Extrinsic Bayesian Optimization (eBO)

• eBO is a framework that integrates expert knowledge and side information with Bayesian optimization to enhance sample efficiency in complex search spaces.
• It extends standard BO by incorporating methodologies like pairwise preference elicitation, abstract property modeling, and manifold embeddings to guide optimization.
• Empirical studies indicate that eBO can reduce evaluation budgets by 30-40% and achieve more robust convergence compared to traditional BO methods.

Extrinsic Bayesian Optimization (eBO) refers to a class of Bayesian optimization (BO) methods that efficiently leverage side information—commonly human-elicited preferences, expert knowledge, or domain-specific structural properties—beyond primary function evaluations to accelerate or enhance global black-box optimization. The defining feature of eBO is the integration of this extrinsic, often inexpensive information in a principled surrogate modeling and acquisition process, yielding marked gains in sample efficiency, robustness, and performance, especially for expensive-to-evaluate objectives and in complex search spaces such as manifolds.

1. Core Principles and Motivation

Standard Bayesian optimization targets the minimization (or maximization) of expensive black-box functions $f:\mathcal X\to\mathbb R$ by iteratively querying, modeling via a surrogate (typically a Gaussian process, GP), and guiding exploration through acquisition functions such as Expected Improvement (EI) or Upper Confidence Bound (UCB). Traditionally, BO "learns everything from scratch" by relying solely on observed $f(x)$ values.

In many practical scenarios, domain experts possess partial knowledge about the optimum, function shape, or high-level properties not directly observed in $f$, but accessible via inexpensive secondary queries. eBO extends BO by formalizing, modeling, and integrating such information—ranging from pairwise preference comparisons to qualitative abstract property judgments or geometric constraints on the search space—into the BO pipeline, thereby improving sample efficiency and broadening the applicability of BO to richer domains and tasks (Huang et al., 2022, V et al., 2024).

2. Methodologies for Expert and Extrinsic Knowledge Integration

2.1 Actively Elicited Pairwise Preferences

Rather than requesting absolute numerical estimates, eBO commonly queries domain experts in the form of “Is $x$ better than $x'$?” over input pairs $(x,x')\in\mathcal X\times\mathcal X$. This data is used to learn a latent expert utility or abstract property function $g:\mathcal X\to\mathbb R$, assumed to correlate with $f$ but potentially biased or on an arbitrary monotonic scale.

In "Bayesian Optimization Augmented with Actively Elicited Expert Knowledge" (Huang et al., 2022), the expert function $g$ is modeled using a Bayesian Siamese neural network. The likelihood is formulated via logistic preference modeling: $$P(y=+1\mid x,x',w_g)=\sigma\bigl(g_{w_g}(x)-g_{w_g}(x')\bigr)$$ where $f(x)$0 is the sigmoid function, and $f(x)$1 are network parameters endowed with a variational Bayesian treatment. The network is optimized to maximize the evidence lower bound (ELBO), capturing both fit to preferences and model uncertainty. Active learning strategies—such as maximizing mutual information (BALD)—select the most informative preference queries.

2.2 Preferential Modeling of Abstract Properties

"Enhanced Bayesian Optimization via Preferential Modeling of Abstract Properties" (V et al., 2024) generalizes eBO to collect and utilize preferential judgments regarding latent, abstract properties $f(x)$2, which may not be measurable by $f(x)$3. Each property is modeled via a rank-Gaussian process, where the posterior is inferred from expert-provided pairwise preferences using a Thurstone likelihood: $f(x)$4 with hyperparameters and posterior mean/variance optimized by maximizing the (approximate) marginal likelihood.

These inferred property predictions $f(x)$5 and associated uncertainties $f(x)$6 are used to augment the BO surrogate’s input vector: $f(x)$7. A spatially varying ARD kernel with the augmented dimensions attenuates unreliable properties when uncertainty is high.

2.3 Multi-task and Architectural Integration

A key technical advance in (Huang et al., 2022) is the multi-task architecture jointly modeling $f(x)$8 and $f(x)$9 via shared feature extractors and independent Bayesian linear heads. The joint loss is a convex combination of ELBOs for $f$0 and $f$1, with a decaying expert-weight parameter $f$2 that phases out expert impact as true $f$3 observations accumulate: $f$4 where $f$5 is a regularization term.

3. Acquisition Functions and Robustness to Expert Bias

A defining element of eBO is the construction of acquisition functions that harness both objective and expert-based model uncertainties. (Huang et al., 2022) proposes an additive hybrid: $f$6 where $f$7 is the standard BO acquisition (e.g., EI), $f$8 is the predictive standard deviation of the expert model, and $f$9 governs exploration in expert-uncertain regions.

(V et al., 2024) develops a two-arm model selection design: at each iteration, two surrogates are fit—standard GP (Arm $x$0) and augmented GP (Arm $x$1). The arm with higher predictive log-likelihood on past data determines which surrogate is used for Thompson sampling and optimization. This mechanism guards against misleading or erroneous expert feedback by reverting to standard BO when augmented surrogate predictions degrade, preserving regret guarantees.

4. Extrinsic Bayesian Optimization on Manifolds

(Fang et al., 2022) generalizes eBO to Riemannian manifolds, where $x$2 lives on a non-Euclidean, smooth domain $x$3. Intrinsic kernel construction is nontrivial due to lack of global coordinate charts and positive definiteness constraints. eBO remedies this via extrinsic Gaussian processes built on equivariant, injective embeddings $x$4 and standard Euclidean kernels on $x$5: $x$6 This method enables the use of canonical acquisition strategies (EI, PI, UCB) over $x$7, with projection, retraction, or exponential map optimization steps to ensure iterates remain on the manifold. Empirical studies on the sphere, Grassmannian, and SPD matrices demonstrate sample-efficient optimization, often outperforming bespoke derivative-free solvers and gradient descent in expensive settings.

5. Empirical Performance, Robustness, and Limitations

Evaluations across (Huang et al., 2022, V et al., 2024), and (Fang et al., 2022) demonstrate that eBO achieves rapid convergence in early and mid-phase optimization—often outperforming standard BO by factors of $x$8 to $x$9 in objective-evaluation budgets when expert accuracy is reliably above random. For instance, in synthetic and real-world cases (battery manufacturing, electrode design), eBO reduces simple regret and objective evaluations required for achieving high-performing designs by $x'$0 (V et al., 2024).

The framework is robust to expert bias: both empirical and algorithmic mechanisms ensure eBO performance reverts to that of standard BO if the extrinsic information is uninformative or misleading. This is achieved via decaying multitask loss weights (Huang et al., 2022), arm selection (V et al., 2024), and posterior variance-based kernel attenuation.

Major limitations include the cognitive load of repeated pairwise queries, cubic GP inference per abstract property (mitigated by inducing-point/sparse GP methods), and the dependence on suitable expert or property models. Extensions to multi-objective, categorical, or top-$x'$1 preference structures and automated property learning are suggested. Theoretical regret analysis for eBO matches that of standard BO in the worst case, with the potential for improvements as the effective eluder dimension drops under informative extrinsic signals (V et al., 2024).

6. Summary Table: eBO Variants and Methodological Distinctions

eBO Framework	Type of Extrinsic Knowledge	Modeling Approach
(Huang et al., 2022)	Human pairwise preferences ($x'$2)	Bayesian Siamese neural net
(V et al., 2024)	Preferences on abstract properties ($x'$3)	Rank-Gaussian process surrogate
(Fang et al., 2022)	Geometric structure (manifold)	Extrinsic GP via embedding

These methodologies differ in the nature and modeling of extrinsic information: from direct pairwise preference over the search domain, to abstract high-level latent functions, to geometric and manifold constraints. Each exploits domain structure for improved sample efficiency and global optimization performance.

7. Theoretical Analysis and Future Directions

Formal regret guarantees for eBO closely follow those derived for standard GP-based BO, with modifications accounting for the augmented dimensionality and kernel structure. Covering number and eluder dimension arguments justify sublinear regret for UCB and Thompson sampling variants. Empirical evidence supports the claim that augmented surrogates, when fed reliable side information, decrease uncertainty faster and concentrate search around optima more aggressively than uninformed surrogates.

Anticipated future directions include more scalable preference modeling for high query volumes, automatic abstraction and learning of extrinsic properties from historical or simulation data, richer elicitation (e.g., $x'$4-way, natural language), and formal convergence guarantees that directly incorporate side information uncertainty and multi-fidelity sources.

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Key references:

• "Bayesian Optimization Augmented with Actively Elicited Expert Knowledge" (Huang et al., 2022)
• "Extrinsic Bayesian Optimizations on Manifolds" (Fang et al., 2022)
• "Enhanced Bayesian Optimization via Preferential Modeling of Abstract Properties" (V et al., 2024)