Bayesian Optimization

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Overview

Bayesian optimization studies how to find good inputs to an expensive or unknown process—such as a simulation, experiment, or model configuration—using as few evaluations as possible. It typically fits a probabilistic surrogate, often a Gaussian process, then uses an acquisition function to balance trying promising inputs with exploring uncertain ones. Current approaches extend this loop with trust regions, multi-fidelity data, constraints, multiple objectives, changing conditions, and human or language-model guidance. A central challenge is choosing models and search strategies that remain reliable in high-dimensional or structured spaces without making inference and candidate selection too costly; theoretical guarantees do not always predict practical performance.

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240 papers45 papers in the last four weeks, up 36% on the four weeks before. 0.3% of all new papers.

Mar 11, 2026

Bayesian Optimization of Partially Known Systems using Hybrid Models

Bayesian optimization (BO) has gained attention as an efficient algorithm for black-box optimization of expensive-to-evaluate systems, where the BO algorithm iteratively queries the system and suggests new trials based on a probabilistic model fitted to previous samples. Still, the standard BO loop may require a prohibitively large number of experiments to converge to the optimum, especially for high-dimensional and nonlinear systems. We present a hybrid model-based BO formulation that combines the iterative Bayesian learning of BO with partially known mechanistic physical models. Instead of learning a direct mapping from inputs to the objective, we write all known equations for a physics-based model and infer expressions for variables missing equations using a probabilistic model, in our case, a Gaussian process (GP). The final formulation then includes the GP as a constraint in the hybrid model, thereby allowing other physics-based nonlinear and implicit model constraints. This hybrid model formulation yields a constrained, nonlinear stochastic program, which we discretize using the sample-average approximation. In an in-silico optimization of a single-stage distillation, the hybrid BO model based on mass conservation laws yields significantly better designs than a standard BO loop. Furthermore, the hybrid model converges in as few as one iteration, depending on the initial samples, whereas, the standard BO does not converge within 25 for any of the seeds. Overall, the proposed hybrid BO scheme presents a promising optimization method for partially known systems, leveraging the strengths of both mechanistic modeling and data-driven optimization.

Eike Cramer, Luis Kutschat, Oliver Stollenwerk +2

University College London, Sargent Centre for Process Systems Engineering, Department of Chemical Engineering, London WC1E 7JE, United Kingdom · RWTH Aachen University, Process Systems Engineering (AVT.SVT), Aachen 52074, Germany · University of Wisconsin–Madison, Department of Chemical & Biological Engineering, Madison, WI 53706, USA +2

Bayesian OptimizationConstrained Optimization
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Mar 11, 2026

A Tutorial Review of Bayesian Optimization with Gaussian Processes to Accelerate Stationary Point Searches

Building local surrogates to accelerate stationary point searches on potential energy surfaces spans decades of effort. Done correctly, surrogates can reduce the number of expensive electronic structure evaluations by roughly an order of magnitude while preserving the accuracy of the underlying theory, with the gain depending on oracle cost, search distance, and the availability of analytical forces. We present a unified Bayesian optimization view of minimization, single-point saddle searches, and double-ended path searches: all three share one six-step surrogate loop and differ only in the inner optimization target and the acquisition criterion. The framework uses Gaussian process regression with derivative observations, inverse-distance kernels, and active learning, and we develop optional extensions for production use, including farthest-point sampling with the Earth Mover's Distance, MAP regularization, an adaptive trust radius, and random Fourier features for scaling. Accompanying pedagogical Rust code demonstrates that all three applications use the same Bayesian optimization loop, bridging the gap between theoretical formulation and practical execution.

Rohit Goswami

Institute IMX and Lab-COSMO, École polytechnique fédérale de Lausanne · Institute IMX and Lab-COSMO, École polytechnique fédérale de Lausanne (EPFL), Station 12, CH-1015 Lausanne, Switzerland

Computational Quantum ChemistryBayesian Optimization
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Mar 10, 2026

Information Theoretic Bayesian Optimization over the Probability Simplex

Bayesian optimization is a data-efficient technique that has been shown to be extremely powerful to optimize expensive, black-box, and possibly noisy objective functions. Many applications involve optimizing probabilities and mixtures which naturally belong to the probability simplex, a constrained non-Euclidean domain defined by non-negative entries summing to one. This paper introduces αα-GaBO, a novel family of Bayesian optimization algorithms over the probability simplex. Our approach is grounded in information geometry, a branch of Riemannian geometry which endows the simplex with a Riemannian metric and a class of connections. Based on information geometry theory, we construct Matérn kernels that reflect the geometry of the probability simplex, as well as a one-parameter family of geometric optimizers for the acquisition function. We validate our method on benchmark functions and on a variety of real-world applications including mixtures of components, mixtures of classifiers, and a robotic control task, showing its increased performance compared to constrained Euclidean approaches.

Federico Pavesi, Antonio Candelieri, Noémie Jaquier

Department of Robotics, Perception and Learning, KTH Royal Institute of Technology, Stockholm, Sweden · Department of Economics, Management and Statistics, University of Milan, Bicocca, Milan, Italy

Bayesian OptimizationConstrained OptimizationRiemannian Optimization
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Mar 10, 2026

On Regret Bounds of Thompson Sampling for Bayesian Optimization

We study a widely used Bayesian optimization method, Gaussian process Thompson sampling (GP-TS), under the assumption that the objective function is a sample path from a GP. Compared with the GP upper confidence bound (GP-UCB) with established high-probability and expected regret bounds, most analyses of GP-TS have been limited to expected regret. Moreover, whether the recent analyses of GP-UCB for the lenient regret and the improved cumulative regret upper bound can be applied to GP-TS remains unclear. To fill these gaps, this paper shows several regret bounds: (i) a regret lower bound for GP-TS, which implies that GP-TS suffers from a polynomial dependence on 1/δ1/δ with probability δδ, (ii) an upper bound of the second moment of cumulative regret, which directly suggests an improved regret upper bound on δδ, (iii) expected lenient regret upper bounds, and (iv) an improved cumulative regret upper bound on the time horizon TT. Along the way, we provide several useful lemmas, including a relaxation of the necessary condition from recent analysis to obtain improved regret upper bounds on TT.

Shion Takeno, Shogo Iwazaki

Nagoya University · MI-6 Ltd.

Bayesian OptimizationThompson Sampling
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Mar 9, 2026

Advancing Automated Algorithm Design via Evolutionary Stagewise Design with LLMs

With the rapid advancement of human science and technology, problems in industrial scenarios are becoming increasingly challenging, bringing significant challenges to traditional algorithm design. Automated algorithm design with LLMs emerges as a promising solution, but the currently adopted black-box modeling deprives LLMs of any awareness of the intrinsic mechanism of the target problem, leading to hallucinated designs. In this paper, we introduce Evolutionary Stagewise Algorithm Design (EvoStage), a novel evolutionary paradigm that bridges the gap between the rigorous demands of industrial-scale algorithm design and the LLM-based algorithm design methods. Drawing inspiration from CoT, EvoStage decomposes the algorithm design process into sequential, manageable stages and integrates real-time intermediate feedback to iteratively refine algorithm design directions. To further reduce the algorithm design space and avoid falling into local optima, we introduce a multi-agent system and a "global-local perspective" mechanism. We apply EvoStage to the design of two types of common optimizers: designing parameter configuration schedules of the Adam optimizer for chip placement, and designing acquisition functions of Bayesian optimization for black-box optimization. Experimental results across open-source benchmarks demonstrate that EvoStage outperforms human-expert designs and existing LLM-based methods within only a couple of evolution steps, even achieving the historically state-of-the-art half-perimeter wire-length results on every tested chip case. Furthermore, when deployed on a commercial-grade 3D chip placement tool, EvoStage significantly surpasses the original performance metrics, achieving record-breaking efficiency. We hope EvoStage can significantly advance automated algorithm design in the real world, helping elevate human productivity.

Chen Lu, Ke Xue, Chengrui Gao +5

National Key Laboratory for Novel Software Technology, Nanjing University, China · School of Artificial Intelligence, Nanjing University, China · Huawei Noah’s Ark Lab, China

Automated Algorithm DiscoveryAutomated Heuristic DesignEvolutionary OptimizationElectronic Design AutomationBayesian Optimization
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Mar 9, 2026

Local Constrained Bayesian Optimization

Bayesian optimization (BO) for high-dimensional constrained problems remains a significant challenge due to the curse of dimensionality. We propose Local Constrained Bayesian Optimization (LCBO), a novel framework tailored for such settings. Unlike trust-region methods that are prone to premature shrinking when confronting tight or complex constraints, LCBO leverages the differentiable landscape of constraint-penalized surrogates to alternate between rapid local descent and uncertainty-driven exploration. Theoretically, we prove that LCBO achieves a convergence rate for the Karush-Kuhn-Tucker (KKT) residual that depends polynomially on the dimension dd for common kernels under mild assumptions, offering a rigorous alternative to global BO where regret bounds typically scale exponentially. Extensive evaluations on high-dimensional benchmarks (up to 100D) demonstrate that LCBO consistently outperforms state-of-the-art baselines.

Jing Jingzhe, Fan Zheyi, Szu Hui Ng +1

State Key Laboratory of Mathematical Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, China · School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, China · Department of Industrial Systems Engineering & Management, National University of Singapore, Singapore.

Surrogate-Assisted OptimizationBayesian OptimizationConstrained Optimization
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Mar 6, 2026

Multi-Robot Trajectory Planning via Constrained Bayesian Optimization and Local Cost Map Learning with STL-Based Conflict Resolution

We address multi-robot motion planning under Signal Temporal Logic (STL) specifications with kinodynamic constraints. Exact approaches face scalability bottlenecks and limited adaptability, while conventional sampling-based methods require excessive samples to construct optimal trajectories. We propose a two-stage framework integrating sampling-based online learning with formal STL reasoning. At the single-robot level, our constrained Bayesian Optimization-based Tree search (cBOT) planner uses a Gaussian process as a surrogate model to learn local cost maps and feasibility constraints, generating shorter collision-free trajectories with fewer samples. At the multi-robot level, our STL-enhanced Kinodynamic Conflict-Based Search (STL-KCBS) algorithm incorporates STL monitoring into conflict detection and resolution, ensuring specification satisfaction while maintaining scalability and probabilistic completeness. Benchmarking demonstrates improved trajectory efficiency and safety over existing methods. Real-world experiments with autonomous surface vehicles validate robustness and practical applicability in uncertain environments. The STLcBOT Planner will be released as an open-source package, and videos of real-world and simulated experiments are available at https://stlbot.github.io/.

Sourav Raxit, Abdullah Al Redwan Newaz, Jose Fuentes +3

Department of Computer Science, University of New Orleans, New Orleans, LA 70148, USA · Knight Foundation School of Computing and Information Sciences, Florida International University, Miami, FL 33199, USA · Providence College, Department of Mathematics & Computer Science, Providence, RI 02918, USA

STL Motion PlanningMulti-Robot Motion PlanningConstrained Motion PlanningKinodynamic Motion PlanningConflict-Based SearchBayesian Optimization
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Mar 3, 2026

LAGO: A Local-Global Optimization Framework Combining Trust Region Methods and Bayesian Optimization

We introduce LAGO, a LocAl-Global Optimization framework coupling Bayesian Optimization (BO) and gradient-based trust region local refinement through an adaptive competition mechanism for smooth expensive-to-evaluate objective functions with available gradients. At each iteration, global and local optimization strategies independently propose candidate points, and the next evaluation is selected based on predicted improvement. LAGO separates global exploration from local refinement at the proposal level: the BO acquisition function is optimized outside the active trust region, while local candidates are proposed within the trust region. Points in the vicinity of the accepted local step are incorporated in the global GP dataset only when satisfying a lengthscale-based minimum-distance criterion, hence reducing the risk of numerical instability during local exploitation. LAGO enhances BO with efficient local refinement when reaching promising regions, and reverts to exploratory behavior when local steps are not competitive.

Eliott Van Dieren, Tommaso Vanzan, Fabio Nobile

Institute of Mathematics, EPFL, Lausanne, Switzerland · Dipartimento di Scienze Matematiche, Politecnico di Torino, Turin, Italy

Black-Box OptimizationBayesian OptimizationTrust-Region Optimization
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Mar 3, 2026

Deep learning-guided evolutionary optimization for protein design

Designing novel proteins with desired characteristics remains a significant challenge due to the large sequence space and the complexity of sequence-function relationships. Efficient exploration of this space to identify sequences that meet specific design criteria is crucial for advancing therapeutics and biotechnology. Here, we present BoGA (Bayesian Optimization Genetic Algorithm), a framework that combines evolutionary search with Bayesian optimization to efficiently navigate the sequence space. By integrating a genetic algorithm as a stochastic proposal generator within a surrogate modeling loop, BoGA prioritizes candidates based on prior evaluations and surrogate model predictions, enabling data-efficient optimization. We demonstrate the utility of BoGA through benchmarking on sequence and structure design tasks, followed by its application in designing peptide binders against pneumolysin, a key virulence factor of Streptococcus pneumoniae. BoGA accelerates the discovery of high-confidence binders, demonstrating the potential for efficient protein design across diverse objectives. The algorithm is implemented within the BoPep suite and is available under an MIT license at GitHub.

Erik Hartman, Di Tang, Johan Malmström

Division of Infection Medicine, Faculty of Medicine, Lund University, Sweden

Protein DesignSurrogate-Assisted OptimizationProtein Binder DesignGenetic AlgorithmsBayesian Optimization
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Mar 2, 2026

Randomized Kriging Believer for Parallel Bayesian Optimization with Regret Bounds

We consider the optimization problem of an expensive-to-evaluate black-box function, in which we can obtain noisy function values in parallel. For this problem, parallel Bayesian optimization (PBO) is a promising approach, which aims to optimize with fewer function evaluations by selecting a diverse input set for parallel evaluation. However, existing PBO methods suffer from poor practical performance or lack theoretical guarantees. In this study, we propose a PBO method, called randomized kriging believer (KB), based on a well-known KB heuristic and inheriting the advantages of the original KB: low computational complexity, a simple implementation, versatility across various BO methods, and applicability to asynchronous parallelization. Furthermore, we show that our randomized KB achieves Bayesian expected regret guarantees. We demonstrate the effectiveness of the proposed method through experiments, including those on real-data emulators.

Shuhei Sugiura, Ichiro Takeuchi, Shion Takeno

Department of Engineering, Nagoya University, Aichi, Japan · RIKEN AIP, Tokyo, Japan

Bayesian Optimization
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Mar 1, 2026

Time-Aware Latent Space Bayesian Optimization

Latent-space Bayesian optimization (LSBO) extends Bayesian optimization to structured domains, such as molecular design, by searching in the continuous latent space of a generative model. However, most LSBO methods assume a fixed objective, whereas real design campaigns often face temporal drift (e.g., evolving preferences or shifting targets). Bringing time-varying BO into LSBO is nontrivial: drift can affect not only the surrogate, but also the latent search space geometry induced by the representation. We propose Time-Aware Latent-space Bayesian Optimization (TALBO), which incorporates time in both the surrogate and the learned generative representation via a GP-prior variational autoencoder, yielding a latent space aligned as objectives evolve. To evaluate timevarying LSBO systematically, we adapt widely used molecular design tasks to drifting multi-property objectives and introduce metrics tailored to changing targets. Across these benchmarks, TALBO consistently outperforms strong LSBO baselines and remains robust across drift speeds and design choices, while remaining competitive under actually time-invariant objectives.

Tuan A. Vu, Julien Martinelli, Harri Lähdesmäki

Aalto University

Molecule DesignBayesian OptimizationLatent Space OptimizationVariational Autoencoders
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Feb 26, 2026

Bayesian Preference Elicitation: Human-In-The-Loop Optimization of An Active Prosthesis

Tuning active prostheses for people with amputation is time-consuming and relies on metrics that may not fully reflect user needs. We introduce a human-in-the-loop optimization (HILO) approach that leverages direct user preferences to personalize a standard four-parameter prosthesis controller efficiently. Our method employs preference-based Multiobjective Bayesian Optimization that uses a state-or-the-art acquisition function especially designed for preference learning, and includes two algorithmic variants: a discrete version (EUBO-LineCoSpar), and a continuous version (BPE4Prost). Simulation results on benchmark functions and real-application trials demonstrate efficient convergence, robust preference elicitation, and measurable biomechanical improvements, illustrating the potential of preference-driven tuning for user-centered prosthesis control.

Sophia Taddei, Wouter Koppen, Eligia Alfio +8

Federated Labs Robotics & AI, Vrije Universiteit Brussel. · Human Physiology and Sports Physiotherapy Research Group, Vrije Universiteit Brussel. · IDLab, Ghent University – imec. +1

Preference ElicitationHuman-in-the-Loop AIBayesian Optimization
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Feb 20, 2026

Gradient-Informed Bayesian and Interior Point Optimization for Efficient Inverse Design in Nanophotonics

Inverse design, particularly geometric shape optimization, provides a systematic approach for developing high-performance nanophotonic devices. While numerous optimization algorithms exist, previous global approaches exhibit slow convergence and conversely local search strategies frequently become trapped in local optima. To address the limitations inherent to both local and global approaches, we introduce BONNI: Bayesian optimization through neural network ensemble surrogates with interior point optimization. It augments global optimization with an efficient incorporation of gradient information to determine optimal sampling points. This capability allows BONNI to circumvent the local optima found in many nanophotonic applications, while capitalizing on the efficiency of gradient-based optimization. We demonstrate BONNI's capabilities in the design of a distributed Bragg reflector as well as a dual-layer grating coupler through an exhaustive comparison against other optimization algorithms commonly used in literature. Using BONNI, we were able to design a 10-layer distributed Bragg reflector with only 4.5% mean spectral error, compared to the previously reported results of 7.8% error with 16 layers. Further designs of a broadband waveguide taper and photonic crystal waveguide transition validate the capabilities of BONNI.

Yannik Mahlau, Yannick Augenstein, Tyler W. Hughes +2

Institute of Information Processing, Leibniz University Hannover, Germany · Flexcompute, Watertown, MA, USA · Institute of Artificial Intelligence, Leibniz University Hannover, Germany +1

Inverse DesignSurrogate-Assisted OptimizationBayesian Optimization
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Feb 16, 2026

Extending Multi-Source Bayesian Optimization With Causality Principles

Multi-Source Bayesian Optimization (MSBO) serves as a variant of the traditional Bayesian Optimization (BO) framework applicable to situations involving optimization of an objective black-box function over multiple information sources such as simulations, surrogate models, or real-world experiments. However, traditional MSBO assumes the input variables of the objective function to be independent and identically distributed, limiting its effectiveness in scenarios where causal information is available and interventions can be performed, such as clinical trials or policy-making. In the single-source domain, Causal Bayesian Optimization (CBO) extends standard BO with the principles of causality, enabling better modeling of variable dependencies. This leads to more accurate optimization, improved decision-making, and more efficient use of low-cost information sources. In this article, we propose a principled integration of the MSBO and CBO methodologies in the multi-source domain, leveraging the strengths of both to enhance optimization efficiency and reduce computational complexity in higher-dimensional problems. We present the theoretical foundations of both Causal and Multi-Source Bayesian Optimization, and demonstrate how their synergy informs our Multi-Source Causal Bayesian Optimization (MSCBO) algorithm. We compare the performance of MSCBO against its foundational counterparts for both synthetic and real-world datasets with varying levels of noise, highlighting the robustness and applicability of MSCBO. Based on our findings, we conclude that integrating MSBO with the causality principles of CBO facilitates dimensionality reduction and lowers operational costs, ultimately improving convergence speed, performance, and scalability.

Luuk Jacobs, Mohammad Ali Javidian

Radboud University Nijmegen, The Netherlands · Appalachian State University Boone, United States

Bayesian OptimizationBlack-Box Optimization
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Feb 16, 2026

Frequentist Regret Analysis of Gaussian Process Thompson Sampling via Fractional Posteriors

We study Gaussian Process Thompson Sampling (GP-TS) for sequential decision-making over compact, continuous action spaces and provide a frequentist regret analysis based on fractional Gaussian process posteriors, without relying on domain discretization as in prior work. We show that the variance inflation commonly assumed in existing analyses of GP-TS can be interpreted as Thompson Sampling with respect to a fractional posterior with tempering parameter α∈(0,1)α\in (0,1). We derive a kernel-agnostic regret bound expressed in terms of the information gain parameter γtγ_t and the posterior contraction rate εtε_t, and identify conditions on the Gaussian process prior under which εtε_t can be controlled. As special cases of our general bound, we recover regret of order O~(T12)\tilde{\mathcal{O}}(T^{\frac{1}{2}}) for the squared exponential kernel, O~(T2ν+3d2(2ν+d))\tilde{\mathcal{O}}(T^{\frac{2ν+3d}{2(2ν+d)}} ) for the Matérn-νν kernel, and a bound of order O~(T2ν+3d2(2ν+d))\tilde{\mathcal{O}}(T^{\frac{2ν+3d}{2(2ν+d)}}) for the rational quadratic kernel. Overall, our analysis provides a unified and discretization-free regret framework for GP-TS that applies broadly across kernel classes.

Somjit Roy, Prateek Jaiswal, Anirban Bhattacharya +2

Department of Statistics, Texas A&M University, College Station, TX 77843, USA · Daniels School of Business, Purdue University, West Lafayette, IN 47907, USA · Department of Statistics, University of Wisconsin-Madison, Madison, WI 53706, USA

Gaussian Process BanditsThompson SamplingBayesian OptimizationGaussian ProcessesPosterior SamplingMulti-Armed Bandits
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Feb 6, 2026

BONSAI: Bayesian Optimization with Natural Simplicity and Interpretability

Bayesian optimization (BO) is a popular technique for sample-efficient optimization of black-box functions. In many applications, the parameters being tuned come with a carefully engineered default configuration, and practitioners only want to deviate from this default when necessary. Standard BO, however, does not aim to minimize deviation from the default and, in practice, often pushes weakly relevant parameters to the boundary of the search space. This makes it difficult to distinguish between important and spurious changes and increases the burden of vetting recommendations when the optimization objective omits relevant operational considerations. We introduce BONSAI, a default-aware BO policy that prunes low-impact deviations from a default configuration while explicitly controlling the loss in acquisition value. BONSAI is compatible with a variety of acquisition functions, including expected improvement and upper confidence bound (GP-UCB). We theoretically bound the regret incurred by BONSAI, showing that, under appropriate conditions, it retains the no-regret property of vanilla GP-UCB and removes irrelevant changes. Across many real-world applications, we empirically find that BONSAI substantially reduces the number of non-default parameters in recommended configurations while maintaining competitive optimization performance with little effect on wall time. Its candidate-generation cost averages only 1.5×1.5\times that of standard BO, compared with 77-34×34\times for prior sparse-BO methods.

Samuel Daulton, David Eriksson, Maximilian Balandat +1

Meta

Bayesian OptimizationBlack-Box Optimization
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Feb 5, 2026

Selecting Hyperparameters for Tree-Boosting

Tree-boosting is a widely used machine learning technique for tabular data. However, its out-of-sample accuracy is critically dependent on multiple hyperparameters. In this article, we empirically compare several popular methods for hyperparameter optimization for tree-boosting including random grid search, the tree-structured Parzen estimator (TPE), Gaussian-process-based Bayesian optimization (GP-BO), Hyperband, the sequential model-based algorithm configuration (SMAC) method, and deterministic full grid search using 5959 regression and binary classification data sets. We find that the SMAC method clearly outperforms all the other considered methods on average, and it gives stable performance across a diverse collection of tabular data sets under a fixed tuning budget, which is relevant for users who cannot afford extensive manual trial-and-error tuning. We further observe that (i) a relatively large number of trials larger than 100100 is typically required for accurate tuning, (ii) using default values for hyperparameters or a full search over a small grid often yields very inaccurate models, (iii) all considered hyperparameters can have a material effect on the accuracy of tree-boosting, i.e., there is no small set of hyperparameters that is more important than others, and (iv) choosing the number of boosting iterations using early stopping yields more accurate results compared to including it in the search space for regression tasks.

Floris Jan Koster, Fabio Sigrist

Seminar for Statistics, ETH Zurich, Switzerland

Gradient-Boosted Decision TreesBayesian OptimizationBinary ClassificationHyperparameter Optimization
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Jan 18, 2026

Tree-Structured Synergy of Large Language Models and Bayesian Optimization for Efficient CASH

To lower the expertise barrier in machine learning, the AutoML community has focused on the CASH problem, which jointly automates algorithm selection and hyperparameter tuning. While traditional methods like Bayesian Optimization (BO) struggle with cold-start issues, Large Language Models (LLMs) can mitigate these through semantic priors. However, existing LLM-based optimizers generalize poorly to high-dimensional, structured CASH spaces. CASH requires coordinated exploration and exploitation both across candidate algorithms and within their algorithm-specific hyperparameter spaces. Yet existing approaches often rely on flat optimization histories that mix observations across algorithms and obscure configuration-refinement trajectories. In this paper, we propose LB-MCTS, a trajectory-structured optimization framework that uses a Monte Carlo Tree Search tree as a shared state for algorithm selection, hyperparameter refinement, and BO-LLM proposer synergy. Within this shared state, BO provides algorithm-specific surrogate modeling for quantitative search, while the LLM exploits path-aware selective memory to generate semantic proposals and reflections. As the surrogate model improves, a reliability-aware proposer policy adaptively shifts from LLM-driven to BO-driven proposals within a unified search trajectory. Experiments on 104 AMLB datasets demonstrate that LB-MCTS consistently outperforms BO-based, LLM-based, and hybrid baselines. Code: https://github.com/PKU-DAIR/open-box.

Beicheng Xu, Weitong Qian, Lingching Tung +2

School of Computer Science, Peking University, Beijing, China

Automated Machine LearningHyperparameter OptimizationLarge Language Model-Guided OptimizationBayesian OptimizationMonte Carlo Tree Search
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Jan 13, 2026

Amortized Acquisition Optimization for Bayesian Optimization with Variational Mutual Information

Bayesian optimization (BO) of expensive black-box functions is traditionally addressed with Gaussian processes (GPs), which scale cubically with observations, or Bayesian neural networks (BNNs), which incur costly posterior sampling and inner-loop acquisition optimization. We propose VBO-MI (Variational Bayesian Optimization with Mutual Information), a fully gradient-based BO framework that requires no explicit GP prior or fixed parametric posterior family over objective function and treats it as a strict black box. An actor-critic architecture pairs an action-net with a variational critic that estimates information gain, eliminating the acquisition optimization bottleneck and achieving up to 102×10^{2}\times fewer FLOPs than BNN-BO baselines. A lightweight surrogate network further reduces real function queries to one batch per iteration. We establish consistency guarantees and evaluate VBO-MI on synthetic benchmarks (Ackley, Levy, Griewank) and real-world tasks (Rover Trajectory, Lunar Lander, Pest Control), demonstrating competitive or superior performance over the baselines.

Farhad Mirkarimi

Bayesian OptimizationBlack-Box Optimization
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Jan 12, 2026

VLM-CAD: VLM-Optimized Collaborative Agent Design Workflow for Analog Circuit Sizing

Vision Language Models (VLMs) have demonstrated remarkable potential in multimodal reasoning. However, they can have spatial blindness and logical hallucinations when interpreting densely structured engineering content, such as analog circuit schematics. To address these challenges, we propose a Vision Language Model-Optimized Collaborative Agent Design Workflow for Analog Circuit Sizing (VLM-CAD) designed to support step-by-step reasoning over multimodal evidence. VLM-CAD bridges the modality gap by integrating a neuro-symbolic structural parsing module, Image2Net, which transforms raw pixels into explicit topological graphs and structured JSON representations to anchor VLM interpretation in deterministic facts. To ensure the reliability required for engineering decisions, we further propose ExTuRBO, an Explainable Trust Region Bayesian Optimization method. ExTuRBO employs agent-generated semantic seeds to warm-start local searches and uses Automatic Relevance Determination to provide sensitivity evidence for the final design report. Experimental results on 12 sizing tasks covering six circuits and four technology platforms show that VLM-CAD achieves a pooled Strict Pass@1 of 23.3% and a Relaxed Pass@1 of 91.7%, while providing sensitivity evidence for final design reports.

Guanyuan Pan, Shuai Wang, Yugui Lin +4

Hangzhou Dianzi University, China · SCBC, Guangdong University of Foreign Studies, China · University of Cambridge, United Kingdom

Bayesian OptimizationElectronic Design AutomationTool-Using Vision-Language AgentsVLM ReasoningAnalog Circuit Sizing
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Dec 10, 2025

AP-BMM: Approximating Capability-Cost Pareto Sets of LLMs via Asynchronous Prior-Guided Bayesian Model Merging

LLM services need to offer a family of models spanning different capability--cost trade-offs to accommodate diverse user preferences. Model merging offers a practical way to construct such a model family by combining a reasoning-enhanced model with an instruction-tuned model. Compared with model-level merging, layer-wise merging offers finer control over the capability--cost trade-off by assigning different merge weights to individual layers. However, it introduces two practical challenges: the layer-wise search space is high-dimensional, and existing methods rarely exploit informative signals from the source models; moreover, the highly variable runtime of LLM evaluations makes synchronous batch optimization inefficient by leaving GPU resources idle while waiting for slow evaluations. To address these challenges, we propose Asynchronous Prior-Guided Bayesian Model Merging (AP-BMM), which formulates layer-wise merging as a multi-objective optimization problem to approximate a Pareto set of merged LLMs, yielding a family of Pareto-optimal merged models with diverse capability--cost trade-offs. AP-BMM leverages parameter and activation discrepancies between the source models to guide the early layer-wise search, employs asynchronous pending-aware Bayesian optimization to maximize GPU utilization through asynchronous evaluations and to select high-quality candidates via pending awareness, and applies lightweight ranking over an oversized candidate pool to improve Pareto-front coverage. Under fixed evaluation budgets, AP-BMM achieves higher hypervolume (HV) and broader Pareto-front coverage than synchronous layer-wise and representative model-level merging methods, while reducing wall-clock time through higher GPU utilization.

Kesheng Chen, Yamin Hu, Zhenqian Zhu +2

Guangdong Provincial Key Laboratory of Novel Security Intelligence Technologies, Institute of Cyberspace Security, School of Computer Science and Technology, Harbin Institute of Technology, Shenzhen 518055, China.

Bayesian OptimizationLanguage Model MergingModel Merging
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Dec 10, 2025

Graph-Based Bayesian Optimization for Quantum Circuit Architecture Search with Uncertainty Calibrated Surrogates

Quantum circuit design is a key bottleneck for practical quantum machine learning on complex, real-world data. We present an automated framework that discovers and refines variational quantum circuits (VQCs) using graph-based Bayesian optimization with a graph neural network (GNN) surrogate. Circuits are represented as graphs and mutated and selected via an expected improvement acquisition function informed by surrogate uncertainty with Monte Carlo dropout. Candidate circuits are evaluated with a hybrid quantum-classical variational classifier on the next generation firewall telemetry and network internet of things (NF-ToN-IoT-V2) cybersecurity dataset, after feature selection and scaling for quantum embedding. We benchmark our pipeline against an MLP-based surrogate, random search, and greedy GNN selection. The GNN-guided optimizer consistently finds circuits with lower complexity and competitive or superior classification accuracy compared to all baselines. Robustness is assessed via a noise study across standard quantum noise channels, including amplitude damping, phase damping, thermal relaxation, depolarizing, and readout bit flip noise. The implementation is fully reproducible, with time benchmarking and export of best found circuits, providing a scalable and interpretable route to automated quantum circuit discovery.

Prashant Kumar Choudhary, Nouhaila Innan, Muhammad Shafique +1

Department of Physics, Indian Institute of Technology (BHU), Varanasi, Uttar Pradesh, India · eBRAIN Lab, Division of Engineering, New York University Abu Dhabi (NYUAD), Abu Dhabi, UAE · Center for Quantum and Topological Systems (CQTS), NYUAD Research Institute, NYUAD, Abu Dhabi, UAE

Bayesian OptimizationVariational Quantum CircuitsQuantum Circuit Optimization
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Nov 29, 2025

No-Regret Gaussian Process Optimization of Time-Varying Functions

Sequential optimization of black-box functions from noisy evaluations has been widely studied, with Gaussian Process bandit algorithms such as GP-UCB guaranteeing no-regret in stationary settings. However, for time-varying objectives, no-regret is unattainable under pure bandit feedback unless strong and often unrealistic assumptions are imposed. We propose a novel method for optimizing time-varying rewards in the frequentist setting, where the objective has bounded RKHS norm almost surely. Time variations are captured through uncertainty injection, enabling heteroscedastic Gaussian process regression that adapts past observations to the current time step. As no-regret is unattainable in general in the strict bandit setting, we relax the latter allowing additional queries on previously observed points. Building on sparse inference and the effect of uncertainty injection on regret, we propose W-SparQ-GP-UCB, an online algorithm that achieves no-regret with a vanishing number of additional queries per iteration. To assess the theoretical limits of this approach, we establish a lower bound on the number of additional queries required for no-regret, proving the efficiency of our method. Finally, we provide a comprehensive analysis linking the temporal regime of the function to achievable regret rates, together with upper and lower bounds on the number of additional queries needed in each regime.

Eliabelle Mauduit, Eloïse Berthier, Andrea Simonetto

Unit´e de Math´ematiques Appliqu´ees ENSTA, Institut Polytechnique de Paris Palaiseau, France · U2IS ENSTA, Institut Polytechnique de Paris Palaiseau, France

Multi-Armed BanditsBayesian OptimizationGaussian Process BanditsRegret MinimizationGaussian Process Regression
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Oct 17, 2025

Nonlinear Dimensionality Reduction Techniques for Bayesian Optimization

Bayesian optimisation (BO) enables sample-efficient global optimisation of expensive black-box functions but remains challenging in high dimensions. We investigate nonlinear dimensionality reduction to a sequence of low-dimensional latent-space BO (LSBO) problems. Early LSBO used linear random and supervised embeddings; building on Grosnit et al., we employ variational autoencoders (VAEs), deep metric loss for structured latent manifolds, and retraining to adapt the encoder-decoder pair to newly sampled regions. We couple LSBO with sequential domain reduction (SDR) directly in latent space (SDR-LSBO), narrowing search domains as evidence accumulates. Implemented in GPU-accelerated BoTorch with Mat'ern-5/2 Gaussian-process surrogates, our methods improve benchmark optimisation quality, and retraining can enhance BO performance. Comparisons with adaptive supervised linear random embeddings demonstrate the effectiveness of VAE-based BO for nonlinear low-dimensional structures. We analyse BO-VAE with a fixed pretrained representation, decomposing ambient-space simple regret into latent BO error and a fixed VAE-induced representation gap. Under a PAC-Bayes-certified reconstruction condition and standard fixed-prior assumptions for expected improvement with a Mat'ern-5/2 kernel, latent BO error vanishes as the evaluation budget increases, whereas the representation gap remains fixed and may impose a non-vanishing error floor. Visualisations empirically assess accessibility of the ambient optimum through the learned decoder. To our knowledge, this is the first study combining SDR with VAE-based LSBO. Our analysis clarifies metric shaping and retraining choices critical for scalable latent-space BO. For reproducibility, source code is available at https://github.com/L-Lok/Nonlinear-Dimensionality-Reduction-Techniques-for-Bayesian-Optimization.git.

Luo Long, Coralia Cartis, Paz Fink Shustin

Mathematical Institute, University of Oxford, Radcliffe Observatory, Andrew Wiles Building, Woodstock Rd, Oxford, OX2 6GG, United Kingdom.

Variational AutoencodersBayesian OptimizationDimensionality ReductionLatent Space Optimization
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Oct 7, 2025

Bayesian Optimization under Uncertainty for Training a Scale Parameter in Stochastic Models

Hyperparameter tuning is a challenging problem especially when the system itself involves uncertainty. Due to noisy function evaluations, optimization under uncertainty can be computationally expensive. In this paper, we present a novel Bayesian optimization framework tailored for hyperparameter tuning under uncertainty, with a focus on optimizing a scale- or precision-type parameter in stochastic models. The proposed method employs a statistical surrogate for the underlying random variable, enabling analytical evaluation of the expectation operator. Moreover, we derive a closed-form expression for the optimizer of the random acquisition function, which significantly reduces computational cost per iteration. Numerical experiments demonstrate substantial gains in optimization efficiency: about 40-fold reduction in data usage relative to a Monte Carlo-based one-dimensional optimization scheme, and an order of magnitude improvement relative to a Gaussian process-based Bayesian optimization baseline.

Akash Yadav, Ruda Zhang

University of Houston

Bayesian OptimizationHyperparameter OptimizationStochastic Optimization
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Jul 9, 2025

Direct Regret Optimization in Bayesian Optimization

Bayesian optimization (BO) is a powerful paradigm for optimizing expensive black-box functions. Traditional BO methods typically rely on separate hand-crafted acquisition functions and surrogate models for the underlying function, and often operate in a myopic manner. In this paper, we propose a novel direct regret optimization approach that jointly learns the optimal model and non-myopic acquisition by distilling from a set of candidate models and acquisitions, and explicitly targets minimizing the multi-step regret. Our framework leverages an ensemble of Gaussian Processes (GPs) with varying hyperparameters to generate simulated BO trajectories, each guided by an acquisition function drawn from a pool of conventional choices and terminated by a Bayesian early stop criterion. These trajectories train an end-to-end Decision Transformer that selects the next query so as to improve the ultimate objective, following a dense training sparse learning paradigm: the transformer is trained on abundant simulated data, while a limited number of real evaluations refine the GPs online. On synthetic and real-world benchmarks, our method attains the best or near-best final simple regret against standard, lookahead, trust-region and amortized BO baselines, with the largest gains in high-dimensional settings. Ablations attribute the gains jointly to region-of-interest filtering and the learned policy, and matched-budget comparisons against explicit two-step lookahead acquisitions show that the advantage is not shared by lookahead alone.

Fengxue Zhang, Yuxin Chen

Department of Computer Science University of Chicago Chicago, IL 60637

Bayesian OptimizationLearning to OptimizeDecision TransformerRegret Minimization
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Feb 26, 2025

Bayesian Optimization for General Reaction Conditions

General chemical reaction conditions that achieve consistently high performance across multiple substrates are important for practical applications such as library synthesis and high-throughput experimentation. However, identifying such conditions efficiently has been a longstanding challenge, as it requires decision making under uncertainty with respect to both conditions and substrates, while minimizing the number of required experiments. Here, we introduce CurryBO, a high-level framework for generality-oriented optimization. By formalizing the problem as Bayesian optimization over curried functions, CurryBO provides a unified framework that accommodates different generality definitions (e.g., mean yield across substrates), and supports a range of substrate and condition selection strategies. We evaluate this framework on four benchmark tasks in experimental reaction optimization, and systematically analyze key algorithmic components. Our results show that efficient experiment planning can be achieved by emphasizing exploration when selecting reaction conditions, followed by the uncertainty-guided prioritization of substrates in a sequential decison-making scheme. Based on these insights, we design and validate an optimization policy that substantially improves sample efficiency relative to previously reported approaches across all benchmarks. Overall, the flexibility and modularity of CurryBO facilitate the integration of generality-oriented optimization into experimental settings, enabling more efficient identification of solutions that perform robustly across diverse tasks.

Stefan P. Schmid, Ella Miray Rajaonson, Cher Tian Ser +6

Institute of Chemical and Bioengineering, Department of Chemistry and Applied Biosciences, ETH Zurich, Zurich CH-8093, Switzerland · NCCR Catalysis, Switzerland · Department of Chemistry, University of Toronto, Toronto, Canada +13

Bayesian OptimizationRobust OptimizationBayesian Experimental Design
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Feb 17, 2025

Enhancing Offline Model-Based RL via Active Model Selection: A Bayesian Optimization Perspective

Offline model-based reinforcement learning (MBRL) serves as a competitive framework that can learn well-performing policies solely from pre-collected data with the help of learned dynamics models. To fully unleash the power of offline MBRL, model selection plays a pivotal role in determining the dynamics model utilized for downstream policy learning. However, offline MBRL conventionally relies on validation or off-policy evaluation, which are rather inaccurate due to the inherent distribution shift in offline RL. To tackle this, we propose BOMS, an active model selection framework that enhances model selection in offline MBRL with only a small online interaction budget, through the lens of Bayesian optimization (BO). Specifically, we recast model selection as BO and enable probabilistic inference in BOMS by proposing a novel model-induced kernel, which is theoretically grounded and computationally efficient. Through extensive experiments, we show that BOMS improves over the baseline methods with a small amount of online interaction comparable to only 1%1\%-2.5%2.5\% of offline training data on various RL tasks.

Yu-Wei Yang, Wei Hung, Yun-Ming Chan +2

Department of Computer Science, National Yang Ming Chiao Tung University, Hsinchu, Taiwan · Applied Machine Learning, Meta AI, Menlo Park, CA, USA

Model SelectionOffline RLModel-Based RLBayesian Optimization
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Jan 31, 2025

Time-Varying Bayesian Optimization Without a Metronome

Time-Varying Bayesian Optimization (TVBO) is the go-to framework for optimizing a time-varying, expensive, noisy black-box function ff. However, most of the asymptotic guarantees offered by TVBO algorithms rely on the assumption that observations are acquired at a constant frequency. As the GP inference complexity scales with the cube of its dataset size, this assumption is unrealistic in the long run. In this paper, we relax this assumption and derive the first upper regret bound that explicitly accounts for changes in the observations sampling frequency. Based on this analysis, we formulate practical recommendations about dataset sizes and stale data policies of TVBO algorithms. We illustrate how an algorithm (BOLT) that follows these recommendations performs better than the state-of-the-art of TVBO through experiments on synthetic and real-world problems.

Anthony Bardou, Patrick Thiran

IC EPFL Lausanne, Switzerland

Sequential Decision MakingBayesian Optimization
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Dec 2, 2024

Classifier-pruned Bayesian optimization for particle accelerator tuning: Exploring temporally structured manifold of 6D beam phase space

Complex dynamical systems, such as particle accelerators, require tuning over high-dimensional, nonlinear state and parameter spaces while experimental measurements and high-fidelity simulations can be expensive. Learned latent representations provide a compact domain for such optimization, but poorly supported regions of the learned manifold may decode into unrealistic physical states and yield deceptively favorable objectives. To address this challenge, we propose the Classifier-pruned Bayesian Optimization-based Latent-space Tuner (CBOL-Tuner), which performs Bayesian optimization over a temporally structured latent representation of 6D beam phase-space dynamics. The CBOL-Tuner integrates a conditional variational autoencoder for latent space representation, a long short-term memory network for temporal dynamics, a lightweight neural network for parameter estimation, and a classifier-pruned Bayesian optimizer to adaptively search and filter the latent space for optimal solutions. This framework enables feasibility-aware exploration of learned scientific representations for accelerator tuning.

Mahindra Rautela, Alan Williams, Alexander Scheinker

Los Alamos National Laboratory NM, United States

Bayesian Optimization
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