Gaussian Processes for Big Data (1309.6835v1)

Abstract: We introduce stochastic variational inference for Gaussian process models. This enables the application of Gaussian process (GP) models to data sets containing millions of data points. We show how GPs can be vari- ationally decomposed to depend on a set of globally relevant inducing variables which factorize the model in the necessary manner to perform variational inference. Our ap- proach is readily extended to models with non-Gaussian likelihoods and latent variable models based around Gaussian processes. We demonstrate the approach on a simple toy problem and two real world data sets.

• The paper introduces stochastic variational inference (SVI) for Gaussian processes, overcoming traditional scalability limits for big data.
• The method leverages inducing variables and natural gradients to ensure efficient optimization and positive definiteness of covariance matrices.
• Empirical validations on real-world datasets demonstrate significant predictive improvements over conventional GP approaches.

Gaussian Processes for Big Data

"Gaussian Processes for Big Data" by James Hensman, Nicolò Fusi, and Neil D. Lawrence introduces stochastic variational inference (SVI) for Gaussian process (GP) models, enabling their application to datasets containing millions of data points. The proposed method achieves significant scalability improvements over traditional GP approaches by leveraging a novel formulation based on inducing variables and a variational decomposition that factorizes the model appropriately for efficient inference.

Introduction

Gaussian processes are a cornerstone of non-parametric Bayesian inference, widely used for regression, classification, and unsupervised learning. Despite their versatility, the computational and storage complexities associated with GPs—O(n^3) and O(n^2), respectively—have traditionally limited their applicability to datasets larger than a few thousand points. This limitation is particularly prohibitive for applications involving large spatiotemporal datasets, high-dimensional video data, extensive social network data, or population-scale medical datasets.

Sparse GPs Revisited

The authors begin by revisiting the variational approach to inducing variables as formulated by Titsias (2009). By introducing a set of inducing variables $u$, the covariance structure required for efficient computation is redefined. However, the core challenge remains: existing techniques that exploit low-rank approximations or partition the data still struggle as datasets grow to millions or billions of points.

Stochastic Variational Inference for GPs

To overcome these limitations, the authors derive a lower bound suitable for stochastic optimization methods. This bound, denoted as $L_3$, is made tractable by leveraging the properties of inducing variables in combination with SVI. Unlike traditional approaches that collapse these variables, this method maintains an explicit representation of inducing variables, which is crucial for the applicability of SVI in GP models.

Natural Gradients

The authors proceed to derive the natural gradients for the variational parameters, a step crucial for the implementation of SVI. The natural gradient ensures positive definiteness of the covariance matrix at each step, circumventing issues typically encountered in standard gradient-based optimizations. This enables efficient updating of both the variational distribution $q(u)$ and kernel hyperparameters.

Extensions to Non-Gaussian Likelihoods and Latent Variable Models

An essential advantage of the proposed method is its flexibility in handling non-Gaussian likelihoods and latent variable models, such as the Gaussian Process Latent Variable Model (GPLVM). The factorization introduced allows the integration of non-Gaussian likelihoods exactly for certain types of likelihoods, thereby broadening the applicability of the method.

Empirical Validation

The authors validate their approach on both toy and real-world datasets. They demonstrate the convergence of their method on small, synthetic datasets before applying it to larger datasets such as UK apartment price data and US airline delays. For the UK apartment price data, the model showed a significant improvement in predictive performance compared to baseline GPs on random subsets of the data. A similar pattern was observed for the airline delay dataset, where the method provided robust estimates of delay times, highlighting the relevance of different features through automatic relevance determination (ARD).

Implications and Future Work

The scalability of the proposed SVI approach holds substantial implications for the application of GPs in big data contexts, offering a method capable of handling much larger datasets than traditional GP methods. This advancement opens avenues for more intricate multi-output models and latent variable models, which were previously infeasible due to computational constraints. The efficient handling of non-Gaussian likelihoods further extends its utility across various domains.

Conclusion

The paper presents a significant advancement in scaling Gaussian processes to handle big data through stochastic variational inference. The derived bounds and natural gradient formulations offer a practical path forward for applying GPs to datasets comprising millions of data points. This methodological contribution sets the stage for further research into more complex GP-based models and their applications in diverse fields such as machine learning, data science, and artificial intelligence.