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On Information Gain and Regret Bounds in Gaussian Process Bandits
Sattar Vakili, Kia Khezeli, Victor Picheny
TL;DR
The paper addresses the challenge of bounding maximal information gain γ_T, which underlies regret guarantees for GP-based Bayesian optimization. It derives eigenvalue-decay-based bounds using finite-dimensional projections, improving information-gain and regret bounds and closing Matérn frequentist gaps up to logarithmic factors.
Problem
Existing regret bounds depend on maximal information gain γ_T, whose upper bounds remain challenging and leave a drastic gap for Matérn kernels.
Method
The paper uses Mercer’s eigenvalue-eigenfeature decomposition and finite-dimensional projections to derive general γ_T bounds from kernel eigendecay.
Results
The resulting bounds improve known γ_T and regret bounds across settings and establish tight frequentist Matérn regret bounds up to logarithmic factors.
Takeaways & Limitations
Because the γ_T bounds are algorithm-independent, they apply to Bayesian and frequentist regret analyses whenever guarantees are expressed in terms of γ_T.
Takeaways & Limitations
The analysis assumes a Mercer kernel with uniformly bounded kernel values and uniformly bounded eigenfeatures.
Abstract
from arXiv · showhide
Consider the sequential optimization of an expensive to evaluate and possibly non-convex objective function $f$ from noisy feedback, that can be considered as a continuum-armed bandit problem. Upper bounds on the regret performance of several learning algorithms (GP-UCB, GP-TS, and their variants) are known under both a Bayesian (when $f$ is a sample from a Gaussian process (GP)) and a frequentist (when $f$ lives in a reproducing kernel Hilbert space) setting. The regret bounds often rely on the maximal information gain $γ_T$ between $T$ observations and the underlying GP (surrogate) model. We provide general bounds on $γ_T$ based on the decay rate of the eigenvalues of the GP kernel, whose specialisation for commonly used kernels, improves the existing bounds on $γ_T$, and subsequently the regret bounds relying on $γ_T$ under numerous settings. For the Matérn family of kernels, where the lower bounds on $γ_T$, and regret under the frequentist setting, are known, our results close a huge polynomial in $T$ gap between the upper and lower bounds (up to logarithmic in $T$ factors).
1 Introduction
The paper develops eigenvalue-decay-based bounds for maximal information gain γT, improving regret guarantees for GP bandit optimization and narrowing known Matérn gaps to logarithmic factors.
- Motivation: Existing GP-UCB, GP-TS, and related regret guarantees depend on maximal information gain γT, whose complete bounding remains challenging.γT is treated identically in Bayesian and frequentist regret analyses, while their confidence bounds differ.
- Motivation: Known Matérn bounds can leave polynomial gaps as large as O(T) for γT and O(T) for regret under particular ν and d configurations.The paper is motivated by reducing these gaps to logarithmic factors.
- Approach: The paper represents GP kernels through Mercer eigenvalue-eigenfeature decompositions and projects their infinite-dimensional RKHS representations onto the first D features.The projection uses the D largest eigenvalues and enables information-gain bounds in terms of D and kernel spectral properties.
- Approach: For polynomial eigendecay λm = O(m^-βp), βp > 1, and exponential eigendecay λm = O(exp(-m^βe)), βe > 0, the paper derives general explicit upper bounds on γT and regret.These bounds apply when the stated eigenvalue-decay conditions hold, rather than only for individually specified kernels.
- Results: For Matérn kernels, the resulting γT and frequentist regret bounds are tight up to logarithmic factors, closing the corresponding lower-bound gap.The paper also applies the results to the Squared Exponential kernel and reports improved bounds across multiple settings.
- Implications: The improved γT bounds directly improve regret guarantees in contextual, safe, and multi-fidelity settings, and related effective-dimension-dependent algorithmic bounds.The effective dimension satisfies D̃T = O(γT), linking information-gain improvements to computational properties of GP-based methods.
2 Problem Formulation and Preliminaries
The paper formulates noisy sequential optimization with GP surrogate models and analyzes Bayesian optimization through information gain and confidence bounds. GP-UCB and GP-TS are considered under Bayesian and frequentist assumptions, with regret bounds expressed using γ_T.
- 2.1 The Sequential Optimization Problem: A learning algorithm sequentially selects points in a compact domain and observes noisy rewards, while regret measures cumulative loss relative to the global maximum.The observation policy maps past histories to new points, and the objective is to minimize cumulative regret over horizon T.
- 2.2 Gaussian Processes: Gaussian processes provide non-parametric surrogate models whose posterior mean and covariance are updated by conditioning on observed histories.The GP is specified by a mean function and positive definite kernel, with posterior quantities defined from covariance matrices.
- 2.2 Gaussian Processes: Matérn and squared exponential kernels are prominent Bayesian optimization choices, with Matérn smoothness controlled by ν and the SE kernel obtained as ν →∞.Variation in ν produces a family of Matérn kernels with different smoothness properties.
- 2.3 Bayesian Optimization Algorithms (GP-UCB and GP-TS): GP-UCB selects points maximizing an optimistic upper-confidence score based on posterior mean, posterior standard deviation, and a scaling parameter α_t.The scaling parameters increase with time to balance exploitation and exploration.
- 2.3 Bayesian Optimization Algorithms (GP-UCB and GP-TS): GP-TS instead samples a function from a scaled posterior GP and selects the point maximizing that sample.Its sampled GP uses the posterior mean and kernel, with α_t controlling the posterior sampling scale.
- 2.4 Regularity Assumptions: The Bayesian setting assumes f is sampled from a GP with Gaussian noise, whereas the frequentist setting assumes f lies in the kernel RKHS with bounded norm and sub-Gaussian noise.These two settings use different regularity assumptions for the objective and observation noise.
- 2.5 The Information Gain and The Upper Bounds on Regret: Regret analysis combines a maximal information-gain bound with a confidence bound, and the information-gain component is shared across Bayesian and frequentist settings.The confidence widths differ by setting because the underlying assumptions differ.
- 2.5 The Information Gain and The Upper Bounds on Regret: Novel bounds on γ_T are derived from kernel eigendecay and immediately yield improved regret bounds for multiple Bayesian optimization settings.The approach uses Mercer’s eigenvalue-eigenfeature representation and finite-dimensional RKHS projections to address infinite-dimensional kernels.
3 Upper Bounds on the Information Gain
The paper bounds maximal information gain by projecting the GP onto its leading eigenfeatures and controlling the residual through the eigenvalue tail. The resulting spectral bound applies broadly and yields tight, improved bounds for common kernels.
- Finite-dimensional projection: The approach projects the GP onto a D-dimensional RKHS formed by the first D eigenfeatures, separating finite-dimensional and orthogonal components.The projected kernel uses the D largest eigenvalues, while the orthogonal kernel captures the residual.
- Residual control: The residual tail quantity δD controls the orthogonal kernel, becoming arbitrarily small when eigenvalues decay sufficiently quickly and D is large enough.This gives kO(x, x′) ≤ δD for all x, x′.
- Information-gain bound: Theorem 3 provides an upper bound on γT based on the kernel spectrum through the projection dimension and eigenvalue-tail residual.The bound is derived by decomposing the covariance log determinant into projected and residual terms.
- Kernel specializations: The spectral bound applies to kernels through their eigendecay and gives tight γT bounds up to log(T) factors for Matérn and squared exponential kernels.The polynomial and exponential eigendecay cases provide corresponding general bounds, with tightness supported by known lower bounds.
- Effective dimension: Choosing D so that D log(T) and TδD have the same order yields γT = O(D log(T)), making the kernel behave like a finite-dimensional kernel up to logarithmic factors.This motivates calling D the effective dimension of the sequential optimization problem.
4 The Improved Regret Bounds for Bayesian Optimization Algorithms
The improved information-gain bounds translate into improved regret guarantees for Bayesian optimization algorithms across Bayesian and frequentist settings. For Matérn and squared exponential kernels, the resulting bounds are stated as tight or improved relative to prior results.
- Algorithmic implications: Applying the new γT bounds improves regret guarantees for GP-UCB and GP-TS under Bayesian and frequentist settings.The bounds also apply to SupKernelUCB and other settings whose regret depends on γT.
- GP-UCB and GP-TS: For GP-UCB and GP-TS, polynomial and exponential eigendecays yield explicit regret bounds, with Matérn and SE kernels as special cases.The paper reports these bounds for both Bayesian optimization algorithms under the considered settings.
- SupKernelUCB: For SupKernelUCB, the new information-gain bounds produce improved regret rates under polynomial and exponential eigendecays.The Matérn and SE specializations reduce the gap with known lower bounds.
5 Conclusion
The paper presents a general eigendecay-based method for bounding information gain and uses it to improve regret guarantees for common kernels and multiple Bayesian optimization settings. It establishes tight Matérn regret bounds under the frequentist setting up to logarithmic factors.
- Contribution: The paper introduces a general approach to bounding information gain using the kernel’s eigendecay.The approach gives explicit bounds in T for γT and regret.
- Implications: The resulting bounds apply directly to Matérn and SE kernels and improve state-of-the-art regret bounds across numerous settings.The improvements arise when existing regret guarantees are expressed in terms of γT.
- Matérn result: For Matérn kernels under the frequentist setting, the paper establishes the first tight regret bounds up to log(T) factors.This also shows that the corresponding information-gain bound is tight up to logarithmic factors.
A (Proof of Theorem 3)
The proof decomposes the information-gain log determinant into projected and orthogonal components, then bounds them using finite-dimensional feature representations, trace inequalities, and the residual kernel bound.
- Kernel decomposition: The covariance kernel is decomposed as k = kP + kO, which separates the finite-dimensional projection from the orthogonal residual.The same decomposition applies to the corresponding covariance matrices.
- Projected term: The projected log determinant is bounded using the D-dimensional feature matrix and the Gram matrix in the projected feature space.The feature matrix stacks projected eigenfeatures evaluated at the observation points.
- Matrix inequalities: The log determinant of a positive definite matrix is bounded in terms of its trace, supporting the finite-dimensional and residual estimates.The proof invokes the arithmetic-geometric mean inequality and a trace-based determinant inequality.
- Residual term: The residual log-determinant term is controlled using the pointwise bound kO(x, x′) ≤ δD and positive-definite matrix trace inequalities.The largest eigenvalue bound and log(1 + z) ≤ z are used in the residual estimate.
- Conclusion of proof: Combining the projected and residual bounds yields the stated upper bound on information gain.The final assembly uses the preceding decomposed terms and estimates.
B (Proof of Corollary 1)
The proof derives bounds on δD and γT from polynomial and exponential eigendecay conditions, treating separate cases for the exponential rate parameter.
- The polynomial eigendecay condition yields a bound on δD from the decay rate of λm.
- The resulting argument gives a growth-rate bound on γT through Theorem 3.
- The exponential eigendecay condition is analyzed separately for βe = 1 and βe ≠ 1.
- Intermediate equalities and inequalities follow from a change of parameter and a standard derivative-based minimization argument.
- For βe = 1, the proof selects the relevant quantity using reasoning analogous to the polynomial eigendecay case.