On Average-Case Error Bounds for Kernel-Based Bayesian Quadrature

Numerical integration underpins many scientific and machine learning tasks, from Bayesian inference to uncertainty quantification. Early works focused on deterministic algorithms with worst-case guarantees, often relying on Sobolev or Hölder spaces. The advent of kernel methods and Gaussian processes introduced probabilistic approaches, notably Bayesian quadrature (BQ), which models the integrand as a Gaussian process (GP). Recent advances have improved error bounds for noiseless settings, but real-world applications often involve noisy data, high dimensions, and model uncertainties. Existing literature includes Oates et al.'s Bayesian Monte Carlo, Gunter et al.'s kernel quadrature, and Fearnhead's random feature approximations. Despite progress, theoretical understanding of average-case errors in noisy environments remains limited, especially for high-dimensional problems.

The core challenge is to develop theoretically grounded, computationally feasible algorithms for Bayesian quadrature that perform well under Gaussian noise, model misspecification, and high-dimensional settings. Traditional methods like grid sampling or repeated resampling are computationally intensive and less effective in noisy environments. Existing bounds are often worst-case, not reflecting typical performance. The difficulty lies in balancing function approximation accuracy with residual estimation, ensuring the overall error diminishes at near-optimal rates. Addressing these issues requires new analytical tools linking function approximation in RKHS to integral error, considering noise and parameter uncertainty.

The paper introduces a two-step meta-algorithm that leverages kernel regularity and Gaussian noise models to relate L2 function approximation errors to integral errors. Key innovations include: 1) using maximum variance sampling for function estimation, 2) employing residual Monte Carlo for correction, 3) deriving general error bounds applicable to Matérn and SE kernels, 4) extending analysis to misspecified smoothness and unknown kernel parameters, 5) establishing algorithm-independent lower bounds. This approach simplifies implementation by avoiding multiple resampling, reduces computational costs, and provides near-optimal theoretical guarantees in noisy, high-dimensional settings.

• �� First, select T/2 samples using maximum variance sampling (e.g., GP posterior variance maximization) to estimate the function in RKHS. • Use the posterior mean as the function estimate, ensuring a controlled L2 approximation error. • Compute the integral approximation by integrating the estimated function against the known density. • Use remaining T/2 samples to estimate the residual integral via Monte Carlo, correcting the initial estimate. • Theoretical analysis relates the L2 approximation error to the overall integral error, incorporating Gaussian noise effects. • Derive bounds for Matérn-ν and SE kernels, considering misspecification and unknown parameters. • Establish lower bounds to validate the near-optimality of the approach, covering algorithm-independent limits.

Synthetic functions generated with Matérn-3/2 and SE kernels tested the algorithm's performance. T=250, repeated 100 trials, measuring mean absolute error against a high-precision trapezoidal reference. Variations in sample split ratio assessed efficiency. Results showed the proposed method consistently outperformed traditional grid and random sampling, closely matching theoretical bounds. The experiments demonstrated robustness to noise and kernel misspecification, confirming the practical viability of the theoretical guarantees. Additional tests on real data and high-dimensional functions are planned for future work.

Empirical results aligned with theoretical predictions, achieving error rates of Θ(T^{−ν/d−1}+σT^{−1/2}) for Matérn kernels, and exponential decay in noiseless SE cases. The method's simplicity and efficiency were validated by reduced sample complexity and improved accuracy compared to baseline methods. The robustness under kernel misspecification and unknown parameters was confirmed, with error bounds remaining near optimal. These findings demonstrate the method's potential for scalable, accurate Bayesian integration in noisy environments.

Applicable to Bayesian inference, uncertainty quantification, sensor data fusion, and scientific computing where high-dimensional integrals are common. The approach enables efficient, accurate estimation with limited samples, crucial in real-time or resource-constrained scenarios. Future integration with deep kernel learning can further enhance adaptability. The method also supports adaptive and multi-task extensions, broadening its utility across machine learning and data science domains.

Assumes Gaussian noise, which may not hold in all applications. High-dimensional kernel behavior can degrade, requiring further scalability improvements. Kernel parameter estimation remains a challenge, especially in non-stationary settings. Theoretical bounds rely on kernel regularity and smoothness assumptions, limiting applicability in highly irregular functions. Future work should address robustness to non-Gaussian noise, adaptive parameter tuning, and scalable implementations.