  Hyper-parameter optimization remains as the core issue of Gaussian process
(GP) for machine learning nowadays. The benchmark method using maximum
likelihood (ML) estimation and gradient descent (GD) is impractical for
processing big data due to its $O(n^3)$ complexity. Many sophisticated global
or local approximation models, for instance, sparse GP, distributed GP, have
been proposed to address such complexity issue. In this paper, we propose two
novel and general-purpose GP hyper-parameter training schemes (GPCV-ADMM) by
replacing ML with cross-validation (CV) as the fitting criterion and replacing
GD 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 conduct
various experiments based on both synthetic and real data sets, wherein the
proposed schemes show excellent performance in terms of convergence,
hyper-parameter estimation accuracy, and computational time in comparison with
the traditional ML based routines given in the GPML toolbox.
