[Paper Review] The Gaussian Process Autoregressive Regression Model (GPAR)

James Requeima, Will Tebbutt+2 more|arXiv (Cornell University)|Feb 20, 2018

Gaussian Processes and Bayesian Inference38 references16 citations

TL;DR

The Gaussian Process Autoregressive Regression (GPAR) model proposes a scalable, tractable approach to multi-output regression by decomposing the joint output distribution into a chain of conditionally independent Gaussian processes, enabling efficient modeling of nonlinear, input-varying output dependencies. GPAR achieves state-of-the-art performance on benchmarks while maintaining computational efficiency.

ABSTRACT

Multi-output regression models must exploit dependencies between outputs to maximise predictive performance. The application of Gaussian processes (GPs) to this setting typically yields models that are computationally demanding and have limited representational power. We present the Gaussian Process Autoregressive Regression (GPAR) model, a scalable multi-output GP model that is able to capture nonlinear, possibly input-varying, dependencies between outputs in a simple and tractable way: the product rule is used to decompose the joint distribution over the outputs into a set of conditionals, each of which is modelled by a standard GP. GPAR's efficacy is demonstrated on a variety of synthetic and real-world problems, outperforming existing GP models and achieving state-of-the-art performance on established benchmarks.

Motivation & Objective

• Address the computational inefficiency and limited representational power of existing multi-output Gaussian process models.
• Enable modeling of complex, nonlinear, and input-dependent dependencies between outputs in a tractable and scalable manner.
• Develop a framework that maintains predictive accuracy while reducing computational cost compared to standard multi-output GP approaches.
• Demonstrate strong empirical performance on both synthetic and real-world multi-output regression tasks.
• Provide a flexible, modular architecture that generalizes well across diverse regression benchmarks.

Proposed method

• Decompose the joint distribution over multiple outputs using the chain rule of probability into a sequence of conditionally independent Gaussian processes.
• Model each output as a GP conditioned on the previous outputs in the sequence, enabling autoregressive structure.
• Use standard GP regression for each conditional distribution, leveraging standard kernels and inference techniques.
• Enable input-varying dependencies by allowing the kernel hyperparameters of each conditional GP to depend on earlier outputs or input features.
• Ensure computational scalability by avoiding full covariance matrix operations across all outputs, instead treating each conditional as a separate GP.
• Support efficient inference and prediction through standard GP approximations and variational inference when needed.

Experimental results

Research questions

• RQ1 Can a structured autoregressive decomposition of the joint GP distribution improve scalability and representational power in multi-output regression?
• RQ2 How well can GPAR model nonlinear and input-varying dependencies between outputs compared to existing multi-output GP models?
• RQ3 Does the autoregressive GP framework achieve state-of-the-art performance on standard multi-output regression benchmarks?
• RQ4 To what extent does GPAR maintain computational efficiency while capturing complex output correlations?
• RQ5 Can GPAR generalize effectively across diverse synthetic and real-world regression tasks?

Key findings

• GPAR outperforms existing multi-output GP models on a range of synthetic and real-world regression tasks, demonstrating superior predictive accuracy.
• The model achieves state-of-the-art performance on established benchmarks, including those from the MUlti-Output Regression (MOR) challenge.
• GPAR maintains computational efficiency by avoiding full joint covariance modeling, enabling scalable inference even with many outputs.
• The autoregressive structure allows effective modeling of nonlinear, input-dependent output dependencies without increasing computational complexity unduly.
• Empirical results show that GPAR generalizes well across diverse data types and output correlation structures.
• The method is robust to varying numbers of outputs and input dimensions, maintaining strong performance across experimental settings.