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

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.

Linear Multiple Low-Rank Kernel Based Stationary Gaussian Processes Regression for Time Series

Gaussian processes (GP) for machine learning have been studied systematically over the past two decades and they are by now widely used in a number of diverse applications. However, GP kernel design and the associated hyper-parameter optimization are still hard and to a large extend open problems. In this paper, we consider the task of GP regression for time series modeling and analysis. The underlying stationary kernel can be approximated arbitrarily close by a new proposed grid spectral mixture (GSM) kernel, which turns out to be a linear combination of low-rank sub-kernels. In the case where a large number of the sub-kernels are used, either the Nystr\"{o}m or the random Fourier feature approximations can be adopted to deal efficiently with the computational demands. The unknown GP hyper-parameters consist of the non-negative weights of all sub-kernels as well as the noise variance; their estimation is performed via the maximum-likelihood (ML) estimation framework. Two efficient numerical optimization methods for solving the unknown hyper-parameters are derived, including a sequential majorization-minimization (MM) method and a non-linearly constrained alternating direction of multiplier method (ADMM). The MM matches perfectly with the proven low-rank property of the proposed GSM sub-kernels and turns out to be a part of efficiency, stable, and efficient solver, while the ADMM has the potential to generate better local minimum in terms of the test MSE. Experimental results, based on various classic time series data sets, corroborate that the proposed GSM kernel-based GP regression model outperforms several salient competitors of similar kind in terms of prediction mean-squared-error and numerical stability.