Graph GP-UCB: Manifold Bayesian Optimization

• GGP-UCB is a Bayesian optimization framework that leverages graph-based Gaussian processes to model expensive black-box functions on unknown compact manifolds.
• It employs spectral kernel approximations and a geometry-corrected UCB acquisition function to efficiently navigate the manifold structure while ensuring query efficiency.
• Empirical evaluations on synthetic and real-world benchmarks demonstrate vanishing regret with fewer queries compared to standard Euclidean GP-UCB methods.

Graph Gaussian Process Upper Confidence Bound (GGP-UCB) is a Bayesian optimization framework designed for maximizing expensive black-box functions defined on unknown compact manifolds. GGP-UCB operates under the minimal informational assumption that only a point-cloud sample from the manifold is available. The method constructs a surrogate Gaussian process model using a graph built over these samples and utilizes a UCB acquisition criterion that corrects for geometric misspecification, offering both theoretical guarantees and empirical robustness across synthetic and inverse-problem benchmarks (Kim et al., 2022).

1. Problem Setting and Motivation

GGP-UCB addresses the scenario of optimizing a function $f$ defined on an $m$-dimensional compact manifold $M \subset \mathbb{R}^d$ when only a finite set $M_N = \{x_1, \ldots, x_N\} \subset M$ is accessible—the full analytic structure of $M$ is unknown. Evaluations of $f$ are costly, motivating the need for query-efficient, geometry-adaptive Bayesian optimization. The framework leverages manifold learning principles to construct a statistical surrogate for $f$ that respects intrinsic geometric constraints, thus ameliorating the limitations of standard Euclidean Gaussian process (GP) surrogates that ignore manifold structure.

2. Graph Construction and Spectral Kernel Approximation

The surrogate model is defined on a weighted graph constructed over $M_N$:

• The weight matrix $W \in \mathbb{R}^{N \times N}$ is defined by

$$W_{ij} = C_m h_N^{-(m+2)} \mathbf{1}_{\|x_i - x_j\| < h_N}$$

where $m$0 normalizes by the unit $m$1-ball's volume, and $m$2 is a connectivity radius.

• The (unnormalized) graph Laplacian is $m$3, with $m$4 the diagonal degree matrix.
• Spectral decomposition yields eigenpairs $m$5, $m$6.

Graph-based approximations to manifold kernels are formed via spectral filtering:

• Graph Matérn kernel (with smoothness $m$7 and inverse length-scale $m$8):

$m$9

• Graph squared-exponential (SE or heat kernel) (with length-scale $M \subset \mathbb{R}^d$0):

$M \subset \mathbb{R}^d$1

Truncation parameter $M \subset \mathbb{R}^d$2 controls the spectral resolution; appropriate scaling of $M \subset \mathbb{R}^d$3 and $M \subset \mathbb{R}^d$4 guarantees uniform convergence to the true manifold covariance functions as $M \subset \mathbb{R}^d$5.

3. Gaussian Process Surrogate and Bayesian Inference

A zero-mean GP prior is placed on $M \subset \mathbb{R}^d$6 with covariance $M \subset \mathbb{R}^d$7. After $M \subset \mathbb{R}^d$8 noisy observations $M \subset \mathbb{R}^d$9 at queries $M_N = \{x_1, \ldots, x_N\} \subset M$0, the GP posterior at $M_N = \{x_1, \ldots, x_N\} \subset M$1 is:

• Posterior mean:

$M_N = \{x_1, \ldots, x_N\} \subset M$2

• Posterior variance:

$M_N = \{x_1, \ldots, x_N\} \subset M$3

where $M_N = \{x_1, \ldots, x_N\} \subset M$4 is the covariance submatrix for the chosen queries.

4. Upper Confidence Bound (UCB) Acquisition and Misspecification Correction

GGP-UCB selects the next query $M_N = \{x_1, \ldots, x_N\} \subset M$5 by maximizing the acquisition function

$M_N = \{x_1, \ldots, x_N\} \subset M$6

over $M_N = \{x_1, \ldots, x_N\} \subset M$7. The UCB parameter $M_N = \{x_1, \ldots, x_N\} \subset M$8 is calibrated to ensure high-probability coverage of the true function under surrogate misspecification. For a graph GP prior approximating the manifold GP with sup-norm error $M_N = \{x_1, \ldots, x_N\} \subset M$9,

$M$0

guarantees with probability at least $M$1 that the UCB envelope covers both the surrogate and true values.

5. Algorithmic Workflow and Computational Costs

The GGP-UCB procedure iterates as follows:

1. Input: point cloud $M$2, graph parameters ($M$3, $M$4, kernel type, smoothness/length-scale), noise $M$5, confidence $M$6, query budget $M$7.
2. Initialization: $M$8, $M$9.
3. For $f$0,
   • Compute $f$1 as above.
   • Choose $f$2.
   • Query $f$3, $f$4.
   • Update posterior.
4. Output $f$5 (alternatively, report the query with maximal $f$6 for small $f$7).

Computational steps involve:

• Building $f$8: $f$9 (sparsifiable).
• Graph Laplacian eigendecomposition: $f$0, or $f$1 for partial spectra.
• GP update (naive): inverting a $f$2 matrix per step, for total $f$3; low-rank GP acceleration is feasible for small $f$4.

6. Theoretical Guarantees and Key Assumptions

Under Assumption 2.1 (manifold is a smooth compact boundaryless $f$5-manifold, $f$6 is an i.i.d. uniform sample), the following are established:

• The sup-norm surrogate error is $f$7 where $f$8 is determined by $f$9, kernel smoothness; for SE kernels, $M_N$0, $M_N$1; for Matérn kernels with $M_N$2 similar rates hold.
• The main regret bound (Theorem 3.1): with probability at least $M_N$3,

$M_N$4

• For $M_N$5 and slowly growing $M_N$6 (e.g., $M_N$7), as $M_N$8 the simple regret $M_N$9.

Key assumptions include:

• The true $W \in \mathbb{R}^{N \times N}$0 is a manifold GP restricted to $W \in \mathbb{R}^{N \times N}$1, with graph GP surrogate incurring sup-norm error $W \in \mathbb{R}^{N \times N}$2.
• Spectral convergence and associated parameter scaling for $W \in \mathbb{R}^{N \times N}$3 and $W \in \mathbb{R}^{N \times N}$4.
• $W \in \mathbb{R}^{N \times N}$5 is properly corrected for surrogate error.

7. Empirical Evaluation and Practical Performance

GGP-UCB has been benchmarked across synthetic and applied settings:

• Unit circle ($W \in \mathbb{R}^{N \times N}$6): GGP-UCB was compared against “oracle” manifold-GP-UCB and a version using marginal-likelihood-estimated (ML) kernel parameters. All variants locate the maximizer using $W \in \mathbb{R}^{N \times N}$7 queries, with GGP-UCB closely matching oracle performance even when $W \in \mathbb{R}^{N \times N}$8 is not observed from a GP.
• Artificial 2D surface (“cow”): On point clouds in $W \in \mathbb{R}^{N \times N}$9 ($$W_{ij} = C_m h_N^{-(m+2)} \mathbf{1}_{\|x_i - x_j\| < h_N}$$0), GGP-UCB achieves lower simple regret than standard Euclidean GP-UCB, reflecting sensitivity to underlying manifold geometry.
• Heat-source inverse problem on the sphere ($$W_{ij} = C_m h_N^{-(m+2)} \mathbf{1}_{\|x_i - x_j\| < h_N}$$1): Given noisy observations of a diffused heat signal over the sphere, GGP-UCB rapidly recovers the source location with high accuracy and few queries, outperforming Euclidean baselines.

These illustrations confirm that GGP-UCB consistently attains vanishing regret with a number of queries far smaller than the sample size $$W_{ij} = C_m h_N^{-(m+2)} \mathbf{1}_{\|x_i - x_j\| < h_N}$$2, robustly handling both exact and approximate manifold function settings (Kim et al., 2022).