LLM-Guided Bayesian Optimization

• LLM-guided BO is a hybrid framework that integrates large language models with Bayesian Optimization to enhance candidate generation, warmstarting, and uncertainty estimation.
• It employs prompt-driven techniques, surrogate modeling with uncertainty quantification, and hybrid acquisition strategies to improve sample efficiency in high-dimensional or low-data regimes.
• Empirical evaluations demonstrate rapid convergence and superior performance across domains such as chemistry, circuit design, and other complex optimization problems.

LLM-guided Bayesian Optimization (LLM-guided BO) refers to the integration of LLMs into the Bayesian Optimization (BO) paradigm for efficient, context-aware optimization of black-box functions across diverse domains. The central premise is to exploit the contextual reasoning, in-context learning, and domain knowledge of LLMs—capabilities accrued via large-scale pretraining and (optionally) domain adaptation—to augment and guide the sampling, acquisition, and surrogate modeling components of BO. This yields hybrid optimization pipelines with improved sample efficiency, rapid warm-starting, and robust search strategies, especially in expensive or high-dimensional problem settings.

1. Frameworks and Methodological Variants

Numerous frameworks instantiate LLM-guided BO through different points of integration:

• Prompt-driven Candidate Generation and Warmstarting:

LLMs are prompted in zero-shot or few-shot settings to generate initial candidate points in the search space, leveraging either general or domain-specific priors. This strategy has been concretely implemented in LLAMBO, where natural language “model cards” and optimization histories are serialized and supplied as prompts, resulting in warmstart candidates that empirically improve early-stage regret relative to uninformed or random initializations (Liu et al., 6 Feb 2024).

• LLM-based Surrogate Modeling:

Several approaches utilize LLMs as surrogate models for the target objective. Discriminative approaches parse optimization histories into text and prompt the LLM for outcome prediction and uncertainty estimation. Generative approaches reformulate surrogate modeling as classification or density ratio estimation (e.g., estimating $p(s \leq \tau | h; D_n)$ for candidate $h$). An example is the Monte Carlo LLM surrogate which uses repeated prompt permutations to approximate mean and standard deviation of predictions, thus addressing uncertainty quantification in low-data regimes (Liu et al., 6 Feb 2024).

• Acquisition and Candidate Sampling via LLMs:

Candidate points are generated from the LLM by conditioning on target scores, e.g., $s' = s_{\min} - \alpha (s_{\max} - s_{\min})$, with the exploration hyperparameter $\alpha$, to encourage the generation of configurations predicted to improve upon the current best (Liu et al., 6 Feb 2024). This mechanism is reflected in both LLAMBO and LLANA (Chen et al., 7 Jun 2024).

• LLM-BO Surrogate Feature Extraction:

In molecular/materials optimization, LLMs serve as fixed or adaptive feature extractors (from domain text representations such as SMILES) for conventional Bayesian surrogates (Gaussian Processes or Laplace-approximated Bayesian NNs), enabling data-efficient, domain-aware exploration of large chemical design spaces (Kristiadi et al., 7 Feb 2024).

• Hybrid Acquisition Functions:

Hybrid acquisition or sampling strategies orchestrate an LLM and a BO algorithm in an explicit loop—e.g., alternately sampling via LLM and the TPE (tree-structured Parzen estimator) as in SLLMBO (Mahammadli et al., 27 Oct 2024), or blending candidate selection between a GP and an LLM per a predefined schedule, as in LLINBO (Chang et al., 20 May 2025).

• Iterative Self-Improving Loops:

BOLT leverages a multi-task feedback loop: after each BO trajectory, the best solutions and trajectory data fine-tune the LLM; the updated LLM then provides stronger initializations for subsequent tasks, thus actualizing transfer learning at unprecedented scale (Zeng et al., 11 Mar 2025).

2. Surrogate Modeling, Uncertainty, and Exploration–Exploitation

LLM-guided BO confronts the canonical challenges of surrogate modeling and the exploration–exploitation dilemma using several mechanisms:

• Uncertainty Quantification:

When LLMs are used directly for prediction, uncertainty is estimated via prompt permutation (Monte Carlo sampling over in-context histories) or through probabilistic output heads (regression/classification score distributions). In frameworks where LLMs are used as feature extractors, uncertainty is inherited from the Bayesian surrogate (e.g., GP posterior over LLM features) or via Laplace network approximations post-parameter-efficient fine-tuning (Kristiadi et al., 7 Feb 2024).

• Balancing Exploration–Exploitation:

Hybrid samplers, such as the LLM-TPE strategy in SLLMBO, introduce randomization to alternate between LLM-driven exploitation (favoring regions encoded as promising by the LLM) and statistical exploration (TPE or GP UCB acquisition), resulting in robust search behavior and mitigation of local trapping or overexploitation. In LLINBO, this is formalized: a probability schedule $p_t$ ensures gradual transition from LLM-driven exploration (low $t$) to GP-driven exploitation (high $t$), with proven sublinear regret bounds (Chang et al., 20 May 2025). LLINBO also enables GP-based “justification” of LLM proposals and constraint-based posteriors to guarantee theoretical tractability.

3. Integration of Domain Knowledge and Transferability

LLMs are uniquely suited to inject domain-specific inductive biases and to facilitate knowledge transfer:

• Domain Priors and Pseudo-experiments:

Frameworks such as ChemBOMAS (Han et al., 10 Sep 2025) employ knowledge-driven, coarse-grained search space decomposition: the LLM, after extracting reaction parameter relationships from literature, induces a hierarchical parameter tree, leveraging physicochemical reasoning to focus Bayesian search in likely fruitful regions. Pseudo-label (synthetic) data generation—where LLMs, fine-tuned on chemical or physical property data, predict outcomes for untried conditions—further bootstraps the BO surrogate and aids in cold-start scenarios.

• Knowledge Summaries and Circuit Design:

LLM-USO (S et al., 4 Feb 2025) formalizes structured, interpretable knowledge representation for analog circuit sizing, enabling the transfer of optimization summaries across circuits sharing sub-structures, and integrating critiques of these summaries via higher-capacity review LLMs. This mechanism closely parallels the cognitive workflow of expert human designers.

• Self-augmenting Multi-task Loops:

In large-scale, multi-task settings (e.g., BOLT (Zeng et al., 11 Mar 2025)), LLMs act as memory-equipped “optimization assistants,” progressively distilled through trajectory feedback to generalize high-quality initializations to new tasks.

4. Empirical Performance and Benchmarking Evidence

Performance evaluations across diverse tasks consistently report improved sample efficiency, convergence, and generalization relative to classical BO:

• Early-stage Superiority and Sample Efficiency:

Zero-shot or few-shot LLM warmstarting yields better initial candidate quality and lower normalized regret in the first trials for hyperparameter optimization (Liu et al., 6 Feb 2024), circuit design (Chen et al., 7 Jun 2024), and chemistry (Han et al., 10 Sep 2025). In low data regimes, LLM-guided surrogate models outperform GPs and other standard surrogates in both prediction quality (NRMSE, log predictive density) and optimization regret.

• Empirical Validation in Real-world and Scientific Domains:

ChemBOMAS achieved a 96% yield in challenging pharmaceutical reaction optimization, in comparison to 15% from domain experts, and consistently accelerated convergence in both synthetic and laboratory settings (Han et al., 10 Sep 2025). BOLT and Reasoning BO demonstrated lower regret and faster convergence across high-dimensional synthetic functions and practical engineering targets (e.g., solar energy, protein/peptide optimization) (Zeng et al., 11 Mar 2025, Yang et al., 19 May 2025).

• Algorithmic Innovation via LLM-driven Design:

LLaMEA-BO employs evolutionary search over LLM-generated Python code templates, yielding novel BO algorithms that outperform state-of-the-art baselines on BBOB and Bayesmark suites, and generalize robustly to higher-dimensional spaces (Li et al., 27 May 2025).

5. Theoretical Guarantees and Interpretability

Recent developments formalize the interplay between LLM-driven and statistical components:

• Regret Guarantees:

Hybrid frameworks such as LLINBO rigorously quantify regret under various LLM-GP collaboration modes, proving that as the optimization progresses, cumulative regret remains sublinear and asymptotically optimal under well-defined schedules (Chang et al., 20 May 2025).

• Human-Interpretable Reasoning:

Systems like BORA (Cissé et al., 27 Jan 2025) and Reasoning BO (Yang et al., 19 May 2025) augment BO with human-centric, LLM-generated commentary, chain-of-thought notes, and real-time hypothesis tracking, fostering interpretability and transparency critical for high-stakes scientific or industrial applications.

6. Comparative Analysis, Limitations, and Future Directions

Comparative evaluations underscore both the strengths and current limitations of LLM-guided BO:

• Comparative Strengths:

LLM integration yields rapid context-aware search, meaningful domain prior incorporation, and robust behavior in low-data and multi-task scenarios. Modular design allows integration with established BO tooling and accelerates transfer of optimization knowledge across tasks.

• Limitations:

LLM-guided methods are sensitive to prompt design, in-context learning quality, and require stochastic sampling (for uncertainty) or hybrid Bayesian surrogates for optimal performance. In some settings, overexploitation or “hallucinations” may arise if LLM-driven sampling is insufficiently constrained. Cost (API, computation) and reproducibility remain active areas of concern, particularly for closed-source or high-capacity models (Mahammadli et al., 27 Oct 2024).

• Future Directions:

Ongoing research targets open-source LLM integration, principled uncertainty estimation, adaptation to complex continuous/discrete or multi-objective domains, and further automation—e.g., via evolutionary strategies (LLaMEA-BO)—of BO algorithm component synthesis. Extensions to self-driving laboratories, automated scientific discovery, resource allocation, and combinatorial design are noted as promising applications.

7. Representative Mathematical Formulations and Algorithmic Flows

Several mathematical expressions recur in LLM-guided BO:

Component	Key Formulation/Procedure	Reference
Surrogate Prediction	$p(s|h; D_n) \approx \frac{1}{K} \sum_k$ LLM($h^{nl}$, $D_n^{nl}$)	(Liu et al., 6 Feb 2024)
Acquisition w/ LLM score	$s' = s_{\min} - \alpha (s_{\max} - s_{\min})$	(Liu et al., 6 Feb 2024)
UCB (Hybrid Selection)	$$\alpha_{UCB}(x, F_{t-1}) = \mu_{t-1}(x) + \beta_t \sigma_{t-1}(x)$$	(Chang et al., 20 May 2025)
BOLT Fine-tuning Loss	$L = -\sum_{i=1}^{|x|} \log \pi(x_i | C, x_{<i})$	(Zeng et al., 11 Mar 2025)
Knowledge-guided UCB	$$UCB_i = \bar{R}_i + C_p \cdot \sqrt{\frac{\log(N_{parent})}{n_i}}$$	(Han et al., 10 Sep 2025)

These formal structures underpin acquisition, selection, feedback, and knowledge transfer steps within state-of-the-art LLM-guided BO pipelines.

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In summary, LLM-guided Bayesian Optimization constitutes a family of modular, hybrid methods leveraging LLMs to improve initialization, surrogate modeling, candidate proposal, and domain knowledge integration in the BO workflow. This integration demonstrably enhances optimization performance in sample-limited, high-dimensional, or knowledge-rich settings and is underpinned by both empirical gains and emerging theoretical analyses across a range of scientific and engineering domains (Liu et al., 6 Feb 2024, Kristiadi et al., 7 Feb 2024, Chen et al., 7 Jun 2024, Mahammadli et al., 27 Oct 2024, Zeng et al., 11 Mar 2025, Yang et al., 19 May 2025, Chang et al., 20 May 2025, Li et al., 27 May 2025, Han et al., 10 Sep 2025).