Unleashing LLMs in Bayesian Optimization: Preference-Guided Framework for Scientific Discovery
Authors: Xinzhe Yuan
Zhuo Chen
Jianshu Zhang
Huan Xiong
Nanyang Ye
Yuqiang Li
Qinying Gu
Venue: ICLR 2026 Poster (2026)
Keywords: Bayesian Optimization; Large Language Models; AI for Science; Scientific Discovery
Primary area: applications to physical sciences (physics, chemistry, biology, etc.)

Abstract: Scientific discovery is increasingly constrained by costly experiments and limited budgets, making efficient optimization essential for AI for science. Bayesian Optimization (BO), while widely adopted for balancing exploration and exploitation, suffers from slow cold-start performance and poor scalability in high-dimensional settings, limiting its effectiveness in real-world scientific applications. To address these challenges, we propose LLM-Guided Bayesian Optimization (LGBO), the first LLM preference-guided BO framework that continuously integrates the semantic reasoning of large language models (LLMs) into the optimization loop. Unlike prior works that use LLMs only for warm-start initialization or candidate generation, LGBO introduces a region-lifted preference mechanism that embeds LLM-driven preferences into every iteration, shifting the surrogate mean in a stable and controllable way. Theoretically, we prove that LGBO is not perform significantly worse than standard BO in the worst case, while achieving significantly faster convergence when preferences align with the objective. Empirically, LGBO achieves consistent improvements across diverse dry benchmarks in physics, chemistry, biology, and materials science.  Most notably, in a new wet-lab optimization of Fe–Cr battery electrolytes, LGBO reaches \textbf{90\% of the best observed value within 6 iterations}, whereas standard BO and existing LLM-augmented baselines require more than 10 iterations. Together, the results suggest that LGBO offers a promising direction for integrating LLMs into scientific optimization workflows.

Published as a conference paper at ICLR 2026
UNLEASHING LLMs IN BAYESIAN OPTIMIZATION: PREFERENCE-GUDED FRAMEWORK FOR SCIENTIFIC DISCOVERY
Xinzhe 2ua Zhuo Chtn
Jianshu Zhang
Huan Xiong?
Nanyang Ye
Yugiang Li Qinying Gu!
i Equal contribution.
:Corresponding authors: (huan couan.xiong xiong math, y
{liyugiang, guginying])ppjaab wt..g.r...........o..o.oo.o....o
'Shanghai Artificial Intelligence Laboratory, Shanghai, China
"Harbin Institute of Technology, Harbin, China
SShnnhaai Jiao Tong University, Shanghai, China
ABSTRACT
Scientific discovery is increasingly constrained by costly experiments and lim- ited resources, underscoring the need for efficient optimization AI for science. Bayesian Optimization (BO), though widely adopted for balancing exploration and exploitation, often exhibits slow cold-start performance and poor scalabil- ity in high-dimensional settings, limiting its applicability in real-world scientific problems. To overcome these challenges, we propose LLM-Guided Bayesian Op- timization (LGBO), the first LLM preference-gee guiidd BO framework that contin- uously integrates the semantic reasoning of large language models (LLMs) into the optimization loop. Unlike prior works that use L.MS only for warm-start ini- tialization candidate generation, LGBO introduces region-lifted preference mechanism that embeds LLM--diven preferences into every iteration, shifting the prove that
surrogate mean in stable and controllable way. Theoretically, LGBO does not perform significantly worse than standard BO in the worst case, while achieving significantly faster convergence when preferences align with the objective. Empirically, LGBO consistently outperforms existing methods across diverse dry benchmarks in physics, chemistry, biology, and materials science. Most notably, in a new wet-lab optimization of Fe-Cr battery electrolytes, L.BBO attains 90% of the best observed value within 6 iterations, whereas standard BO and existing LLM-augmented baselines require more than 10. Together, these results suggest that LGBO offers a promising direction for integrating LLMs into scientific optimization workflows.
INTRODUCTION
Traditional experimental approaches have long driven scientific discovery. However, their labor- intensive and time-consuming nature continues to limit the pace at which new insights can emerge (Xie al. 2023 Tom al. 2024; Seifrid et al., 2022). To alleviate these bottlenecks, self-driving laboratories combine robotic execution with artificial intelligence, enabling systematic and high- throughput exploration (Chen et al. 22044] Ai et al. 2024). Within these platforms, Bayesian Op- timization (BO) has emerged powerful strategy, balancing exploration and exploitation while offering principled uncertainty quantification (Guo et al. 2023 Shields et al. 2021 Shahriari et al. 2015).
BO has achieved success across diverse scientific domains, but it faces two well-known challenges. First, the scarcity of early experimental data makes cold-start optimization slow and inefficient. Second, high- digh-dimensional parameter spaces introduce the curse dimensionality. further limiting scalability (Guo et al. 2023). These issues often hinder BO in precisely the settings where it is most

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needed-ext exploratooo expensive scientific tasks. Several acceleration strategies have been proposed. including transfer learning (Chowdhury & Gopalan 2021). embedding methods (Nayebi et al. 2019 Moriconi al. 2020), and adaptive partitioning [Wang al. 2020). Yet, none of these methods fully addresses the cold-start bottleneck.
Recent efforts have attempted to bring large language models (LLMs) into BO. A first line of work uses LLMs for warm-up, letting the model propose the initial design points before the optimization loop begins. This strategy can provide strong prior when the task is well aligned with the model's knowledge, accelerating the otherwise slow cold-start phase (Liu et al.). A second line of work leverages LLMs to propose candidate points during the loop. which are then re-ranked or filtered by standard acquisition function (Chang et al.] [025 Yin et al.] 2024] Yang et al., (025). These approaches demonstrate the promise of linguistic priors, they also share critical limitation: the L.M's guidance remains only loosely embedded in the optimization loop-either injected once at initialization or later overridden by the acquisition function. This motivates a central question:
Can we design BO framework that more fully integrates LLM preferences throughout the optimization process, rather than confining them to warm-up initialization or auxiliary candi- date proposals?
To this end, we present the L.MM-Guided Bayesian Optimization (LGBO) framework. Rather than restricting LLMs to one-off suggestions, LGBO treats them expert priors that provide preferences and directional guidance throughout the acquisition process. This allows the optimizer to leverage LLM reasoning when informative, while remaining robust weak noisy signals.
We also evaluate this framework across diverse scientific optimization tasks. Our results show that LLM guidance substantially accelerate BO when informative, while performance remains com- petitive under weaker signals. Furthermore, we provide theoretical guarantees ensuring the consis- tency of the framework. Our contributions
follows:
1. We propose an LLM-preference BO framework that incorporates LLM expertise into Bayesian optimization through continuous preference guidance, moving beyond existing warm-start or candidate-proposal paradigms.
2. We provide theoretical guarantees showing that aligned LLM guidance can substantially accelerate convergence, while even under misleading guidance the additional cost remains provably bounded.
3. We validate the framework on a broad suite of scientific optimization tasks, spanning di- verse dry benchmarks and wet-lab electrolyte optimization experiment, demonstrating both generality and practical utility.
2 RELATED WORK
In this section, we review existing studies and analyze their limitations. We begin with approaches that incorporate large language models into Bayesian optimization, then turn to human-in -aetn--i-tehtoooto methods where expert preferences guide the search, and finally discuss how these two lines connect our framework.
2.1 LLMS BAYESIAN OPTIMIZATION
Before reviewing how LLMs have been integrated, it is helpful to first recall what BO is. BO provides a way to search for the best solution when directly testing the target is expensive and time- consuming. The idea is to treat the true objective observe the outcomes, but we do not know its exact formula. To make the search more efficient, BO builds a surrogate model-often a Gaussian Process (GP)-that approximates the black box based on the points already tested. At each step, this surrogate is updated with new observations, and the acquisition function, defined on the surrogate, proposes the next point evaluate. The new outcome then obtained from the real black box, after which the surrogate is updated and the cycle repeats.
"black box": we can try different inputs and
Recently, studies have explored using large language models to assist this BO process. LLAMBO (Liu et a leverages LLMs to warm-start the optimization with informative initial points and to gen- erate candidate samples conditioned past evaluations. However, the final decision each iteration
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is still made by standard acquisition function over surrogate, leaving the LLM's reasoning only indirectly involved. Subsequent studies explore similar ideas in domain-specific settings. ADO- LLM (Yin et al. (2024) applied LLM-assisted proposals to analog circuit design, and ReasoningBO Yang et al. 2025) highlighted how contextual reasoning can enrich candidate generation. Most recently, L. IINO (Chang et al. 2025) proposed more systematic -it---n-tte-loopp framework, designed for the early exploration stage of BO to address the cold-start problem. In addition, beyond the scientific discovery domain, several LLM-BO approaches have also emerged in other areas. For instance, BOPRO (Agarwal et al. 2025) is method that leverages large language models as im- plicit Bayesian priors through co-tott learning. CAKE (Suwandi et al. (025), in contrast, utilizes LLMs to construct kernel functions tailored to specific tasks, enabling structured and data-efficient optimization over complex input spaces.
Across these frameworks, however, LLMs are treated auxiliary components rather than integral parts of the loop, with their guidance incorporated only partially and indirectly. This gap moti- vates the development of approaches that continuously and systematically embed evolving model preferences into the optimization loop.
2.2 HUMAN EXPERTS IN BAYESIAN OPTIMIZATION
A natural source of inspiration is human-in-the-loop Bayesian optimization, where algorithms ex- ploit the fact that experts often find it easier to provide preferences or qualitative guidance rather than precise numerical labels. Early work on preference- baseed BO formalized how such compar- ative judgments can be integrated into acquisition strategies (Brochu al.] 2010; González et al.) 2017]. More recent systems extend this idea by modeling expert reliability (Xu et al.) (222)) or pro- gressively eliciting knowledge through targeted queries (Huang et al. 2022), showing that expert feedback can substantially improve sample efficiency even when partial or noisy.
ColaBO (Hvarfner et al., 2024) provides general framework in which experts specify a preference, and the optimizer reweights sampled paths based on how well the best candidates identified along those paths align with that preference. This design is effective with human experts, whose input is typically stable and given once initialization. With LLMs experts, however, two challenges emerge. First, since ColaBO accepts only an initial preference, it cannot naturally incorporate up- dated guidance during the optimization process, leaving the LLM's capacity for ongoing reasoning underutilized. Second, if preferences are revised over time, the required reweighting of sampled paths would need to be updated continually, which may introduce instability or even divergence. These factors make ColaBO less directly applicable to LLM-guides BO and point to the need for frameworks that can embed evolving model preferences in stable manner.
3 LLM-GUIDED BAYESIAN OPTIMIZATION
We present 1.LM-Guidee Bayesian Optimization, framework that integrates large language models into Bayesian optimization in a stable and tractable way. The key challenge is to incorporate ex- ternal preferences without sacrificing mathematical rigor: standard BO relies purely data-driven posteriors, while preference-ber based extensions for human experts assume sparse, stable, or one-off feedback. LLM guidance, by contrast, is iterative, coarse, and potentially noisy. Directly injecting such signals into the acquisition function can destabilize the optimization loop.
LGBO addresses this challenge through the region-lifted preference, mechanism that translates LLM suggestions into lightweight priors which shift the surrogate mean but leave its covariance unchanged. This design allows semantic guidance be incorporated continuously while preserv- ing the statistical properties of BO. Section provides the preliminaries of standard BO and prior expert-guided BO, Section 3.2 introduces Region-Lifted Preference as the core part of the LGBO framework, Section 3.3 details the optimization loop, and Section 3.4 presents theoretical guaran- tees.
3.1 PRELIMINARIES
We first recall the basic setup of Bayesian optimization and the challenges of incorporating external preferences.
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Standard BO. Given observed data D, {( Y-)] BO maintains posterior distribution over the objective f X' + I. typically modeled with a Gaussian process (p, k). where μ is mean function and k is kernel (covariance) function. This posterior provides both mean predic- tions and uncertainty estimates. An acquisition function (2(2) (e.g.. El or UCB) then selects the next query by balancing exploration and exploitation to efficiently approach the global optimum Shahriari et al. 2015).
Preference a functional. Most expert-guided Bayesian optimization methods can be viewed through preference functional p(f), which reweights the posterior toward functions that better match external guidance. For example, a statement like "x1 preferred ø2" can be modeled as p(J) = (x(1 - (I2)), where is the sigmoid function. The standard BO posterior p(f D.) represents our belief about the black-box function after observing data Dr, and multiplying it with p(f) produces an adjusted posterior
pl De,p) & p(f D(),
which incorporates additional preferences directly into the BO loop. In practice, however, such pos- teriors are often intractable, especially when p(f) depends on global properties of f. For this reason, existing human-in-the- the acquisition function. While this strategy effective under stable human feedback, letting LLMs directly alter the acquisition function is risky, it can cause instability even divergence of the optimization process.
optimization methods usually inject preferences indirectly by modifying
3.2
CORE TOOL: REGION-LIFTED PREFERENCE
We argue that LLMs are naturally better at providing coarse, semantic guidance-such identi- fying promising regions (e.g., "try near high temperature and low pressure") rather than precise pairwise comparisons. To leverage this strength while preserving tractability, we introduce a region- lifted preference scheme: the LLM specifies preferred region R C X. and the preference functional defined as a linear exponential lift over a discretization 9 = {rg)
G
pl) exp Σ (
exp(Aa Fg),
where a [a loi.....ag ag (f.x))...,
RC with ag 2 (eg. uniform ag = 1/G), Fg
g. and À 0 controls the guidance strength. This functional encourages higher function values in R without explicitly modeling optimality, thereby avoiding the intractabil- ity of argmax-dependent preferences.
Although this regional lift appears complex, proposition [ shows that it can be implemented ex- actly and tractably, as a mean shift of the GP prior, preserving the Gaussian structure and enabling seamless integration into standard BO pipelines.
Proposition 1 (Exponential lift equals mean shift). Let f~ GP(,, k) and consider linear func- lional over a grid G {zg)g= GG with weights E IRG. Lifting the posterior distribution by
exp(Xa FF)
yields new Gaussian process with the same covariance kernel k, but a shifted mean function
(x(x) u(z) + 0g kg),
Equivalently, for any finite query set X, the marginal distribution is
Fx Mux + XExaa Σxx).
In particular, the expected regional lift is
A Exja Fgl Eja Fgl Xa Egga.
Here N Σ) denotes the multivariate normal distribution. This result is direct finite- dimenssonal consequence of Theorem Appendix[A] which establishes the general Hameron-Martee form of exponential lifting for Gaussian measures.
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Region-lifted tool. Building Proposition 1 the region-lifted preference provides a practical interface for large language models. In the system prompt, the model's output is strictly constrained to one of two formats:
1) Point mode: [point, [xl, x2, xdl, ccc),
2) Region mode:
(region, [[1bl,
1bd], [ubl,
ubd] 1, ccc].
Here, cao confidence score in [0, 1]. region mode, the model specifies hyper-rectangle through 1b/ub; we discretize it using Sobol grid G. In point mode, the lift is localized: we construct a as a smoothly decaying weight vector centered at the suggested point, so that at Fg captures soft neighborhood average. This structured interface allows the LLM to express coarse semantic preferences while ensuring that its influence remains mathematically tractable.
Calibrating the guidance strength. To ensure that the lift is well-scaled, we map the LLM's confidence E (0, 1] directly to the guidance strength. From Proposition under exponential tilting, the effect of a lift is equivalent to shifting the regional functional a Fg by
A = Xa Egga,
so the choice of À directly determines the effective magnitude of this shift. We set
Va Egga
where Σgg is the GP posterior covariance on G. and a T Egga is the posterior variance of the regional functional. This normalization calibrates the lift to a consistent c-sttnddard-deviation scale-regions with low posterior uncertainty receive a small effective shift, while uncertain re- gions receive a larger one. In this way, the LLM's preference integrates smoothly with the model's uncertainty without introducing extra hyperparameters or risking premature over-exploitation.
3.3 LGBO FRAMEWORK
LGBO integrates these region-lifted preferences into BO in continuous loop. Instead of altering the acquisition function directly, LLM outputs reshape the surrogate model at each iteration. This maintains robustness while allowing semantic guidance to evolve alongside experimental data.
From warm-start to continuous guidance. Earlier approaches have used LLMs in limited ways: either to provide one-off initialization auxiliary candidate proposals that later filtered by the acquisition function. In both cases, the guidance is only loosely connected to the optimization loop. LGBO departs from these approaches by embedding LLM preferences every iteration. At every iteration, the surrogate is informed not only by newly collected experimental data but also by updated LLM preferences, which are systematically incorporated through region-lifted priors. This design allows semantic signals, such as domain heuristics or qualitative trends, to shape the surrogate throughout the process, ensuring that LLM reasoning evolves tandem with data and becomes persistent part of the optimization loop.
Workflow. The LGBO loop. illustrated in Figure I follows the structure of standard BO with one key modification: region-lifted updates are injected at each iteration. The process begins with initialization, where the LLM provides starting points based on prior knowledge. After each eval- uation, the GP surrogate is retrained, and the LLM is queried with structured prompt with fixed components (domain knowledge, constraints, and the required output format) and a dynamic user prompt summarizing experimental history plus the reasoning trace from the previous round. The model then outputs either a point or a region with a confidence score, which is converted into a region-lifted preference and applied as mean shift of the surrogate mean while leaving the covari- ance unchanged. The acquisition function, such as EI or UCB, then selects the next experiment based this updated surrogate. By repeating this cycle, LGBO integrates semantic guidance tinuously, while the acquisition function retains full control over Decision-making The full prompt specification is provided in Appendix B
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Semantic Warm Start
LGBO Loop
LLM Region Preference
Region Tilt Tool
Prior Knowledge (literature, domain nales,
propose Rogion Thigh preferance....
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Initial GP
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Figure I: LGBO framework. The proposed LLM-Guided Bayesian Optimization integrates prior knowledge and LLM reasoning warm-start the search, and iteratively combines GP surrogate with region-lifted preferences. At each round, the LLM provides coarse regional guidance, imple- mented as a mean shift the surrogate, while the acquisition function selects the next experiment. This design ensures stability in the worst case while enabling acceleration when the guidance is informative.
Advantages of semantic guidance. This design allows LGBO to harness aspects of LLM rea- soning that are otherwise inaccessible to purely data-driven methods. For instance, the model can highlight regions suggested by chemical intuition (e.g. higher temperature and lower concentra- tion may accelerate reaction yield) or by cross-domain analogies (e.g., drawing parallels between material compositions). By embedding such regional signals into the surrogate, LGBO guides ex- ploration toward scientifically meaningful areas without abandoning the statistical rigor of BO.
Guarantees. Our framework provides two complementary guarantees: in the worst case of irrel- evant misleading guidance, the induced lift only enlarges surrogate's norm bound, so regret remains within constant factor of standard BO; when guidance aligns with the true objective, the effective radius shrinks, yielding strictly tighter regret bounds (see Section (.4). Thus LGBO is provably safe, yet can substantially accelerate convergence when preferences informative.
3.4 THFORETICAL GUARANTER
We now provide regret guarantees for LGBO. In the full LGBO algorithm, the LLM proposes new region at every BO iteration, producing an adaptive sequence of lifts {Tr}. The theoretical analysis is not intended characterize this fully dynamic process. Instead, we study a simplified but mathematically tractable variant in which single preference direction is fixed at the beginning of optimization, and the corresponding lift rax) = >g(x)) applied consistently throughout the T rounds.
This "frozen-lift" setting is a reasonable abstraction because, in scientific optimization tasks, domain-informed LLM often exhibits structurally coherent behavior: once the model identifies fea- tures associated with near-optimal or high-quality regions, its suggested regions tend to remain clustered around similar areas rather than fluctuating dramatically across iterations. A fixed direc- tion therefore serves as a representative surrogate for a stable class of regional preferences while avoiding the technical complications of analyzing fully adaptive sequence.

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Under this setting, the lifted objective f'= = remains single element of the RKHS, allow- ing direct application of classical GP-UCB theory. The analysis isolates the mechanism-level effect of regional preference direction: when the direction is aligned with the true objective, the ef- fective RKHS radius contracts and the regret bound becomes strictly tighter; when the direction is misaligned, the radius increases only by constant amount, and the worst-case rate matches stan- dard GP-UCB. Thus, although the fixed-lift variant is not an exact description of the full LGBO dynamics, provides principled and verifiable abstraction of the core preference mechanism.
Lift representation. Given a finite grid {z,} and non-negative weights the structured region or point suggested by the LLM mapped, under the LGBO lifting rule defined in Section 3.2 to the kernel-induced function
(ax) Σ (ak((,,)))
=(x) >g(x))
This yields an equivalent mean- mean-shiffee representation of the prior G
Regularity assumptions.
We follow the classical GP-UCB assumptions (Sinniaas et al. 2009
- Hk. |lf - pll S Bo.
where ||
|| denotes the RKHS norm associated with kernel k and Sub-Gaussian noise.
Alignment coefficient.
The analysis uses an alignment coefficient
(f T, g)H,
1]],
| T 1l, 1|l||
which measures how well the lifted direction g agrees with the centered objective ff - in the RKHS. Building the fixed lift and the alignment structure introduced above, we now state the corresponding regret bounds.
k) satisfies pu H4 I Bo, then under the fixed lift Txx) = >g(x)) defined above. Run GP-UCB on the residual labels yi yr. T(xe) with confidence parameter Br (-) and let "r denote the maximal information gain. Then, with probability
Theorem 1 (Lifted regret bounds). If f GP(H,,)
at least -
Rr S VST BB( TT
Bout = Bo + Allg||H
and, when the alignment coefficient satisfies >> 0,
Ri S
8T By 8T
Bin) YI,
B = Bu Bu 1
All hidden constants are universal. Proof given in Appendix
Remark (Reading the bounds). For standard BO, the complexity is controlled by Ba, yielding re- gret on the order of /ST (T yr (Srinivas et al. 2009), Under LGBO, the misaligned case sim- ply replaces Bo by Bout = Bo + AllggT so the bound remains comparable up to a lift-dependent cost. In the aligned case, the effective radius shrinks to Bmn = BovI- 2², which can be much smaller than Bo whenever is large, leading strictly tighter regret. Thus LGBO is not perform significantly worse than standard BO, and can be substantially better when the suggested region is informative.
4 EXPERIMENTS
We evaluate LGBO in two settings: dry experiments, which use pre-collected public datasets, and a wet experiment conducted in a real laboratory. Dry experiments provide a finite set design- response pairs, from which offline optimum can be identified. While this optimum may not coincide with the true global optimum of the underlying physical system, serves as a consistent reference for comparing algorithms. In contrast, the wet experiment involves ongoing task with limited data, measurement noise, and an unknown optimum, representing realistic scenario where optimization must proceed under laboratory constraints.

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GPBO
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Figure 2: LNP3 results. Left: LGBO achieves faster convergence and higher final performance than GPBO and LLAMBO. Right: trajectory heatmaps show that LGBO converges more rapidly and consistently across seeds.
4.1 EXPERIMENT SETTING
Baselines. Two representative methods are used for comparison: GPBO, the canonical Gaussian-proces Bayesian optimization framework with Matérn-5/2 kernel; (ii) LLAMBO (Liu jet al.j. state-of-the-ard L.MM-augmened BO approach that leverages language models for warm- start initialization and candidate generation.
Implementation details. All methods the surrogate model (GP with Matérn-5/2 kernel), and the log-şEI acquisition function. GP hyperparameters are re-optimized after each iteration via marginal likelihood. Each run begins with two initial evaluations: GPBO uses Sobol initializa- tion, while LLAMBO and LGBO share the same LLM- LLM-suggested points for fairness. Results are reported as mean + variance deviation over five seeds. For LLL.-based methods, we use Intern-S1- 241B, pretrained large model in the scientific domain (Bai et al.) (025). Prompts are designed so that no task-specific objectives or seeds are exposed, ensuring that the LLM cannot rely on memo- rized knowledge.
Dry experiment datasets. Four benchmarks are used assess generality across scientific do- mains. Detailed specifications are given in Appendix C
1) LNP3 (Chen et al. 2022): : lipid nanoparticle formulations for cannabidiol delivery, optimizing for drug loading, encapsulation efficiency, particle size. The search space has five design variables: four mixture inputs (three continuous, one discrete) and one categorical lipid type.
2) Cross-barrel (Gongora al., 2020): 3D-printed crossed-barre structures, optimized for me- chanical toughness. The design variables are four continuous geometry parameters (n, 0, t).
3) Concrete (Tripathi 2020): mixture design of cement, aggregates, water, and admixtures, with the objective of maximizing compressive strength under realistic composition constraints.
4) HPLC (Häse et al. 2021): automated high performance liquid chromatography, where six pro- cess parameters (e.g., loop size, flow rate, pump settings) are tuned to maximize chromatographic peak area.
Fe-Cr redox flow battery (wet experiment). The wet experiment focuses on optimizing elec- trolyte formulations for Fe-Cr redox flow batteries, system for which no benchmark dataset exists and where the true optimum is unknown. Thus, LLMs cannot rely on memorized training data and must instead draw on general scientific knowledge. Key ion concentrations (HCI, FeClz/3. CrCly/2) jointly determine viscosity and conductivity, and all measurements are aggregated into a weighted scalar objective reflecting practical application priorities. Compared to dry experiments, this setting offers few observations and is subject to laboratory noise and measurement uncertainty, providing a stringent test of whether LGBO can qualitative priors to guide exploration under scarce data.
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Figure 3: Convergence traces on the three dry benchmark tasks. From left to right: HPLC, Cross- barrel, and Concrete.
4.2 MAIN RESULTS
Dry experiments. Figure 2 shows the main results of LNP3. The performance traces (left) show that LGBO converges faster and achieves higher final objectives than both baselines: GPBO suffers from cold-start, while LLAMBO improves early progress but soon plateaus. The trajectory heatmaps (right) further reveal that LGBO yields much more consistent convergence across seeds, whereas GPBO and LLAMBO exhibit large variability. This stability is not accidental: by continuously incorporating domain-informen search directions and avoids uninformative exploration. For completeness, Appendix [.1 provides pairplots of sampled points, showing that LGBO avoids large regions of uninformative exploration, consistent with its reduced variance and faster convergence.
LLM preferences through region lifting, LGBO maintains coherent
Across the other benchmarks (Figure 3 LGBO consistently achieves stronger performance. On HPLC, the task is highly noisy and all methods fluctuate considerably, yet LGBO still maintains the highest final performance. On Cross-barrel, the low-dimensional design space allows GPBO perform reasonably well, yet LGBO achieves the best toughness and explores around the optimum. On Concrete, both LGBO and LLAMBO benefit from LLM warm-starts, but LGBO further escapes local optima more quickly, ultimately attaining higher compressive strength. Together these results highlight the robustness and generality of LGBO across noisy, simple, and practical engineering settings.
effectively
Wet experiment. In the Fe-Cr electrolyte optimization (Figure LGBO quickly identifies promising regions and by the sixth iteration surpasses 90% of the best observed value, while the baselines require substantially more evaluations. As optimization progresses, LGBO steadily fines its surrogate and concentrates on high-performing regions, yielding higher final performance and lower variance optimal regions; the latter is further constrained by acquisition filtering, which prevents it from leveraging LLM guidance when the GP surrogate is crude. Together these results show that LGBO can effectively translate domain knowledge into accelerated and more reliable convergence.
runs. By contrast, GPBO and LLAMBO explore scattered, often sub-
4.3 ABLATION EXPERIENCES
To disentangle the contributions of different components in LGBO, conduct two ablation exper- iments on the HPLC dataset, which serves as a representative benchmark beyond the main results.
Different LLM backbones. To evaluate robustness, we replace Intern-S1 with backbones of dif- ferent sizes and capabilities: Intern-S1-minn-7BB, as well as two larger general-purpose large lan- guage models, Owen3-235B-Instrun models exhibit stronger stability, which further enhanced when pretrained on scientific domains. Meanwhile, Thinking variants tend to achieve higher final values but converge more slowly, likely because their instruction-fol foolloiinn--olooiing ability is weaker under strict prompting compared to Instruct models. These results demonstrate that LGBO is not tied to specific LLM: larger domain- pretrained models can provide richer guidance, but the framework itself remains broadly applicable across diverse backbones.
and Swen3-233BBThinkkee
As shown in Figure [5] larger

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Figure 4: Wet-lab experiment results. Left: convergence traces across iteration rounds, showing observed values and best objective performance. The hollow markers indicate the historical best lutions observed up to that round. Right: 3D optimization trajectories in the chemical concentration space (Fe, Cr, additive), with color indicating the best- objective.
Convergence speed
performance
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randevelio
200%
Qwedkine Qwodk-tikk
Intro-S St
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Rusual
Figure 5: Ablation study results. Left: convergence traces comparing LGBO, GPBO, and Random region lifting BO on the HPLC task. Middle and Right: Different LLM backbones ablation exper- iment, where dots represent individual test runs and bars denote the corresponding means.
Random region lifting. We replace LLM suggestions with randomly lifted regions of matched size and confidence. To eliminate warm-start effects, LGBO is also initialized from Sobol points, identical to GPBO. As shown in Figure [ random guidance not only prevents fast convergence in the early stage but also forces the optimizer to spend more rounds on exploration. This confirms that LGBO's advantage arises from informative semantic cues provided by the LLM, rather than from the lifting mechanism alone.
On initialization. Both LGBO and LLAMBO start from the same LLM-suggested initialization points, common practice in recent literature. Yet, LLAMBO does not achieve the same acceleration as LGBO, indicating that the gain is not from initialization alone but from LGBO's continuous integration LLM preferences.
CONCLUSION
This work introduces L.Ldeed Bayesian Optimization, a preference-driven framework that in- tegrates LLMs into BO through novel region-lifted preference mechanism. LGBO provides con- tinuous guidance while preserving the rigor of BO. We establish regret bounds showing that LGBO matches BO in the worst case and accelerates convergence when guidance aligns. Ablation studies confirm that its gains stem from the continuous incorporation of LLM preferences via region lifting, and the framework broadly applicable across different LLM backbones. On four dry benchmarks and a new wet-lab Fe-Cr electrolyte task, LGBO achieves faster convergence, lower variance, and stronger robustness than GPBO and ILAMBO,, demonstrating its promise for scientific discovery.
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ACKNOWLEDGEMENTS
This work is supported by New Generation Artificial Intelligence-National Science and Technology Major Project(20357D0121180))
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A THEORETICAL DETAILS
Theorem (Gaussian measure under exponential linear lift), Let (E, B) be separable real topo- logical vector space, E' its continuous dual, and IP = N N(m, C) Gaussian measure with mean m E E and covariance operator C E - E. For any continuous linear functional E E" and
any X R, define the lifted measure
dP dIP
=xp= ((1), (AA) = log Ee N.L)
Then [l is still Gaussian with unchanged covariance C, and mean shifted by m - m + XCE.
Moreover, the expected lift of the functional
= Er, - [ = (CE, = À Varp(e(f))-
Thus a prescribed increment 8 is achievable [a'Var [([]] > 0, in which case X = A A/Vanm(e([))
Proof. We present the argument via fiite-ddimensional projections. For any finite set of functionals O = (éi..... .it......ne 2m) C E", the vector
Y = (f1(f)..... (f). {(f))
is multivariate Gaussian under P. Lifting the density by exp(Ae(f)) yields, by standard Gaussian completion-of-squares or cumulant-generatinn function arguments, that
(0) ..... Tm)) ~ Mus + +CCov((()(()))) Eo).
Here μa (d((m)); and Σq ((Cor, Φ1)) But
Covy (ovz()(f), (()) (CE, o),
so the new mean is exactly (m. + XCE)); while the covariance remains Σg. By Kolmogorov consistency, this determines a Gaussian measure N(m + XCE, C) on E.
Finally,
A {(m + ACE) {(m) = X (CE, l), which equals À (arr(((f)), yielding the desired characterization of X.
Lemma 1 ((Chowdhury & Gopalan 2017 Theorem 2)). Let X be compact and k X x X - R continuous kernel scaled so that xr x) & 1. Assume the (possibly mean-centered) target function satisfies f E Hg with ||f ll,, B, and observations follow
Y1
f(x)) + Et, 1,1,2....
where {er} are independent Rasus-Ga noise variables. Let K₁-1 = [k(a,, xz ) kx-)(x)= [k(01, x)] and define the GP posterior (with noise variance o² and prior mean 0) by
Pt ( (x) (x- (x-1(z)((KK-))) + 0²1) 11--1
(x) = k(x1,)) ( (K,-1 + o²T) (x),
where y1--1 [yh....
11-1)" Let the (maximal) information gain be
1 log det (1 =2
= ACX,IA|=
Then for any E (0, 1), with probability at least 5, simultuneously for all 2 and all x X,
[(x)) n4 -=1(xx) < B R RV 2(m +11 81 (x).
Equivalently, defining
3c(B) = B² + 2R ( + 1+1++
we have
- - < / Be( (z
(I)
uniformly all 1 with probability at least 8.
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Lemma 2 ( (riniias et al. 2009
Lemma 5.3 & Lemma 5.4)). Under the same setting and notation as in Lemma with k v, 2) & 1), for any chosen sequence (at the following hold:
(Information gain decomposition)
T 1 3
(y =
Σ t=1
log
2. (Capped variance sum)
Σ ming 1, 0- = S 2TT.
3. (Width-sum bound) Consequently,
Σ ( S 2T
Theorem (Lifted-GP-UCBB high-probability regret bound). Let X' be compact and continuous with k(x, x) S. 1. Let the true function f E Hy with || | < Bo. and observations = f((x)++ where } are i..dd R-sub-Gaussian Fix a grid & { and weights IRC and define
g(x) Σ 0g kg,
7(x) >g(x),
11 a E'ggaa
where (Egg) k(zz ). Consider running GP regression on the residual labels v. == 7 (xte) (same kernel le and noise variance o²), yielding posterior mean مور and standard deviation 81-1, and choose
3t. E arg max (z)) + v Be(B) 11-1
TEX
where BB( = - + 2R (7-11 + 1 + In }) and Yt is the maximal information gain. Define the "in-region" " and conservative radii
Bun f He-
Bout Bu + 1lllg
Then, with probability at least 1 8, for all T 2 the cumulative regret Rr = -E= (f(x*) (x:)) satisfies
T
Rr S 2 Br(Bnn)) Bm))
Σ
8t-1 8 S / 8T Bia) rr:
1=1
and, without assuming alignment,
RT S / 8T Br (Bout) Yr. Yr.
Proof. Let (=
= f(x) - -(x). -xx). By Lemma for any 2 such that || $ B,
ff'()) (x) s (z)) Vo X. 1
holds with probability least 6. By the selection rule,
f(x²) s (x') + (B) (x-1(x)) + s,)
and
f((x)) 2 (x)) /3,(B)s,,((21)
Thus the instantancous regret
r f(x') f(xx) ((xx)) (x(x) s 2 2 B(B) 811( $ 2 (rBrrB) 81 8 (xx)-
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Summing over t and applying Lemma gives
Rr S BT
Σ
( (B)) 2T nr
V8T (r( r7.
When the LLM provides unreasonable regions, we have
Rr s 2/ BT 2T .T YT V8T ( A1gll)> nT.
(h.o) |
Now, we focus on the bound of Bin- Let h = - -
LLM provides a suitable lifting
area, we have > Let À
cBe
we have Tlol
B = 15 T| = (Allall - cBo)2 + - B3
So, have
Pr S (Bm) 22
VST ByVi1-
B PROMPT
To ensure stability, reprodueibility, and transparency, we design the prompt in modular fashion, consisting of fixed system prompt and dynamic user prompt. The system prompt encodes general scientific principles and strict output rules that remain invariant across tasks, while the user prompt provides experiment-specific background and historical context. This separation allows LGBO (i) maintain consistent reasoning discipline, (ii) incorporate domain knowledge explicitly, and (iii) adapt flexibly to different experimental datasets.
User prompt. The user prompt augments the fixed rules with dataset-specific context. It defines experimental background, parameter order, optimization objectives, and constraints. In addition, it incorporates both the experimental history (latest observations) and the reasoning trace from the previous round, enabling iterative refinement while avoiding overfitting to past data. To illustrate the structure, schematic excerpt of the user prompt is shown below:
[Background]
Experiment type purpose:
Parameter order (d=):...
Objective: Minimize f() (single ab objective).
Constraints: All parameters real-valued, dimensionless;
respect declared order and bounds; do not normalize.
Bounas:
[Review]
Historical data (newest first):
-Thinking from the previous round:...
Adoption note: Suggestions were used as guidance;
actual tested points may differ.
System prompt. The system prompt frames the LLM scientist specializing in experimental optimization. establishes an evidence hierarchy caaurccy--iioriiiiinng mechanistic background knowledge over historical data-and specifies strict output formats to ensure valid and auditable responses. The output must be either point or a region with confidence score, following the declared parameter order and physical units. To mitigate collapse, the prompt explicitly forbids anchoring past best points unless mechanistically justified. Below is the core structure:
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Evicence hierarchy (critical)
PRIMARY: Background knowledge, physical/chemi calmical Techanisms, constraints, and units.
SECONDARY (auxiliary only): Histcrical trial points/dbservation. and thinking the review.
If background implications conflict with historical points, SIDE WITH BACXGROUND.
Background (fixed across rounds)
We run iterative chemistry experiments (e.g., polymerraattooohhyrroooo. /organic synthesis)..
Parameters are physical and NOT normalized. Always use the declared units order.
Modes (pick exactly ONE)
1) [point, 1xl, x2, .... xd), ccc|
2) [region, [[1bl, 1b2, .... 1bd],
[ubl, ub2, cee S\inø [0,1].
ubd]l, ccc]
In region mode, each dimensicn must have (lb 8\leqS ub) and follow the declared parameter order & units.
For categorical variables, output their literal value (e.g., "DMF") in both point and region.
If a category is fixed in regicn, set lb=ub to that same literal value.
How to reason (prioritize background over past points)
Start from first principles: mechanis sn-aniend trends, feasible/unsafe ranges, known monoton cies,, interactions.
Use historical data ONLY as weak corroboration or disproof of background-based hypothesis.
NOT anchor on previous best/nearest points; avoid preposing point merely because appeared before.
If historical points cluster narrowly, consider a background-justifiee exploratory move (o.g., shift in a mechanism-zelovene
factor).
Frefer REGION when background suggests multiple nearby settings could satisfy the mechanistic target; choose POINT only when backgroundt data imply a sharp optimum.
# Output protocol (two plocks)
1) Thinking:
Be concise but informative, this order:
(a) background-based rationale (mechanissn/constrriitts)) that leads to your proposal.
(b) How (if at all) historical data supports/contradicts this mechanism ($\leq2 sentences).
(c) Why point vs region given the mechanism and uncertainty.
2) Final Answer:
Strict structure with no extra words:
[point, [x1, x2, ... xd], ccc)
OR
[region, [[1bl, 1b2, 1bd], ubd]], ccc]
[ubl, ub2,
Eard constraints
Do NOT normalize or re-order parameters.
Keep units consistent with the declared parameter order.
No extra commentary in Final Answer beyond the bracketed structure. Anti-collapse checks
Never center region or point a past observation unless mechanistically justified.
If you reuse past setting, explicitly state the mechanism that makes it optimal (in Thinking).
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Summary. Together, the system and user prompts enforce a disciplined reasoning protocol: the system ensures scientific validity and structural consistency, while the user prompt injects contextual knowledge and history. This modularity makes LGBO's prompting scheme both transparent and easily transferable to new experimental domains.
C DRY EXPERIMENT DATASETS
We provide detailed specifications of the four dry-experiment datasets used in our study. Each dataset is summarized with variable types and ranges, and the corresponding optimization objec- tive. Together they span formulation science, structural design, materials engineering, and process control, offering diverse and representative testbed for Bayesian optimization.
Lipid Nanoparticle Formulation) (Chen et al., 2022). This dataset contains 768 samples de- scribing lipid nanoparticle (LNP) formulations for cannabidiol (CBD) delivery. It has 5 input vari- ables and 3 objectives. We perform 2D one-hot interpolation for solid lipid based whether it is an acid. Because the dataset fully enumerates all variable combinations, no interpolation is required.
Table 1: LNP3 dataset specification.
Variable
Type
Range Options
drug_inpu
discrete
{6, 12, 24, 48}
solid_lipid
categorical
(Stearic_aeee accid Compritol m
solid lipid input
discrete
(120, 108, 96, 72)
liquid lipid input surfactant_input
discrete
(0, 12, 24, 48)
discrete
(0, 0.0025, 0.005, 0.01)
drug loading
objective
maximize
encapsulation efficiency
objective
maximize
particle_diameter ddiameter
objective
minimize
In our experiments, each objective is normalized to [0, 1] by its observed range, and the three nor- malized objectives are summed to define single scalar objective.
3D-printed structural design Cross-barrel (Gongora et al., 2020 This dataset consists 600 samples characterizing the performance of crossed- barrel 3D printed structures. Interpolation (mul- tilinear grid or kNN+IDW fallback) is applied to enable continuous evaluation.
Table 2: Cross-barrel dataset specification.
Variable
Type
Range
continuous
[6.0, 12.0]
continuous
[0.0, 200.0]
continuous
[1.5, 2.5]
t
continuous
[0.7, 1.4]
toughness
objective
maximize
Concrete compressive strength (Tripathi, 2020). This dataset contains 1,030 samples describ- ing mixture design for compressive strength optmiization Age is excluded, leaving 7 continuous mixture variables. Interpolation (multilinear grid or kNN+IDW fallback) is used build the oracle.
HPLC Hase et al. 2021). This dataset contains 1,386 samples of high-perffrmance liquid chro- matography (HPLC) experiments. The objective is maximize chromatographic peak area. Inter- polation (multilinear grid or KNN+IDW fallback) is applied to extend the dataset to continuous oracle.
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Table Concrete dataset specification.
Variable
Type
Range
cement
continuous
[102.0, 540.0]
blast furnace slag
continuous
[0.0, 359.4]
fly ash
continuous
[0.0, 200.1]
water
continuous
[121.8, 247.0]
superplasticizer coarse aggregate fine aggregate
continuous
[0.0, 32.2]
continuous
[801.0 10, 1145.0) )
continuous
[594.0, 992.6]
compressive strength
objective
maximize
Table 4: HPL.C dataset specification.
Variable
Type
Range
sample loop additional additioonllvolume volume
continuous
[0.00, 0.08) ml
continuous
[0.00, 0.06] ml
tubing.volume
continuous
[0.10, 0.90] ml
sample flow
continuous
[0.50, 2.50] ml/min
push.speed wait_time
continuous
[80.0, 150.0] Hz
continuous
[1.0, 10.0]
peak area
objective
maximize
Oracle construction protocol. For consistency, datasets are treated black-box objectives over their observed parameter ranges. LNP3 requires no interpolation since it fully enumerates all pa- rameter combinations. For Cross-barrel, Concrete, and HPLC, if complete grid is detected, N-dimensional multilinear interpolation is applied; otherwise, scattered data interpolation with KNN+IDW (k 12, exponent p = 2.0, tolerance E = 10 (2) is used.
D ADDITIONAL EXPERIMENTAL RESULTS
D.1 LNP3
Figures 88 present the scatter plots of sampled points for the three methods on the LNP3 task. Each plot pairplot the normalized input space: off-diagonal panels show pairwise projections of the six variables, with points colored by optimization round (darker earlier, lighter = later), while diagonal panels display marginal histograms. This visualization enables us to directly compare how different methods explore the search space.
Compared with GPBO, which scatters samples broadly across the space and exhibits large run-to- variability, and LLAMBO, which provides warmer start but still explores many uninformative regions, LGBO under expert LLM guidance concentrates sampling much faster into promising subregions. This behavior avoids wasting evaluations in irrelevant areas and explains why LGBO achieves both lower variance across runs and faster convergence, even though the exact optimum is unknown in real-world experiments.
D.2 HPLC
As the LNP3 case, Figure 13 presents the convergence traces and trajectory heatmaps on the HPLC task. The overall trend similar: LGBO converges faster and achieves higher final perfor- mance than the baselines. The main difference is that this experiment is considerably noisier, leading to much larger fluctuations across all methods.
Figures 10 12 show the pairplots of sampled points for the three methods. Compared with GPBO, which scatters samples broadly and exhibits high variability, and LLAMBO, which provides
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Pairplot of Sampled Points GP8O (normalized space)
count
x2
round
0.5
05
0.
XS
x2
x5
Figure 6: Pairplot of sampled points for GPBO LNP3 (normalized space).
Pairplot Sampled Points- LLAMBO (nermalized space)
count
X4
0.
x2
x4
x5
Figure 7: Pairplot of sampled points for LLAMBO on LNP3 (normalized space).
warmer start but still explores many uninformative regions, LGBO quickly concentrates sampling
into promising subregions under expert LLM guidance. This targeted exploration avoids wasting evaluations and explains why LGBO achieves both faster convergence and lower variance across runs, even under noisy real-world conditions.
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Pairplot of Sampled Points LG80 (nermalized space)
count
X2
round
0.5
x2
x5
Figure 8: Pairplot of sampled points for LGBO (ours) on LNP3 (normalized space).
Trojectory Heatmap - LGBO(Ours)
29
Seed
Best- 100-000-00
28
27-
Round
Trajectory Healmap LLAMBO
Best Best-se-far
55
15
CPRO LS53 LUAVBO
Seed
Trajectory Heatmap GPBO
Rest-so-fa fat
mard
Round
Figure 9: LNP3 results. Left: LGBO achieves faster convergence and higher final performance than GPBO and LLAMBO. Right: trajectory heatmaps show that LGBO converges more rapidly and consistently across seeds.
D.3 CROSS BARREL
For the Cross-barre task 3D-printed structural optimization with only four variables), the search space is relatively low-dimenssonal and thus easier for Bayesian optimization. As shown Fig- ure 14 even pure GPBO quickly reaches strong performance. Nevertheless, LGBO consistently achieves the best results across runs. In Figures 17 one interesting observation is that the scat- ter plots appear more dispersed in later rounds: this is because the search rapidly approaches the optimum, after which additional rounds tend to explore more broadly around the high perforrefonnng region.
D.4 CONCRETE
For the Concrete mixture design task, in Figure [8] a widely used engineering benchmark, the LLM warm-up already lifts the initial performance to high level. Interestingly, LGBO appears slightly
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Pairlet of Sampled Points GPBD (nommalized space)
x2
x5
Figure 10: Pairplot of sampled points for GPBO on HPLC (normalized space).
Pairplat of Sampled Points- LLAMBO (ncemalised spsce)
count
x2
x4
x5
Figure 11: Pairplot of sampled points for LLAMBO on HPLC (normalized space).
weaker than LLAMBO in the very early rounds. We attribute this to the fact that LGBO, guided by the LLM, is directed toward higher-value regions rather than over-exploiting the initial local optima. As a result, although LLAMBO temporarily maintains an advantage, LGBO rapidly catches up and ultimately surpasses both baselines, achieving the highest final objective values.
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Prirplot Sampled Poims LGBO (noemalized
006TT
x2
x2
x5
Figure 12: Pairplot of sampled points for LGBO (ours) HPLC (normalized space).
100
Trajectory Heatmap LGBO(Ouus)
90
Seed
Dest sox
Best Objective (%)
80
Round
70-
Trujectory lleatmap LLAMBO
Dost: sp Ear
60-
50
40
30
LGBO
Trajectory Heatmap GPBO
Bestss-fi ftr
LLAMBO
20
GPBO
10
Round
0
5
10
15
20
25
30
Round
Figure 13: HPLC results. Left: mean performance traces (mean + variance over five runs). Right:
trajectory heatmaps showing convergence dynamics across seeds.
LARGE LANGUAGE MODEL USAGE STATEMENT
In accordance with the ICLR 2026 Author Guidelines on the use of large language models, we acknowledge that large language models were used for phrasing refinement and grammar correction during manuscript preparation. All scientific ideas, algorithmic designs, and experimental results solely the work of the authors.
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100
Trajectory Heatmap LGBOOOurr)
Bel-so- far
90
Best Objective (%)
Round
80
Trujectory Heutmap
70
Round
60
LGBO
Trajectory Heatmap- GPBO
Bost- Gtr
LLAMBO
50
GPBO
Round
0
5
10
15 Round
20
25
30
Figure 14: Cross Barrel results. Left: LGBO achieves faster convergence and higher final perfor- mance than GPBO and LLAMBO. Right: trajectory heatmaps show that LGBO converges more rapidly and consistently across seeds.
Paiplet Samplal space))
x2
x5
Figure 15: Pairplot sampled points for GPBO on Cross Barrel (normalized space).
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Prieplot of Sampled Points- AMBB (neemalived space)
x5
x2
x5
Figure 16: Pairplot of sampled points for LLAMBO on Cross Barrel (normalized space).
Prispiat o'Suapled Points- LGBO (nommalizel spaze)
count
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Figure 17: Pairplot of sampled points for LGBO (ours) on Cross Barrel (normalized space).
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Trajectory lleatmup LGBl0(Oens)
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Figure 18: Concrete results. Left: LGBO achieves faster convergence and higher final performance than GPBO and LLAMBO. Right: trajectory heatmaps show that LGBO converges more rapidly and consistently across seeds.
Paiplee Sampled Points GPRO (noemallinel spead)
Figure 19: Pairplot of sampled points for GPBO on Concrete (normalized space).
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Published as a conference paper at ICLR 2026
Prirplot Sampled ALAMMO (nommalized space)
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Figure 20: Pairplot of sampled points for LLAMBO on Concrete (normalized space).
Paisplet Sampled Points- LGBO (smmaliied space))
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Figure 21: Pairplot of sampled points for LGBO (ours) on Concrete (normalized space).
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Published as a conference paper at ICLR 2026
E MORE EXPERIMENTS
To further evaluate the robustness and generalization capability of LGBO in scientific discovery tasks, extend our experimental analysis with two additional baselines and higher-dimensional benchmark. Specifically, we incorporate two recently proposed LLM-augmented Bayesian opti- mization methods and conduct experiments
challenging 14-dimensional scientific dataset. And two newly LLM-BO baselines are compared with LGBO.
We conduct fair comparisons with these methods the original benchmark datasets and further validate LGBO on a more complex 14-dimensional scientific exploration task. All extended exper- iments consistently demonstrate that LGBO maintains clear performance advantage- only in low-dimensional settings but also in high-dimensional, real-world scientific scenarios scenarios- highlighting its effectiveness, scalability, and superiority scientific discovery applications.
E.1
MORE DATASETS
COF (Gantzler et al., 2023 The COF dataset consists of 608 covalent organic framework (COF) candidates evaluated for their performance in xenon/krypton separations. Each material is encoded 14-dimensional feature vector that captures its pore geometry (e.g.. pore diameter, void fraction), textural properties (e.g., surface area, crystal density). and elemental composition. The optimization objective is the Xe/Kr selectivity, central metric in adsorption-base -based noble-gas separations that quantifies a material's 's thermodynamic preference for xenon relative to krypton. Selectivity is defined as the ratio of xenon uptake krypton uptake under identical pressure and temperature conditions, and higher values indicate stronger differential affinity toward xenon. This property is particularly important for applications such as the capture of radioactive xenon released during nuclear fuel reprocessing, off- gas treatment, and environmental monitoring, where efficient enrichment of xenon over krypton is critical.
Table 5: COF dataset specification
Variable
Type
Range
pore diremeieeter
continuous
[3.51, 56.40]
void_fraction
continuous
[0.164, 0.928]
surface area
continuous
[1996.63, 6357.01] m
crystal density
continuous
[102.72, 1610.70] kg
B
continuous
[0.000, 0.182) mol fraction
O
continuous
[0.000, 0.250) mol fraction
C
continuous
[0.325, 0.667] mol fraction
H
continuous
[0.000, 0.500] mol fraction
Si
continuous
[0.000, 0.0295] mol fraction
N
continuous
[0.000, 0.333] mol fraction
S
continuous
[0.000, 0.143] mol fraction
P
continuous
[0.000, 0.0667] mol fraction
halogens
continuous
[0.000, 0.286] mol fraction
metals
continuous
[0.000, 0.0238] mol fraction
Xe/Kr selectivity
objective
maximize
E.2 MORE RESULTS
We compare with two more LLM-BO algorithms below:
Co (Hvarfner et al. 2024): An enhanced variant of the classical colaBO framework that integrates large language models (LLMs) to model expert preferences and employs an LLM- warm-up strategy to generate high-quality initial candidates, thereby acceler- ating convergence.
BOPRO (Agarwal et al. 2025) : A method that leverages large language models as im- plicit Bayesian priors through in-context learning. BOPRO constructs prompts containing
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previously observed input-outpur pairs and uses the LLM's contextual reasoning ability to extrapolate patterns or trends the objective function.
CAKE (Suwandi et al., 2025) method that accelerates Bayesian optimization by design- ing kernel functions informed by large language models.
Across all five scientific datasets (Figure 22 26). LGBO consistently achieves the highest or near- highest objective values, particularly demonstrating strong early-stage convergence and stable final performance.
On the Cross-barrel and LNP3 tasks, LGBO rapidly surpasses other methods within the first 10 rounds, showing clear sample efficiency under limited evaluations. Although ColaLLM and LLAMBO also leverage LLM guidance, their convergence is slower and exhibits greater variance, suggesting less effective utilization of model priors.
In HPLC, LGBO steadily outperforms other approaches throughout the entire optimization process, with narrow confidence bands indicating robustness. On the COF dataset-the highest-dimensional task (--14)LGBB again achieves the best overall performance, highlighting its scalability and effectiveness in more challenging, high-dimensional search spaces.
The Concrete dataset appears relatively saturated across methods, yet LGBO still maintains a late- stage advantage, pointing to better exploitation capability in the final rounds.
GPBO, the only method without L.M guidance, generally lags behind the LLM-aided methods across most tasks, particularly where prior knowledge is essential or the search space is complex. While not uniformly the worst, its performance gap illustrates the tangible benefit of incorporating lightweight model-derived preferences.
Overall, LGBO shows the most consistent, sample-efficien,, and scalable optimization behavior, making it well-suited for complex scientific design settings.
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LGBO
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LLAMBO
GPBO
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BOPRO
ColaLLM
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Figure 22: Performance comparison of LGBO against LLAMA, GPBO, ColaLLM and BOPRO on the Cross-barrel scientific dataset.
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Figure 23: Performance comparison of LGBO against LLAMA, GPBO, ColaLLM and BOPRO on the LNP3 scientific dataset.
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Figure 24: Performance comparison of L.BBO against I.AMM,, GPBO, ColaLLM and BOPRO on the HPLC scientific dataset.
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Published as a conference paper at ICLR 2026
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Figure 25: Performance comparison of LGBO against LLAMA, GPBO, ColaLLM and BOPRO on the Concrete scientific dataset.
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Figure 26: Performance comparison of LGBO against LLAMA, GPBO, ColaLLM and BOPRO on the COF scientific dataset.
31


Reviews:
[{"review_id": "ChkaFyd3QM", "reviewer": "Reviewer_935h", "summary": "This paper proposes a way to integrate prior information in large language models into Bayesian optimization.\nAt its core, the method is based on a simple idea.\nIt prompts the language model to generate a promising region in the search space.\nThen, the proposed method updates the GP mean function so that the mean function has higher values in the promising region.\nExperiments on benchmark test functions across several areas show promising results against conventional BO.", "questions": "1. What's the exact definition of misaligned lift in Theorem 1? Isn't the misaligned case the same as \\\\(c = 0\\\\), i.e., small cosine similarity?\n\n1. Line 293: Is the norm in theorem 1 the RKHS norm?\nIf so, it would be better to make it explicit.\n\n1. Line 162: \"Intuitively, \\\\(p(\\cdot)\\\\) here denotes a probability distribution...\".\nShouldn't it be \\\\(\\rho\\\\)?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. This paper works in the intersection of Bayesian optimization and large language models, and shows how to effective utilize the prior information stored in LLMs.\nThose prior information is often hard to encode into conventional Bayesian optimization by kernel designs.\n\n1. Using the lifting functional to encode the prior information from the language model makes sense to me.\nThough I have not checked the math, the part that the functional in the exponential form turns out to shift the mean function makes sense to me, as it is similar to exponential tilting for Gaussian distributions.", "weaknesses": "1. The interpretation of Theorem 1 seems to be over claimed.\nThe correct interpretation should be weaker than what's claimed in the paper.\nThe theorem assumes the lift is given in advanced, and **does not change** during BO.\nHowever, the method proposed in this paper actually utilizes language models interactively in that the lifted region and or points get updated in each iteration of BO.\nThus, if the language model's prediction is bad, the regret bound could be unbounded.\n\n1. Many important technical details are missing.\n    - How to set the guidance strength parameters \\\\(a_g\\\\)?\n    Do you estimate them during GP model fitting with maximum likelihood, or do you set them to fixed numbers?\n    - How the discretization \\\\(x_g\\\\) is chosen in a lifted region?\n    - How does the confidence scores generated by LLMs affect the lifting functional?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "moAo0ARehj", "reviewer": "Reviewer_W7ee", "summary": "This paper introduces LLM-Guided Bayesian Optimization (LGBO), a framework that integrates priors from large language models (LLMs) into Bayesian Optimization (BO) for scientific experimentation. Unlike previous approaches that use LLMs only for the initialization of BO exploration or candidate generation, LGBO continuously incorporates LLM-derived region-lifted preferences into every iteration of the optimization loop. The authors show that this mechanism effectively shifts the Gaussian process (GP) surrogate’s mean without affecting its covariance, allowing semantic LLM guidance while retaining BO’s statistical guarantees. Theoretical results prove that LGBO is not worse than standard BO in the worst case and can achieve faster convergence when LLM guidance aligns with the true objective. Experiments on four “dry” scientific benchmarks (LNP3, Cross-barrel, Concrete, HPLC) and one “wet-lab” Fe–Cr battery optimization demonstrate consistent acceleration and improved stability over GPBO and LLAMBO baselines.", "questions": "- What is really the motivation for choosing preference optimisation for this setting? It is not immediately obvious to me. Better motivation as context for the approach would be helpful.\n\n- The region-lifted preference is formalized as an exponential mean shift. Could similar effects be achieved using other functional forms (e.g., linear or kernel-weighted lifts), and what motivated the specific exponential choice?\n\n- Since LGBO queries the LLM at every iteration, what is the computational or latency overhead compared to traditional BO? Could lightweight surrogates or cached reasoning traces mitigate this cost?\n\n- How sensitive are the results to the exact prompt design or LLM choice? While Appendix B provides structured prompts, have you quantified performance variance across alternative phrasing or reasoning styles?\n\n- Have you analyzed the semantic quality of the LLM-suggested regions—do they align with known scientific heuristics or physical laws?", "limitations": "", "rating": 8, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "- Novel integration: Proposes a principled and theoretically grounded method to embed LLM preferences directly into the BO surrogate, moving beyond heuristic or warm-start use of LLMs.\n\n- Theoretical formulation: Provides formal regret bounds showing bounded degradation under misalignment.\n\n- Evaluation: Including both simulation and real-world experiments, showing convincing empirical improvements across diverse domains.\n\n- Reproducibility: Clearly describes prompt templates, datasets, and experimental protocols; theoretical and implementation details are provided.\n\n- Scientific relevance: Demonstrates the potential for LLMs to accelerate experimental optimization in time-limited domains.\n\n- Stable framework design: The region-lifted preference mechanism elegantly maintains GP structure and stability, avoiding instability issues typical of preference-based methods.", "weaknesses": "- Evaluation baselines: Evaluation omits other preference-based or human-in-the-loop BO methods (e.g., ColaBO, Preferential BO with human experts), which could contextualize LGBO’s relative contribution.\n\n- Dependence on prompt engineering: Performance and robustness may depend strongly on prompt quality and LLM capabilities, but sensitivity analyses on prompt design are limited.\n\n- Scalability questions: Experiments focus on low- to medium-dimensional tasks (≤6 variables); it remains unclear how LGBO scales to high-dimensional or multi-objective settings.\n\n- Computational cost: Continuous LLM querying at each iteration may incur significant computational or latency overhead; this is not quantified or mentioned in the paper. \n\n- Interpretability of LLM Guidance: While the framework embeds preferences mathematically, the semantic validity or interpretability of the generated regions is not thoroughly analyzed.\n\n- Limited real-world task diversity: Only one wet-lab experiment is included; broader real-world validations would strengthen generality claims. It is also not clear why these particular tasks were selected for the evaluation.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "BYPkgVGNcz", "reviewer": "Reviewer_M7Dg", "summary": "This paper introduces LLM-Guided Bayesian Optimization (LGBO), a new framework that continuously integrates LLM preferences into the BO process. Unlike prior work that uses LLMs only for warm-start initialization or candidate proposal, LGBO employs a region-lifted preference mechanism that adjusts the GP surrogate mean based on LLM-specified regions of interest. The authors show that this modification maintains theoretical rigor, preserving covariance structure and provide regret bounds demonstrating that LGBO performs no worse than standard BO in the worst case and converges faster when LLM preferences align with the true objective.\n\nEmpirically, LGBO is evaluated on four dry scientific benchmarks (LNP3, Cross-barrel, Concrete, HPLC) and a new wet-lab experiment on Fe–Cr battery electrolytes. Across all tasks, LGBO outperforms both standard GP-based BO and LLMABO, showing faster convergence, higher final performance, and lower variance. Ablation studies further confirm that performance gains arise from the continous preference integration rather than initialization or random region lifting.", "questions": "1. Why $a_g$ are chosen to be greater than or equal 0? ( why don't we define a general LLM-based adjustment, not just lifting potential points, but also penalizing undesirable queries by allowing negative $a_g$s?)\n2. The novelty of the region-lifted preference is not emphasized until conclusion (by searching the key word \"novel\"). Could the authors re-confirm this is a novel idea?", "limitations": "", "rating": 2, "confidence": 3, "soundness": 2, "presentation": 1, "contribution": 3, "strengths": "- Novel mechanism for incorporating LLM guidance into BO: the region-lifted preference is mathematically principled and computationally tractable. Making UCB-type bound natural to derive. \n- This paper is well-motivated, with a wide range of real-world scientific discovery problems tested in the experiments. \n- The methodology in this paper provides new insights about how **continuously** incorporating LLM preference could stabilize BO convergence compared to LLMABO, which integrates LLM preference indirectly, with robust ablation study.\n- The framework is general and modular, compatible with standard GP surrogates and acquisition functions.", "weaknesses": "**The writing can be improved a lot** \n- The definitions and assumptions are not stated in the main paper. The definitions of $||\\cdot||$ in Line 293 and $R_T$ in Line 302 are missing, and the assumption for GP regression in the RKHS setting is not clearly stated in the main paper.\n    \n- The statement _“running GP-UCB on residual labels”_ in Line 295 is vague. Moreover, how it is equivalent to executing the proposed LGBO algorithm may be unclear to readers at first glance.\n    \n- The definition of _alignment_ being equivalent to $c > 0$ in Theorem 1 is not clearly stated and explained. Its definition appears only in the proof of Theorem 1 in Appendix A.\n    \n- In addition, there is a major typo in Theorem 1. According to the proof,  \n    $c = \\frac{\\langle f - \\tau, g \\rangle}{||f - \\tau||\\cdot||g||}$,  \n    while in the main text it is written as  \n    $c = \\frac{\\langle f - \\mu, g \\rangle}{||f - \\mu||\\cdot||g||}$.  \n    This is misleading, since $\\mu$ is typically the GP mean, whereas $\\tau$ is related to the lifting function.\n    \n**Theoretical result is not consistent with the proposed method.** Theorem 2 only describes the algorithm’s behavior when the mean adjustment is a fixed  \n    $g(\\cdot) = \\sum_{i=1} a_i  k(x_i, \\cdot)$ through out all iterations \n    whereas the proposed algorithm proposes a different $g$ through the LLM at each BO iteration. Thus, the theoretical results only partially explain the performance of LGBO.\n    \n**Lack of baseline comparison.** Only two baselines are included, and among them, only LLAMBO is an LLM-based BO method. Other LLM-integrated BO methods mentioned in the related work, such as ColaBO and LLINBO, are not included. Therefore, the statement about the instability and potential divergence of ColaBO in Line 128 is not well-supported by evidence.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "Sm6Z1WAyN9", "reviewer": "Reviewer_oNUD", "summary": "This paper introduces LLM-Guided Bayesian Optimization (LGBO), a novel framework for integrating LLMs into the BO loop. The core contribution is a \"region-lifted preference\" mechanism, where the LLM provides continuous guidance by suggesting promising regions in the search space. This guidance is incorporated as a stable mean shift in the GP surrogate model, leaving the covariance structure intact. The authors provide theoretical regret bounds demonstrating that the method is safe in the worst case (i.e., with misaligned guidance) and can significantly accelerate convergence when the LLM's guidance is accurate. The framework is evaluated on several \"dry\" scientific optimization benchmarks and a \"wet-lab\" experiment, showing improved performance over standard GPBO and a prior LLM-augmented method, LLAMBO.", "questions": "- How does LGBO handle optimization landscapes where the promising regions are have complex geometries that cannot be well-approximated by this format? Does this structural constraint limit the framework's applicability?\n- How do we translate the LLM's confidence score into the guidance strength $\\lambda$ and the region's properties (e.g., radius in point mode)? Since the parameter $\\lambda$ is critical to the regret bounds and practical performance, the authors should provide more discussion on its selection, tuning, or sensitivity\n- Based on my experience, a strong mean shift in an incorrect region without a corresponding increase in uncertainty might overly encourage the acquisition function to exploit that area. Since the proposed mechanism only shifts the surrogate's mean while leaving the covariance unchanged, could this lead to premature convergence if the LLM is confidently wrong?", "limitations": "", "rating": 4, "confidence": 5, "soundness": 2, "presentation": 3, "contribution": 3, "strengths": "- The core idea of using an exponential lift on a discretized region, and showing its equivalence to a GP mean shift, is novel and technically sound\n- Theorem 1 is useful, as it formally establishes that the framework is robust to poor/misleading LLM guidance (worst-case) and can provably accelerate convergence when guidance is informative\n- The wet experiment on Fe-Cr battery electrolyte optimization is a great example to demonstrate the method's applicability to real scientific discovery problems beyond simulated benchmarks", "weaknesses": "- The paper motivates the work by citing the poor scalability of BO in high-dimensional settings. However, the experiments are conducted in search spaces of relatively low dimensionality (i.e., all experiments are conducted in low-dimensional settings with $d \\leq 7$). Hence, the effectiveness of the proposed method in a truly high-dimensional problem is not sufficiently demonstrated\n- The choice of baselines is far from adequate. The authors cite LLINBO and ReasoningBO as more systematic \"LLM-in-the-loop\" frameworks, yet fail to include them in the comparison. Comparisons with other related LLM-driven BO approaches, such as BOPRO [1] and CAKE [2], are also missing\n- To the best of my understanding, the LLM's guidance is limited to a single point or a hyper-rectangular region. I believe this format may be too simplistic for real-world problems where promising regions could be complex (e.g., non-convex, disjoint, or like a manifold). Hence, it is not clear if this method can be applied to problems with more complex geometries\n\n[1] D. Agarwal et al., \"Searching for optimal solutions with LLMs via Bayesian optimization,\" ICLR, 2025.\n\n[2] R. C. Suwandi et al., \"Adaptive kernel design for Bayesian optimization is a piece of CAKE with LLMs,\" arXiv preprint arXiv:2509.17998, 2025.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)