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Gaussian Process Regression Networks
Andrew Gordon Wilson, David A. Knowles, Zoubin Ghahramani
TL;DR
GPRN addresses multi-output regression by combining Bayesian-neural-network structure with Gaussian-process flexibility, allowing input-dependent correlations and covariance properties. It provides MCMC and variational Bayes inference and shows strong performance across gene-expression, geostatistics, and volatility benchmarks, while inference can mix poorly in some settings and future applications remain open.
Problem
Existing Gaussian-process extensions address multi-output correlations, changing noise, non-stationary covariance properties, and heavy tails, but the paper targets a unified regression framework for vector-valued responses with input-dependent behavior.
Method
GPRN combines Bayesian-neural-network structure with Gaussian-process flexibility as an adaptive mixture, with inference based on MCMC and variational Bayes.
Results
GPRN significantly outperforms the other models on the 50D and 1000D gene-expression datasets, achieving between 46% and 68% of the SMSE and similarly strong MSLL results.
Takeaways & Limitations
GPRN supports adaptive signal and noise correlations, input-dependent length-scales and amplitudes, heavy-tailed predictive distributions, and multivariate volatility modeling across benchmark applications.
Takeaways & Limitations
MCMC can mix poorly because of strong weight-node prior correlations and costly likelihood matrix inversions, with mixing difficulties more evident in high dimensions.
Abstract
from arXiv · showhide
We introduce a new regression framework, Gaussian process regression networks (GPRN), which combines the structural properties of Bayesian neural networks with the non-parametric flexibility of Gaussian processes. This model accommodates input dependent signal and noise correlations between multiple response variables, input dependent length-scales and amplitudes, and heavy-tailed predictive distributions. We derive both efficient Markov chain Monte Carlo and variational Bayes inference procedures for this model. We apply GPRN as a multiple output regression and multivariate volatility model, demonstrating substantially improved performance over eight popular multiple output (multi-task) Gaussian process models and three multivariate volatility models on benchmark datasets, including a 1000 dimensional gene expression dataset.
1 Introduction
Gaussian processes offer flexible, interpretable regression, but traditional models and recent extensions address different aspects of correlated outputs, changing noise, non-stationarity, and heavy-tailed predictions. GPRN combines Bayesian-neural-network structure with Gaussian-process flexibility and supports efficient inference and broad empirical comparisons.
- Gaussian processes provide expressive, interpretable, non-overfitting models with strong predictive performance for nonlinear regression and classification.
- Neural networks motivated multi-output modeling through adaptive hidden units shared across responses, a capability inherited by GPRN.
- Fixed output correlations in multi-task Gaussian processes can improve predictions by incorporating measurements from related response variables.
- Prior Gaussian-process extensions separately model input-dependent noise variances and noise correlations between multiple outputs.
- Other extensions address non-stationary amplitudes and length-scales, as well as heavy-tailed predictive distributions for outlier rejection.
- GPRN combines Bayesian-neural-network structure with Gaussian-process flexibility as an adaptive mixture that accommodates changing correlations, covariance properties, and heavy-tailed predictions.The framework is designed to avoid expensive or numerically unstable computations.
- The paper develops efficient MCMC and variational Bayes inference, then compares GPRN with eight multiple-output GP models and multivariate volatility models on benchmark datasets.
2 Gaussian Process Regression Networks
GPRN predicts vector-valued responses while allowing signal and noise correlations to vary with the input. It constructs outputs by combining latent Gaussian-process nodes with input-dependent Gaussian-process weights, yielding adaptive covariance and volatility structures.
- The model predicts the conditional mean and covariance of vector-valued observations at a test input while allowing signal and noise correlations to change with the input.Examples include gene expression over time and heavy-metal concentrations across geographic locations.
- GPRN forms outputs from a vector of independent latent Gaussian processes with additive noise and a matrix of independent Gaussian-process weight functions.The latent nodes use kernels k_fi, while the weights use kernel k_w.
- The network is adaptive because its input-dependent weights can strengthen, weaken, disappear, or form new connections between latent nodes and outputs.
- Conditioning on the weight functions makes each output a Gaussian process whose covariance reflects the learned data-dependent coupling between outputs.
- The resulting covariance can have input-dependent amplitudes and effective length-scales, and can switch between different kernel structures such as periodic, squared-exponential, and Brownian-motion behavior.
- The same construction also represents a multivariate volatility model through input-dependent noise covariance, with node count q controlling how signal and noise correlations are modeled.The framework can disable changing correlations and multivariate volatility by setting σ_f^2 = 0 and using large fixed weight-function length-scales.
3 Inference
GPRN inference uses both MCMC and variational Bayes to estimate predictive distributions and hyperparameters. The paper addresses sampling challenges from tightly correlated priors while retaining favorable computational complexity relative to many multi-task GP models.
- Inference approaches: GPRN estimates predictive distributions using both variational Bayes and Markov chain Monte Carlo.Variational Bayes also estimates the model hyperparameters.
- Markov chain Monte Carlo: Elliptical Slice Sampling jointly updates tightly correlated Gaussian-process variables without free parameters.The authors report that it mixes well and avoids costly or numerically unstable likelihood matrix inversions.
- Predictive distribution: Even with Gaussian observations, GPRN produces an infinite Gaussian mixture that is generally heavy-tailed and robust to outliers.The predictive distribution is constructed from joint samples of the weight and node processes.
- Variational Bayes: Variational EM fits an approximate posterior by minimizing KL(q||p), using variational message passing in the E-step and gradient-based optimization in the M-step.The M-step optimizes log length-scale parameters with gradient descent and line search.
- Computational complexity: GPRN’s main computational limitation is Cholesky decomposition of the block-diagonal C_B matrix, while shared covariance structure reduces repeated work.Under variational Bayes, exploiting equal weight-function covariances reduces the stated inversion cost from O(qpN^3) to O(qN^3).
- Computational complexity: GPRN accounts for input-dependent signal correlations with computational demands favorable to multi-task GP models commonly requiring O(p^3N^3) complexity.Its likelihood evaluation also avoids the repeated p×p covariance inversions typical of multivariate volatility models.
4 Experiments
Experiments evaluate GPRN on gene-expression, geostatistical, and multivariate-volatility tasks against established multi-task GP and volatility models. GPRN achieves strong predictive performance while learning input-dependent correlations and retaining scalability in high dimensions.
- Gene Expression: GPRN significantly outperforms the other gene-expression models, achieving 46%–68% of their SMSE and similarly strong MSLL results.On the 50D dataset, MCMC and VB are comparable; on 1000D, VB noticeably outperforms MCMC, while MCMC still outperforms all other methods.
- Gene Expression: The marginal likelihood is sharply peaked at q = 1 for both gene-expression datasets, supporting one transcription factor controlling the selected genes.
- Gene Expression: Typical GPRN (VB) runtimes are 12 seconds for the 50D dataset and 330 seconds for the 1000D dataset.
- Jura Geostatistics: The Jura model selects q = 2 latent node functions and learns spatially varying correlations among cadmium, nickel, and zinc.Cadmium–zinc correlations are generally positive around 0.6 but drop noticeably in one region.
- Jura Geostatistics: For Jura cadmium prediction, GPRN outperforms SLFM and the other multiple-output methods on mean absolute error.All multiple-output methods achieve lower MAE than an independent GP.
- Computational Comparison: GPRN’s complexity scales as O(Nqp), unlike competing methods scaling as O(N^3p^3), enabling its use on the 1000-dimensional dataset.The improved Jura performance comes at the cost of a slightly greater runtime.
- Multivariate Volatility: GPRN models multivariate volatility by predicting time-dependent covariance matrices for currency-exchange and equity-index returns.Its performance is comparable to GWP; GPRN with VB often outperforms GPRN with MCMC on these volatility tasks.
5 Discussion
GPRN combines an interpretable, adaptive structure with scalable inference and strong benchmark performance. The authors identify broader adaptive covariance structures and classification as future directions.
- GPRN has a simple, interpretable structure that generalizes recent Gaussian-process extensions.
- GPRN accommodates input-dependent correlations, heavy-tailed predictions, nonstationary length-scales and amplitudes, and adaptive covariance functions.
- GPRN combines scalable inference procedures with strong empirical performance on several benchmark datasets.
- Future work includes adaptive covariance structures beyond those explored and applications of GPRN to classification.
7 Appendix
The appendix reviews Gaussian-process foundations, from function distributions and kernel properties through standard regression prediction. It then describes predictive and noise-covariance calculations for GPRN inference, including MCMC and variational Bayes treatments.
- Gaussian-process foundations: A Gaussian process defines a distribution over functions in which every finite collection of function values is jointly Gaussian.
- Gaussian-process foundations: Kernel functions determine properties such as smoothness and periodicity, while the squared-exponential kernel uses amplitude A and length-scale l.
- Gaussian-process foundations: The Ornstein-Uhlenbeck kernel gives a continuous-time AR(1) Gaussian process with Markovian dynamics, and is a special Matérn-kernel case.
- Standard regression: Gaussian-process prediction conditions the joint distribution of a test function value and training observations on the observed data.
- Standard regression: Standard Gaussian-process regression assumes Gaussian observation and predictive distributions, constant noise, and constant amplitude and length-scale.
- GPRN inference: Positive-weight constraints can be implemented by exponentiation for MCMC or truncated Gaussians for variational Bayes.
- GPRN inference: Variational Bayes approximates the posterior over weights and latent functions with q(W)q(f), then integrates this approximation to obtain predictions.
- GPRN inference: The GPRN noise is formed from weighted latent-process and observation-noise terms, yielding a noise covariance used for multivariate volatility modeling.