A General $\mathcal{O}(n^2)$ Hyper-Parameter Optimization for Gaussian Process Regression with Cross-Validation and Non-linearly Constrained ADMM
This work addresses a computational bottleneck for researchers and practitioners using Gaussian processes on large datasets, though it is incremental as it builds on existing approximation methods.
The authors tackled the high computational complexity of hyper-parameter optimization in Gaussian process regression by proposing two novel training schemes that reduce complexity from O(n^3) to O(n^2), achieving faster convergence and improved accuracy compared to traditional methods.
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.