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Hyper-parameter optimization remains as the core issue of Gaussian process(GP) for machine learning nowadays. The benchmark method using maximumlikelihood (ML) estimation and gradient descent (GD) is impractical forprocessing big data due to its $O(n^3)$ complexity. Many sophisticated globalor local approximation models, for instance, sparse GP, distributed GP, havebeen proposed to address such complexity issue. In this paper, we propose twonovel and general-purpose GP hyper-parameter training schemes (GPCV-ADMM) byreplacing ML with cross-validation (CV) as the fitting criterion and replacingGD with a non-linearly constrained alternating direction method of multipliers(ADMM) as the optimization method. The proposed schemes are of $O(n^2)$complexity for any covariance matrix without special structure. We conductvarious experiments based on both synthetic and real data sets, wherein theproposed schemes show excellent performance in terms of convergence,hyper-parameter estimation accuracy, and computational time in comparison withthe traditional ML based routines given in the GPML toolbox.

A General $\mathcal{O}(n^2)$ Hyper-Parameter Optimization for Gaussian Process Regression with Cross-Validation and Non-linearly Constrained ADMM