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

## TL;DR

This paper introduces two novel hyper-parameter optimization schemes for Gaussian process regression that achieve quadratic complexity using cross-validation and ADMM, outperforming traditional methods in accuracy and efficiency.

## Contribution

It proposes general-purpose GP hyper-parameter training schemes replacing maximum likelihood with cross-validation and gradient descent with ADMM, reducing complexity to O(n^2).

## Key findings

- Schemes demonstrate excellent convergence and accuracy.
- Significant reduction in computational time.
- Effective on both synthetic and real datasets.

## Abstract

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.