promising solution for such expensive, black-box problems: it fits probabilistic surrogates and selects new designs via an acquisition function that balances exploration and exploitation. In practice, it requires tailored choices of surrogate and acquisition that rarely transfer to the next problem, is myopic when multi-step planning is often required, and adds refitting overhead, particularly in parallel or time-sensitive loops. We present TAMO, a fully amortized, universal policy for multi-objective black-box optimization. TAMO uses a transformer architecture that operates across varying input and objective dimensions, enabling pretraining on diverse corpora and transfer to new problems without retraining: at test time, the pretrained model proposes the next design with a single forward pass. We pretrain the policy with reinforcement learning to maximize cumulative hypervolume improvement over full trajectories, conditioning on the entire query history to approximate the Pareto frontier. Across synthetic benchmarks and real tasks, TAMO produces fast proposals, reducing proposal time by 50–1000× versus alternatives while matching or improving Pareto quality under tight evaluation budgets. These results show that transformers can perform multi-objective optimization entirely in-context, eliminating per-task surrogate fitting and acquisition engineering, and open a path to foundation-style, plug-and-play optimizers for scientific discovery workflows.

Theory · Probabilistic Methods

Jingru Huang, Haijie Xu, Jie Guo, manrui jiang, Chen Zhang

Bayesian optimization (BO) has been widely used to optimize expensive and black-box functions across various domains. However, existing BO methods have not addressed tensor-output functions. To fill this gap, we propose a novel tensor-output BO framework. Specifically, we first introduce a tensor-output Gaussian process (TOGP) with two classes of tensor-output kernels as a surrogate model of the tensor-output function, which can effectively capture the structural dependencies within the tensor. Based on it, we develop an upper confidence bound (UCB) acquisition function to select query points. Furthermore, we introduce a more practical and challenging problem setting, termed combinatorial bandit Bayesian optimization (CBBO), where only a subset of the tensor outputs can be selected to contribute to the objective. To tackle this, we propose a tensor-output CBBO method, which extends TOGP to handle partially observed tensor outputs, and accordingly design a novel combinatorial multi-arm bandit-UCB2 (CMAB-UCB2) criterion to sequentially select both the query points and the output subset. We establish theoretical regret bounds for both methods, guaranteeing sublinear regret. Extensive experiments on synthetic and real-world datasets demonstrate the superiority of our methods.

General Machine Learning · Probabilistic Methods

Brendan Ross, Noël Vouitsis, Atiyeh Ashari Ghomi, Rasa Hosseinzadeh, Ji Xin, Zhaoyan Liu, Yi Sui, Shiyi Hou, Kin Kwan Leung, Gabriel Loaiza-Ganem 等

Although large language models (LLMs) are becoming increasingly capable of solving challenging real-world tasks, accurately quantifying their uncertainty remains a critical open problem—one that limits their applicability in high-stakes domains. This challenge is further compounded by the closed-source, black-box nature of many state-of-the-art LLMs. Moreover, LLM-based systems can be highly sensitive to the prompts that bind them together, which often require significant manual tuning (i.e., prompt engineering). In this work, we address these challenges by viewing LLM-based systems through a Bayesian lens. We interpret prompts as textual parameters in a statistical model, allowing us to use a small training dataset to perform Bayesian inference over these prompts. This novel perspective enables principled uncertainty quantification over both the model’s textual parameters and its downstream predictions, while also incorporating prior beliefs about these parameters expressed in free-form text. To perform Bayesian inference— a difficult problem even for well-studied data modalities—we introduce Metropolis-Hastings through LLM Proposals (MHLP), a novel Markov chain Monte Carlo (MCMC) algorithm that combines prompt optimization techniques with standard MCMC methods. MHLP is a turnkey modification to existing LLM pipelines, including those that rely exclusively on closed-source models. Empirically, we demonstrate that our method yields improvements in both predictive accuracy and uncertainty quantification (UQ) on a range of LLM benchmarks and UQ tasks. More broadly, our work demonstrates a viable path for incorporating methods from the rich Bayesian literature into the era of LLMs, paving the way for more reliable and calibrated LLM-based systems.

Optimization · Zero-order and Black-box Optimization

Nicolas Menet, Aleksandar Terzic, Michael Hersche, Andreas Krause, Abbas Rahimi

Bayesian optimization in large unstructured discrete spaces is often hindered by the computational cost of maximizing acquisition functions due to the absence of gradients. We propose a scalable alternative based on Thompson sampling that eliminates the need for acquisition function maximization by directly parameterizing the probability that a candidate yields the maximum reward. Our approach, *Thompson Sampling via Fine-Tuning* (ToSFiT) leverages the prior knowledge embedded in prompt-conditioned large language models, and incrementally adapts them toward the posterior. Theoretically, we derive a novel regret bound for a variational formulation of Thompson Sampling that matches the strong guarantees of its standard counterpart. Our analysis reveals the critical role of careful adaptation to the posterior probability of maximality—a principle that underpins our ToSFiT algorithm. Empirically, we validate our method on three diverse tasks: FAQ response refinement, thermally stable protein search, and quantum circuit design. Within a collection of methods covering in-context Bayesian optimization, reinforcement learning, and evolutionary search, ToSFiT exhibits both state-of-the-art sample efficiency and computational efficiency.

Optimization · Zero-order and Black-box Optimization

Giang Ngo, Dat Phan Trong, Dang Nguyen, Sunil Gupta, Svetha Venkatesh

Bayesian Optimization critically depends on the choice of acquisition function, but no single strategy is universally optimal; the best choice is non-stationary and problem-dependent. Existing adaptive portfolio methods often base their decisions on past function values while ignoring richer information like remaining budget or surrogate model characteristics. To address this, we introduce LMABO, a novel framework that casts a pre-trained Large Language Model (LLM) as a zero-shot, online strategist for the BO process. At each iteration, LMABO uses a structured state representation to prompt the LLM to select the most suitable acquisition function from a diverse portfolio. In an evaluation across 50 benchmark problems, LMABO demonstrates a significant performance improvement over strong static, adaptive portfolio, and other LLM-based baselines. We show that the LLM's behavior is a comprehensive strategy that adapts to real-time progress, proving its advantage stems from its ability to process and synthesize the complete optimization state into an effective, adaptive policy.

Junjie Huang, Jiarui Qin, Jianghao Lin, Ziming Feng, Weinan Zhang 0001, Yong Yu 0001

Recommender systems (RS) are pivotal in managing information overload in modern digital services. A key challenge in RS is efficiently processing vast item pools to deliver highly personalized recommendations under strict latency constraints. Multi-stage cascade ranking addresses this by employing computationally efficient retrieval methods to cover diverse user interests, followed by more precise ranking models to refine the results. In the retrieval stage, multi-channel retrieval is often used to generate distinct item subsets from different candidate generators, leveraging the complementary strengths of these methods to maximize coverage. However, forwarding all retrieved items overwhelms downstream rankers, necessitating truncation. Despite advancements in individual retrieval methods, multi-channel fusion, the process of efficiently merging multi-channel retrieval results, remains underexplored. We are the first to identify and systematically investigate multi-channel fusion in the retrieval stage. Current industry practices often rely on heuristic approaches and manual designs, which often lead to suboptimal performance. Moreover, traditional gradient-based methods like SGD are unsuitable for this task due to the non-differentiable nature of the selection process. In this paper, we explore advanced channel fusion strategies by assigning systematically optimized weights to each channel. We utilize black-box optimization techniques, including the Cross Entropy Method and Bayesian Optimization for global weight optimization, alongside policy gradient-based approaches for personalized merging. Our methods enhance both personalization and flexibility, achieving significant performance improvements across multiple datasets and yielding substantial gains in real-world deployments, offering a scalable solution for optimizing multi-channel fusion in retrieval.

Tianxing Man, Xingchen Li, Zhaogeng Liu, Haozhen Zhang, Bin Gu 0001, Yi Chang 0001

Black-box attacks have emerged as a significant threat to deep neural networks. This challenge is particularly difficult in discrete sequential data compared to continuous data. Recently, the Blockwise Bayesian Attack (BBA) leveraging discrete Bayesian optimization with an adapted RBF kernel has gained prominence as a cutting-edge solution. However, it relies solely on alignment information (i.e., positional differences) within the RBF kernel, which may not fully capture the information (such as statistical, structural, and semantic information) inherent in discrete sequential data and potentially lacks the desired inductive bias necessary to approximate the target function accurately. To overcome this limitation, this paper proposes a novel bilevel Bayesian optimization approach to adaptively learn a hybrid space that better captures the similarity between discrete sequences. Specifically, we introduce a multi-kernel mechanism that incorporates multiple types of information, creating a more comprehensive similarity measure. Moreover, we develop a bilevel Bayesian optimization algorithm, where the outer-level objective determines the optimal weights of the multiple kernels, while the inner-level objective identifies the optimal adversarial sequence. Extensive experiments conducted on discrete sequential data demonstrate that our approach ensures secure multi-kernel selection and achieves a higher attack success rate with only a few additional queries, compared to BBA and other traditional optimization strategies.

Xiang Xia, Xiang Shu, Shuo Liu, Yiyi Zhu, Yijie Zhou, Weiye Wang, Bingdong Li, Hong Qian

In black-box optimization, when directly evaluating the function values of solutions is very costly or infeasible, access to the objective function is often limited to comparing pairs of solutions, which yields dueling black-box optimization. Dueling optimization is solely based on pairwise preferences, and thus notably reduces cost compared with function value based methods. However, the optimization performance of dueling optimization is often limited due to that most existing dueling optimization methods do not make full use of the pairwise preferences collected. To better utilize these preferences, this paper proposes relation-augmented dueling Bayesian optimization (RADBO) via preference propagation. By considering solution similarity, RADBO aims to uncover the potential dueling relations between solutions within different preferences through the proposed preference propagation technique. Specifically, RADBO first clusters solutions using a Gaussian mixture model. After obtaining the solution set with the highest intra-cluster similarity, RADBO utilizes a directed hypergraph to model the potential dueling relations between solutions, thereby realizing relation augmentation. Extensive experiments are conducted on both synthetic functions and real-world tasks such as motion control, car cab design and spacecraft trajectory optimization. The experimental results disclose the satisfactory accuracy of augmented preferences in RADBO, and show the superiority of RADBO compared with existing dueling optimization methods. Notably, it is verified that, under the same evaluation cost budget, RADBO can be competitive with or even surpass the function value based Bayesian optimization methods with respect to optimization performance.

Vu Viet Hoang, Quoc Anh Hoang Nguyen, Hung The Tran

Bayesian Optimization (BO) is a widely-used method for optimizing expensive-to-evaluate black-box functions. Traditional BO assumes that the learner has full control over all query variables without additional constraints. However, in many real-world scenarios, controlling certain query variables may incur costs. Therefore, the learner needs to balance the selection of informative subsets for targeted learning against leaving some variables to be randomly sampled to minimize costs. This problem is known as Bayesian Optimization with cost-varying variable subsets (BOCVS). While the goal of BOCVS is to identify the optimal solution with minimal cost, previous works have only guaranteed finding the optimal solution without considering the total costs incurred. Moreover, these works assume precise knowledge of the cost for each subset, which is often unrealistic. In this paper, we propose a novel algorithm for the extension of the BOCVS problem with random and unknown costs that separates the process into exploration and exploitation phases. The exploration phase will filter out low-quality variable subsets, while the exploitation phase will leverage high-quality ones. Furthermore, we theoretically demonstrate that our algorithm achieves a sub-linear rate in both quality regret and cost regret, addressing the objective of the BOCVS problem more effectively than previous analyses. Finally, we show that our proposed algorithm outperforms comparable baselines across a wide range of benchmarks.

Haitao Li, Qingyao Ai, Jia Chen, Qian Dong, Zhijing Wu, Yiqun Liu

Large Language Models (LLMs) like ChatGPT and GPT-4 are versatile and capable of addressing open-domain question-answering(QA) tasks effectively. However, general LLMs, which are developed on open-domain data, may lack the domain-specific knowledge essential for tasks in vertical domains, such as legal, medical, etc. To address this issue, previous approaches either conduct continuous pre-training with domain-specific data or employ retrieval augmentation to support general LLMs in handling QA tasks. Unfortunately, these strategies are either cost-intensive or unreliable in practical applications. To this end, we present a novel framework named BLADE, which enhances Black-box LArge language models with small Domain-spEcific models. BLADE consists of a black-box LLM and a small domain-specific LM. The small LM preserves domain-specific knowledge and offers specialized insights, while the general LLM contributes robust language comprehension and reasoning capabilities. Specifically, our method involves three steps: 1) pre-training the small LM with domain-specific data, 2) fine-tuning this model using knowledge instruction data, and 3) joint Bayesian optimization of the general LLM and the small LM. In our experiments, we verify the effectiveness of BLADE on diverse LLMs and datasets across different domains. This shows the potential of BLADE as an effective and cost-efficient solution in adapting general LLMs for vertical domains.

Lam Ngo, Huong Ha, Jeffrey Chan, Hongyu Zhang

When it comes to expensive black-box optimization problems, Bayesian Optimization (BO) is a well-known and powerful solution. Many real-world applications involve a large number of dimensions, hence scaling BO to high dimension is of much interest. However, state-of-the-art high-dimensional BO methods still suffer from the curse of dimensionality, highlighting the need for further improvements. In this work, we introduce BOIDS, a novel high-dimensional BO algorithm that guides optimization by a sequence of one-dimensional direction lines using a novel tailored line-based optimization procedure. To improve the efficiency, we also propose an adaptive selection technique to identify most optimal lines for each round of line-based optimization. Additionally, we incorporate a subspace embedding technique for better scaling to high-dimensional spaces. We further provide theoretical analysis of our proposed method to analyze its convergence property. Our extensive experimental results show that BOIDS outperforms state-of-the-art baselines on various synthetic and real-world benchmark problems.

Jiaxing Li, Wei Liu, Chao Xue, Yibing Zhan, Xiaoxing Wang, Weifeng Liu, Dacheng Tao

Bayesian Optimization (BO) is a sample-efficient black-box optimizer commonly used in search spaces where hyperparameters are independent. However, in many practical AutoML scenarios, there will be dependencies among hyperparameters, forming a conditional search space, which can be partitioned into structurally distinct subspaces. The structure and dimensionality of hyperparameter configurations vary across these subspaces, challenging the application of BO. Some previous BO works have proposed solutions to develop multiple Gaussian Process models in these subspaces. However, these approaches tend to be inefficient as they require a substantial number of observations to guarantee each GP's performance and cannot capture relationships between hyperparameters across different subspaces. To address these issues, this paper proposes a novel approach to model the response surfaces of all subspaces in one, which can model the relationships between hyperparameters elegantly via a self-attention mechanism. Concretely, we design a structure-aware hyperparameter embedding to preserve the structural information. Then, we introduce an attention-based deep feature extractor, capable of projecting configurations with different structures from various subspaces into a unified feature space, where the response surfaces can be formulated using a single standard Gaussian Process. The empirical results on a simulation function, various real-world tasks, and HPO-B benchmark demonstrate that our proposed approach improves the efficacy and efficiency of BO within conditional search spaces.

Yuanhao Lai, Pengfei Zheng, Chenpeng Ji, Cheng Qiu, Tingkai Wang, Songhan Zhang, Zhengang Wang, Yunfei Du

Bayesian optimization (BO) is a key technique for solving black-box optimization problems. This study extends the scope of BO from conventional applications (e.g., AutoML and robotics learning) to automated tuning of software systems. Despite GP (Gaussian Process) implementing a foundation formalism for exploitation and exploration in BO, its limited predictive power and unrealistic assumptions (e.g., continuity and Gaussianity) can severely affect its effectiveness and efficiency in tuning complex software systems. To overcome these limitations, we propose a BO framework CoffeeBoost, which implements exploitation and exploration with a GBDT-native distribution-free probabilistic surrogate model. CoffeeBoost constructs surrogate models via stochastic gradient boosting ensembles (SGBE) and quantifies probabilistic distributions via distribution-free conformal predictive systems. Moreover, CoffeeBoost leverages the residual paths in SGBE to improve the local adaptiveness of the resulting predictive distributions in a GBDT-native manner. Across eight auto-tuning benchmarks for database management systems (DBMS), we evaluate CoffeeBoost and show its superior learnability and optimizability against existing GP-based and tree-ensemble-based BO schemes. Detailed analysis further shows CoffeeBoost's predictive distributions excel in both coverage and tightness.

Shuyang Wang, Somayeh Moazeni, Diego Klabjan

Designing effective prompts is essential to guiding large language models (LLMs) toward desired responses. Automated prompt engineering aims to reduce reliance on manual efforts by streamlining the design, refinement, and optimization of natural language prompts. This paper proposes an optimal learning framework for automated prompt engineering for black-box models, designed to sequentially identify effective prompt features under limited evaluation budgets. We introduce a feature-based method to express prompt templates, which significantly broadens the search space. Bayesian regression is employed to utilize correlations among similar prompts, accelerating the learning process. To efficiently explore the large space of prompt features, we adopt the forward-looking Knowledge-Gradient (KG) policy for sequential optimal learning efficiently by solving mixed-integer second-order cone optimization problems, making it scalable and capable of accommodating prompts characterized only through constraints. Our method significantly outperforms a set of benchmark strategies assessed on instruction induction tasks within limited iterations of prompt evaluations, showing the potential of optimal learning for efficient prompt learning.