Cost-Aware Bayesian Optimization

• Cost-aware Bayesian optimization is a methodology that integrates heterogeneous evaluation costs into Bayesian decision-making to efficiently optimize black-box functions under resource constraints.
• It employs diverse acquisition functions such as EI per unit cost and dynamic penalization to balance utility and cost, adapting to variable subsets, multi-fidelity sources, and switching scenarios.
• Empirical applications in hyperparameter tuning, experimental design, and materials discovery demonstrate significant cost savings and improved performance in resource-constrained environments.

Cost-aware Bayesian optimization (CA-BO) is a broad class of Bayesian optimization (BO) methodologies that explicitly incorporate heterogeneity in evaluation cost into decision-making and policy design. In contrast to classical BO, which counts sample queries and assumes uniform cost, CA-BO seeks to maximize a utility function or minimize a black-box objective under constraints such as total spend, wall-clock time, or other resource budgets. Diverse research efforts in CA-BO address practical scenarios where costs may be associated with subsets of variables, levels of fidelity, switching between system modules, or physical experiment constraints. CA-BO methods subsume a collection of acquisition strategies, model architectures, regret guarantees, and stopping policies, and are implemented in a wide variety of domains, including hyperparameter tuning, materials discovery, and experimental design.

1. Formal Problem Settings and Cost Structures

The prototypical CA-BO problem is formulated as follows: given a search domain $X \subset \mathbb{R}^d$, an unknown objective $f: X \to \mathbb{R}$, and a cost function $c: X \to \mathbb{R}_+$ (possibly stochastic, dynamic, or history-dependent), the goal is to find $x^* = \arg\max_{x \in X} f(x)$ (or minimize $f$) while respecting a cost budget constraint $\sum_{t=1}^T c(x_t) \leq B$. Several generalizations arise:

• Variable subset costs: Only a subset of variables is controlled at each round; setting certain coordinates incurs cost, while the remainder are cheap or random. The Bayesian Optimization with Cost-Varying Subsets (BOCVS) framework defines costs over subsets $I = \{ I_1, \ldots, I_m \}$ and optimizes over both subset selection and query value (Hoang et al., 2024).
• Multi-fidelity costs: Parallel sources (e.g., low- and high-fidelity simulators) with different accuracy/cost trade-offs are available. The objective is to allocate queries across sources to maximize information gain or minimize regret under the budget (Foumani et al., 2022, Foumani et al., 3 Mar 2025, Tang et al., 2024).
• Switching costs and modularity: In sequential experimental settings or modular pipelines, changing certain parameters or modules is disproportionately expensive (e.g., changing a hardware setup) (Lin et al., 2020, Pricopie et al., 2024).
• History-dependent and dynamic costs: Costs may depend on the current state or prior actions (e.g., travel or reconfiguration costs), with generalizations to Markovian or even combinatorial update settings (Truong et al., 10 Jan 2026, Chawla et al., 21 Nov 2025).

Cost-aware acquisition and sampling policies must, therefore, track heterogeneous, often stochastic, and sometimes history-dependent cost landscapes.

2. Methodologies: Acquisition Functions and Stopping Rules

The design of cost-aware acquisition functions (CA-AFs) is a central theme in CA-BO. The prevailing approaches include:

• EI per unit cost (EIpu): The classical adaptation divides Expected Improvement by cost, i.e., $\alpha_{\mathrm{EIpu}}(x) = \mathrm{EI}(x)/c(x)$ (Lee et al., 2020). While computationally simple, it can over-penalize high-cost regions, especially when the global optimum is expensive.
• Cost cooling / dynamic penalization: Methods such as CArBO use $$\alpha(x) = \mathrm{EI}(x) / [\hat{c}(x)]^{\lambda}$$, with $\lambda$ annealed from 1 to 0 as the budget is expended, smoothly transitioning from cost prioritization to value prioritization (Lee et al., 2020).
• Pareto-efficient and multi-objective acquisition: Pareto-efficient acquisition frameworks, including Contextual EI (CEI) and explicit EI/c$^\alpha$ scalarizations, directly trace the cost-utility Pareto front, providing a set-valued or parametrically adaptive trade-off mechanism (Guinet et al., 2020).
• Pandora’s Box Gittins Index (PBGI): The solution to a cost-aware one-step optimal stopping problem (Pandora’s Box) is cast as solving $\mathrm{EI}_f(x;g) = c(x)$ for each x, and selecting the maximizer of $g$ (i.e., the "fair price" threshold at which opening x is marginally worth the cost) (Xie et al., 2024). This acquisition is Bayes-optimal in the independent-arm limit and robust under weak model assumptions.
• Hybrid, LLM-evolved, and information-theoretic acquisition functions: Recent frameworks such as EvolCAF employ LLMs to evolve CA-AFs that combine history-heavy uncertainty quantification, budget/cost scaling, and explicit diversity bonuses (Yao et al., 2024). Information-based CA-AFs such as cost-aware gradient entropy search (CAGES) maximize the expected reduction in parameter entropy per unit cost (Tang et al., 2024).
• Stopping rules: Cost-aware stopping is achieved via budget-adaptive thresholds, such as halting when no point offers expected improvement exceeding its cost (as in PBGI/LogEIPC) (Xie et al., 16 Jul 2025).

3. Theoretical Guarantees and Regret Analysis

CA-BO methods are increasingly supported by precise regret and convergence analyses, parameterizing both quality and cumulative cost:

• Dual regret bounds: In the cost-varying variable subset regime (Hoang et al., 2024), two regret terms are defined: quality regret $R_T^f$ and cost regret $R_T^c$. Sublinear bounds in both are proved, with $R_T^f, R_T^c = o(T)$ achievable for properly chosen phase lengths and kernels satisfying bounded information gain.
• Optimality of index strategies: Pandora’s Box Gittins Index policies are Bayes-optimal in independent-arm models, and (with adjustable Lagrange multipliers) are also optimal under expected budget constraints (Xie et al., 2024). Theoretical cost bounds guarantee that expected cumulative cost does not exceed the sum of initial cost and a problem-dependent constant (Xie et al., 16 Jul 2025).
• Pareto-efficiency: Scalarizations and contextual acquisition rules are proven to yield selections along the cost-utility Pareto front, with convergence mirroring the underlying BO strategy (Guinet et al., 2020). Regret analyses for movement-regularized settings (see LaMBO) prove sublinear growth with the number of steps (Lin et al., 2020).
• Exploration–exploitation balance: Multi-fidelity cost-aware acquisitions ensure that high-fidelity exploitation cannot be indefinitely starved by cheap exploratory queries: utility ratios involving Gaussian PDF/CDF asymptotically force selection of high-fidelity sources as model confidence increases (Foumani et al., 2022).
• No-regret switching: In modular/switching-cost settings, lazy modular BO architectures provably achieve vanishing movement-regularized regret for reasonable cost scaling (Lin et al., 2020).

Formal regret guarantees often rely on bounded RKHS norm, kernel information gain, and cost function regularity.

4. Model and Algorithmic Innovations

Progress in CA-BO is tightly linked with developments in surrogate modeling and algorithmic architecture:

• Surrogate costs and modeling: Both GP and low-variance linear surrogates for log-cost or cost are prevalent (Lee et al., 2020, Guinet et al., 2020). Empirically, low-variance models yield superior cost prediction in limited data regimes, reducing cumulative error and improving acquisition efficiency.
• Phase-based approaches: Explore-then-commit methodology cycles through all variable control subsets, filters out low-value/high-cost subsets, and then aggressively exploits the best candidate, updating cost means with lower confidence bounds (Hoang et al., 2024).
• Modular and tree-based bandit embeddings: Tree metric structures (MSETs) encode switching costs in modular pipelines and are coupled with lazy, multiplicative-weights-based sampling to minimize both function and movement regret (Lin et al., 2020).
• Batch and parallelization: Heterotopic and isotopic querying strategies in batch experimental workflows allocate queries adaptively based on cost and surrogate model uncertainty, leveraging deep GP surrogates for high-dimensional applications (Alvi et al., 17 Sep 2025).
• Nonmyopic long-horizon planning: Neural amortization of multi-step lookahead, as in LookaHES, brings tractable nonmyopic acquisition planning to dynamic cost settings, with policies learned over combinatorial histories (Truong et al., 10 Jan 2026).
• LLM-driven AF discovery: The EvolCAF framework automates the design of CA-AFs via evolutionary code-editing with LLMs, discovering hybrid formulas that incorporate budget, history, and coverage components and outperform expert baselines across diverse tasks (Yao et al., 2024).

5. Representative Applications and Empirical Results

Empirical validation of CA-BO methodologies spans domains and complexity levels:

• Hyperparameter optimization: CArBO, PBGI, and Pareto-efficient CA-BO methods deliver 20–50% cost savings in learning tasks, with minimal loss in test accuracy versus cost-agnostic BO (Lee et al., 2020, Guinet et al., 2020, Xie et al., 2024, Xie et al., 16 Jul 2025).
• Experimental design and prototyping: Real-world device prototyping achieves the same utility with only 55–70% of the cost compared to standard BO, and cost-aware methods adapt sampling in real time as component (hardware/software) costs change (Langerak et al., 2 Feb 2026).
• Materials discovery and automated experimentation: Heteroskedastic GPs with embedded cost models and hierarchical scheduling reach mapping precision of grid-based methods at 1/30th the time, and deep GP batch CA-BO unlocks efficient parallelism in high-entropy alloy discovery (Chawla et al., 21 Nov 2025, Alvi et al., 17 Sep 2025).
• Multi-fidelity settings: Multi-fidelity cost-aware BO reduces total cost by 30–70% relative to single-fidelity baselines in both analytic (e.g., Borehole, Rosenbrock) and real-world (alloys, perovskites) settings, while automatically excluding unreliable or biased low-fidelity data through latent map distance metrics (Foumani et al., 2022, Foumani et al., 3 Mar 2025, Tang et al., 2024).
• Sequencing and pipeline optimization: Modular and switching-aware BO architectures outperform per-unit-cost heuristics and prior cost-agnostic BO by up to 4× in sequential neuroimaging pipelines or simulated process control (Lin et al., 2020, Pricopie et al., 2024).

Careful matching of cost model and experimental regime is crucial for empirical performance at scale.

6. Extensions, Limitations, and Practical Guidelines

While CA-BO is advancing rapidly, several limitations and guidelines are recognized:

• Surrogate model sensitivity: Model miscalibration and limited-data regimes can degrade cost prediction and thus acquisition effectiveness; hybrid surrogates or transfer learning can mitigate early-stage variance (Lee et al., 2020, Guinet et al., 2020).
• Choice and adaptation of trade-off parameters: Static penalization (scalarization exponents, fixed λ) can suboptimally balance cost and utility; contextual or budget-adaptive schedules tend to perform better in real workloads (Guinet et al., 2020, Lee et al., 2020, Yao et al., 2024).
• Nonstationary and dynamic costs: Several domains require real-time or adaptive cost tracking and model updating; CA-BO frameworks remain robust when history- or state-dependent cost surrogates are incorporated (Truong et al., 10 Jan 2026, Chawla et al., 21 Nov 2025).
• Complex action spaces and constraints: Large combinatorial, modular, or constrained variable spaces challenge naive batch or scalarization methods; specialized topology- or tree-embedding strategies are necessary (Lin et al., 2020, Pricopie et al., 2024).
• Safety and bias in multi-fidelity sources: Latent space distance metrics for source correlation (e.g., LMGP) are crucial to automatically exclude uninformative or misleading low-fidelity data, especially in materials or engineering design (Foumani et al., 2022, Foumani et al., 3 Mar 2025, Tang et al., 2024).

Recommended practice is to exploit simple, low-variance cost surrogates where possible, apply contextual acquisition functions that adapt penalization throughout optimization, and use hybrid and batch querying strategies to handle high-dimensional and resource-constrained regimes.

7. Prospective Directions

Emerging research points to several key directions:

• Automated AF generation: LLM-driven evolutionary and code-based design of cost-aware acquisition functions will reduce reliance on human intuition for new problem settings (Yao et al., 2024).
• Nonmyopic, long-horizon planning: Integrated neural or amortized multi-step lookahead offers a scalable path for CA-BO in history-dependent and dynamically constrained environments (Truong et al., 10 Jan 2026).
• Multi-objective and multi-budget optimization: Extension to true multi-objective and multi-budget (e.g., time, money, labor) models expands applicability to realistic project planning (Guinet et al., 2020, Langerak et al., 2 Feb 2026).
• Constraints and reliability: Systematic treatment of constraints (box, black-box, or modular) within CA-BO, and robust treatment of surrogate model uncertainty, remain as active research areas (Foumani et al., 3 Mar 2025).
• Empirical validation in new domains: Continued validation and benchmarking of CA-BO in next-generation scientific workflows, autonomous laboratories, and physical experiment pipelines is anticipated to solidify practical best practices.

The literature demonstrates that cost-awareness, when systematically embedded into BO methodology, offers substantial performance and resource gains across domains, and continues to motivate rich theoretical and practical innovation.