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Gaussian Process Optimization in the Bandit Setting: No Regret and Experimental Design

Niranjan Srinivas, Andreas Krause, Sham M. Kakade, Matthias Seeger

arXiv:0912.3995v4cs.LG

TL;DR

The paper addresses how to optimize expensive, noisy functions in a nonparametric GP bandit setting, where prior regret guarantees were limited. It analyzes GP-UCB through information gain and derives sublinear regret bounds for common kernels, supported by favorable comparisons on sensor data.

  • Problem

    The paper asks how to obtain regret bounds and convergence rates for optimizing expensive noisy functions when the objective is modeled by a GP or has low RKHS norm.

  • Method

    The paper analyzes GP-UCB and bounds its cumulative regret through maximal information gain, which is further controlled using kernel spectra and submodularity.

  • Results

    The paper establishes the first sublinear regret bounds for GP optimization with commonly used kernels, including Squared Exponential and Matérn kernels.

  • Takeaways & Limitations

    The results connect GP optimization with experimental design and provide convergence guarantees for GP-UCB in the covered kernel settings.

  • Takeaways & Limitations

    The regret theorem assumes a smooth kernel; it excludes highly erratic sample paths such as those from the Ornstein–Uhlenbeck kernel, for which the authors conjecture the result does not hold.

Abstract

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Many applications require optimizing an unknown, noisy function that is expensive to evaluate. We formalize this task as a multi-armed bandit problem, where the payoff function is either sampled from a Gaussian process (GP) or has low RKHS norm. We resolve the important open problem of deriving regret bounds for this setting, which imply novel convergence rates for GP optimization. We analyze GP-UCB, an intuitive upper-confidence based algorithm, and bound its cumulative regret in terms of maximal information gain, establishing a novel connection between GP optimization and experimental design. Moreover, by bounding the latter in terms of operator spectra, we obtain explicit sublinear regret bounds for many commonly used covariance functions. In some important cases, our bounds have surprisingly weak dependence on the dimensionality. In our experiments on real sensor data, GP-UCB compares favorably with other heuristical GP optimization approaches.

1. Introduction

The paper studies expensive noisy-function optimization as a GP bandit problem and develops regret guarantees for GP-UCB. Its analysis connects cumulative regret to information gain, yielding sublinear bounds for common kernels and favorable sensor-data performance.

  • Motivation: The paper models optimization as a multi-armed bandit problem balancing exploration and exploitation when function evaluations are expensive.The unknown function is estimated from noisy samples while optimization occurs over a potentially high-dimensional input space.
  • Problem setting: GP-UCB is analyzed when the objective is sampled from a known GP or has low RKHS norm.These assumptions provide alternative ways to characterize the function’s complexity through a kernel.
  • Core analysis: Cumulative regret is bounded using information gain, establishing a technical connection between bandit optimization and experimental design.The information-gain quantity measures how quickly sampling can reveal the function.
  • Theoretical results: Sublinear regret bounds are obtained for popular kernel classes, with rates depending finely on kernel choice and parameters.The analysis uses information-gain bounds for common covariance functions, including Squared Exponential and Matérn kernels.
  • Theoretical results: The bounds provide convergence rates for GP optimization and can have weak dependence on dimensionality in important cases.The paper highlights explicit regret bounds for linear, radial basis, and Matérn kernels, expressed up to polylogarithmic factors.
  • Experiments: On sensor network data, GP-UCB compares favorably with existing heuristic GP optimization algorithms.The evaluation concerns real sensor data and compares GP-UCB with other optimization criteria.

2. Problem Statement and Background

The paper frames noisy optimization as sequentially sampling an unknown function to maximize cumulative reward while balancing estimation and optimization. It models functions with Gaussian processes or low RKHS complexity and connects information gain from experimental design to regret analysis.

  • Problem formulation: The objective is to choose points sequentially and maximize cumulative reward while approaching the unknown function’s maximizer.Observations are noisy, and neither instantaneous nor cumulative regret is revealed to the algorithm.
  • Problem formulation: Cumulative regret measures the loss from selecting points below the function maximum, and no-regret means R_T/T tends to zero.Average-regret bounds also provide convergence rates for GP optimization.
  • Gaussian processes and RKHS: Gaussian processes encode smoothness through kernels and provide analytic posterior means and covariances after noisy observations.The posterior remains a GP with mean and covariance determined by the kernel matrix and observation noise.
  • Gaussian processes and RKHS: The paper also studies functions with low RKHS norm, which measures smoothness relative to the chosen kernel without requiring a GP prior.This setting uses distribution-free assumptions on the noise process.
  • Information gain and experimental design: Information gain is mutual information between sampled observations and the function, quantifying uncertainty reduction from a sampling set.For Gaussian observations, it equals 1/2 log |I + σ^-2K_A|, and greedy maximization receives a constant-factor guarantee because information gain is submodular.
  • Information gain and experimental design: Experimental-design sampling reduces uncertainty globally but ignores observed rewards, so its maximum information gain instead becomes a quantity in the regret bounds.This establishes the paper’s connection between experimental design and GP optimization.

3. GP-UCB Algorithm

GP-UCB combines posterior reward estimates with uncertainty to balance exploitation and exploration. The paper analyzes this rule theoretically, compares it with alternative GP optimization heuristics, and notes practical challenges in infinite domains.

  • Selection rule: Posterior-mean maximization alone can become greedy too early and get stuck in shallow local optima, motivating the combined GP-UCB strategy.The rule is motivated by classical UCB algorithms for multi-armed bandits.
  • Selection rule: GP-UCB selects points using an upper-confidence objective that favors both high posterior mean and high posterior uncertainty.The rule interprets the selected value as an upper quantile of the marginal posterior and uses β_t to control the tradeoff.
  • Practical considerations: When the input space is infinite, maximizing the GP-UCB index can be difficult because the index is generally multimodal.Global search heuristics are reported to be effective in practice, while function evaluation is assumed more costly than index maximization.
  • Algorithm: GP-UCB repeatedly samples y_t = f(x_t) + ε_t and performs a Bayesian update to obtain the posterior mean and uncertainty.The algorithm takes the input space, GP prior, and kernel as inputs.
  • Theory and practice: Before this work, GP optimization heuristics had practical use, but convergence rates were not known; the paper supplies theoretical results for GP-UCB.The paper also reports that GP-UCB compares favorably with alternative search heuristics in practice.

4. Regret Bounds

The paper derives high-probability cumulative-regret bounds for GP-UCB under GP-prior and agnostic RKHS settings. These bounds are governed by maximum information gain, connecting GP optimization with experimental design while requiring smoothness assumptions for general domains.

  • Problem setting: GP optimization extends stochastic linear optimization to nonlinear functions whose complexity is characterized by kernel-dependent information gain rather than finite dimensionality.The maximum information gain γT replaces the linear setting’s dimensionality term in the regret analysis.
  • GP-prior regret bounds: The proof relates cumulative regret to confidence-ellipsoid growth and shows that this growth is characterized by maximum information gain.This establishes the paper’s technical connection between GP optimization and experimental design.
  • GP-prior regret bounds: For compact convex domains, GP-UCB achieves high-probability regret O∗(√dTγT) under a high-probability derivative bound on GP sample paths.The derivative condition is imposed coordinatewise through constants a and b, and the analysis covers general compact convex D.
  • Assumptions and limitations: The general-domain analysis excludes highly erratic sample paths, including the Ornstein-Uhlenbeck kernel, and conjectures that the stated theorem does not hold there.The smoothness condition applies to four-times-differentiable stationary kernels such as Squared Exponential and Matérn kernels with ν > 2.
  • RKHS regret bounds: In the agnostic setting, GP-UCB retains a high-probability regret bound O∗(T(B√γT + γT)) for RKHS functions with bounded norm and bounded martingale-difference noise.The same algorithm is used despite misspecified prior and noise models; the theorem is uniform over functions with finite RKHS norm.
  • Assumptions and limitations: The GP-prior and RKHS results are complementary: GP sample paths have infinite RKHS norm almost surely, while the RKHS theorem assumes a finite norm and requires an upper-bound parameter.Guess-and-doubling can handle the absence of a known norm bound, but neither theorem encompasses the other.

5. Bounding the Information Gain

The paper bounds maximum information gain using submodularity, greedy experimental design, and kernel spectra. Spectral decay then yields explicit sublinear-regret guarantees for common kernels, with especially weak dimensional dependence for the Squared Exponential kernel.

  • Submodularity and greedy maximization: Bounding γT reduces to maximizing the submodular information-gain function over T-point subsets, enabling a near-greedy experimental-design upper bound.The greedy design does not depend on observations and can therefore be run without evaluating the objective function.
  • Spectral bounds: The greedy bound supports both numerical, problem-specific estimates of γT and simpler analytical bounds based on T and kernel parameters.The latter are obtained from kernel-matrix or operator-spectrum information.
  • Spectral bounds: Rapid spectral decay makes information gain grow slowly: when the first T∗ = o(T) eigenvalues contain most total spectral mass, γT is small.Figure 3 illustrates the relationship between spectral decay and information-gain bounds across kernel classes.
  • Bounds for common kernels: For finite-dimensional linear, Squared Exponential, and Matérn kernels, the information-gain bounds imply sublinear regret for GP-UCB.The bounds depend on the covariance function and its parameters, and match known lower bounds up to log factors in K-armed and linear cases.
  • Bounds for common kernels: The kernel families compared include independent, linear, Squared Exponential, and Matérn kernels, with Matérn smoothness controlled by ν.The figure compares their spectral decay and resulting information-gain bounds under equal trace.
  • Bounds for common kernels: For the Squared Exponential kernel, dimensionality appears only as an exponent of log T in the regret bound.The paper attributes this weak dependence to the kernel’s high degree of sample-path smoothness.

6. Experiments

The experiments compare GP-UCB with heuristic GP optimization methods on synthetic and sensor-network data, finding competitive or favorable performance across the evaluated settings.

  • Experimental setup: GP-UCB is compared with EI, MPI, maximum-mean, and maximum-variance methods on synthetic and real sensor-network data.The comparison uses both heuristic acquisition criteria and naive sampling rules.
  • Sensor-network data: Temperature data use 46 sensors, with T = 46 and results averaged over 2000 objective-function choices.The covariance matrix is estimated from two-thirds of the data and testing uses the remaining third.
  • Synthetic data: Figure 4 shows GP samples generated with linear, squared exponential, and Matérn kernels with ν = 2.5.
  • Sensor-network data: Traffic data use 357 highway sensors, with T = 357 and results averaged over 900 runs.The task is to identify the most congested highway portion by finding the minimum-speed location.
  • Results: For temperature data, GP-UCB and EI clearly outperform the other methods without a significant difference between them; MPI performs equally well on synthetic and traffic data.Overall, GP-UCB performs at least on par with existing approaches lacking regret bounds.

7. Conclusions

The paper establishes regret guarantees for GP optimization by analyzing GP-UCB through information gain and kernel spectra. Experiments indicate that GP-UCB performs at least on par with competing criteria while offering theoretical bounds.

  • The paper proves the first sublinear regret bounds for GP optimization with commonly used kernels for GP-sampled functions and low-RKHS-norm functions.
  • GP-UCB is analyzed as a Bayesian upper-confidence-bound sampling rule whose regret depends on information gain from sampling.
  • Bounding information gain through kernel spectra provides a general methodology for deriving regret bounds for kernels of interest.
  • Experiments on real sensor-network data indicate that GP-UCB performs at least on par with competing GP optimization criteria for which regret bounds are not known.

A. Regret Bounds for Target Function Sampled from GP

The finite-decision-set analysis establishes high-probability confidence bounds for GP predictions and converts them into cumulative-regret control through information gain and Cauchy–Schwarz.

  • Confidence bounds: The finite-case analysis begins by establishing uniform confidence over the finite decision set D.
  • Confidence bounds: With high probability, GP prediction errors are bounded by β_t^1/2 σ_t−1(x) simultaneously over finite decisions and time.The proof uses Gaussian tail bounds and a union bound over decisions and rounds.
  • Regret bound: The instantaneous regret is bounded by twice the confidence width, 2β_t^1/2 σ_t−1(x_t), on the confidence event.
  • Information gain: The information gain from selected points is expressed through their predictive variances.

A.2. General Decision Set

For compact general decision sets, the proof discretizes the domain and uses smoothness assumptions to transfer confidence bounds from the discretization to the continuous optimum.

  • Generalization: The general-decision-set theorem extends the finite-case result to compact D ⊂ R^d but requires assumptions on the kernel.The Ornstein–Uhlenbeck kernel illustrates why continuity alone is insufficient for this extension.
  • Discretization: The analysis uses time-indexed discretizations D_t to obtain confidence intervals for the unknown continuous maximizer.The discretizations become dense as t increases and use uniformly spaced coordinates as a sufficient construction.
  • Confidence bounds: Confidence bounds are established on the discretized sets using β_t = 2 log(|D_t|π_t/δ) and Gaussian tail arguments.
  • Regret bound: The discretization and smoothness arguments yield a high-probability regret bound for all t ∈ N.The proof combines the confidence events with the regret argument and Cauchy–Schwarz.
  • Smoothness: For sufficiently smooth kernels, including Gaussian and Matérn kernels with ν > 2, the sample paths satisfy the differentiability conditions used by the proof.The resulting constant obeys K_1 = O((log δ−1)^1/2).

B. Regret Bound for Target Function in RKHS

The theorem extends GP-UCB analysis to arbitrary target functions with bounded RKHS norm and more general bounded martingale noise. This agnostic setting supports regret guarantees without requiring the Bayesian model to match how the function and observations are generated.

  • The analysis assumes only that the target function belongs to Hk(D) with bounded RKHS norm, rather than being sampled from the GP.
  • The true noise may be a uniformly bounded martingale difference sequence, although GP-UCB assumes independent Gaussian noise.
  • Theorem 6 provides a high-probability concentration result when noise variables are uniformly bounded by σ.
  • The proof uses Freedman’s Bernstein-type inequality for martingale differences to control concentration under dependent noise.

B.2. Proof of Theorem 6

The proof controls posterior approximation error through information gain and martingale concentration, then bounds information gain using discretization, submodularity, and kernel spectra. These steps connect the high-probability confidence analysis to explicit complexity bounds.

  • B.2. Proof of Theorem 6: The proof analyzes ZT, the RKHS-norm error between the posterior mean µT and target function f.
  • B.2. Proof of Theorem 6: Information gain controls the growth of the posterior approximation error and supports the proof’s inductive concentration argument.
  • B.2. Proof of Theorem 6: A martingale difference sequence is constructed from normalized noise and posterior uncertainty, then bounded with high probability through Lemma 7.3.
  • B.2. Proof of Theorem 6: Freedman’s inequality is used because controlling conditional variances avoids the T^3/4 bound produced by a standard Hoeffding-Azuma argument.
  • C. Bounds on Information Gain: The discretization argument assumes k(x,x)=1 and uses Lipschitz continuity to relate continuous-domain and discretized information gain.
  • C. Bounds on Information Gain: The information-gain bound proceeds from submodularity and greedy maximization to discretization, empirical kernel spectra, and operator spectra.

C.3. Proof of Theorem 5

Theorem 5 specializes the information-gain analysis to squared exponential and Matérn kernels. For squared exponential kernels, the resulting information gain grows polylogarithmically, while Matérn kernels yield sublinear regret for ν > 2.

  • Squared Exponential Kernel: Squared exponential kernels yield γT = O((log T)^(d+1)) on compact domains.
  • Squared Exponential Kernel: For squared exponential kernels, the derived maximum information gain matches the leading-order expected information gain under an independent Gaussian design.
  • Matérn Kernel: Matérn-kernel analysis uses eigenvalue decay λs ≤ cs^−(2ν+d)/d to control the information-gain spectrum.
  • Matérn Kernel: For any ν > 2 and any dimension d, GP-UCB has regret RT = O*(T^(1−η)) up to logarithmic factors and is no-regret with arbitrarily high probability.
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