Variational inference for sparse spectrum Gaussian process regression

Abstract: We develop a fast variational approximation scheme for Gaussian process (GP) regression, where the spectrum of the covariance function is subjected to a sparse approximation. Our approach enables uncertainty in covariance function hyperparameters to be treated without using Monte Carlo methods and is robust to overfitting. Our article makes three contributions. First, we present a variational Bayes algorithm for fitting sparse spectrum GP regression models that uses nonconjugate variational message passing to derive fast and efficient updates. Second, we propose a novel adaptive neighbourhood technique for obtaining predictive inference that is effective in dealing with nonstationarity. Regression is performed locally at each point to be predicted and the neighbourhood is determined using a measure defined based on lengthscales estimated from an initial fit. Weighting dimensions according to lengthscales, this downweights variables of little relevance, leading to automatic variable selection and improved prediction. Third, we introduce a technique for accelerating convergence in nonconjugate variational message passing by adapting step sizes in the direction of the natural gradient of the lower bound. Our adaptive strategy can be easily implemented and empirical results indicate significant speedups.

• The paper presents a novel variational Bayesian approach that reduces computational demands by using a sparse spectral representation in GP regression.
• The paper employs variational inference to update spectral frequencies efficiently, thereby robustly addressing hyperparameter uncertainty.
• The paper’s algorithm accelerates convergence via adaptive natural gradient steps, enabling scalable and precise regression on large datasets.

Variational Inference for Sparse Spectrum Gaussian Process Regression

Introduction

The paper, "Variational Inference for Sparse Spectrum Gaussian Process Regression," discusses a variational Bayesian approach for Gaussian Process (GP) regression models with a sparse spectrum. This work specifically addresses the drawbacks of traditional methods which require intensive computational resources, and offers an alternative that exploits a variational approximation, alleviating overfitting issues while efficiently managing hyperparameter uncertainty.

Sparse Spectrum Gaussian Process Regression

The model proposed in this paper builds upon the traditional GP regression framework by introducing a sparse spectral representation of the covariance function. Typically, Gaussian Processes handle regression by defining a covariance matrix that scales quadratically and cubically in space and time with respect to the number of samples ($n$). Reducing this to a sparse spectrum resolves the primary computational bottleneck by focusing on a sparser set of spectral points, which is crucial for scaling to large datasets.

The innovation here relies on the work by L ázaro-Gredilla et al. (2010), which assumes a latent function that can be approximated by a linear combination of cosine and sine functions whose coefficients are drawn from a Gaussian distribution. By sampling spectral points from a normal distribution, the covariance matrix only needs to be computed over this reduced dimensionality, thus optimizing computational efficiency.

Variational Inference

Variational inference approximates the posterior distribution for the regression model parameters, given the data, by a simpler distribution, thereby simplifying the inferential calculations. This method contrasts with Monte Carlo approaches by sidestepping the resource-intensive computation associated with sampling. Instead, the paper proposes a Variational Bayesian (VB) algorithm that leverages nonconjugate variational message passing as a fast, efficient means for computing updates. This methodology is not only faster but importantly, it handles the uncertainty associated with covariance function hyperparameters more robustly.

The Algorithm

The specific algorithmic advancement here is in the update steps for the variational parameters. By refining updates for parameters such as the Gaussian process spectral frequencies and the diagonal covariance terms in the frequency space, the model retains probabilistic interpretation of hyperparameter uncertainty without the computational overhead of full posterior sampling techniques. A key contribution is the introduction of an algorithmic step to manage nonstationary data by modifying the neighborhood selection surrounding each test point—this local fitting respects the variability inherent in the data, leading to automatic variable selection and increased prediction precision.

Accelerating Convergence

An innovative adaptation technique enhances convergence speeds within the variational message-passing framework. By adjusting the step size in the direction of the natural gradient, the algorithm converges more rapidly to stable solutions. This technique derives inspiration from adaptive optimization strategies and ensures that each iteration leads to an increase in the bound optimization criterion—a vital component of ensuring algorithmic stability and efficiency in practice.

Practical and Theoretical Implications

Practically speaking, this approach facilitates the deployment of Gaussian Process regression models in real-world scenarios with large or challenging datasets. It balances complexity, computational demand, and model fidelity in a manner that is pragmatically attractive for commercial and scientific applications.

Theoretically, the fusion of variational approaches with sparsity in Gaussian Processes opens avenues for more scalable and interpretable machine learning models. By integrating sparse spectrum approximations with variational Bayes, not only is hyperparameter uncertainty better quantified, but the associated calculation is also streamlined to significant computational advantage.

Conclusion

Overall, "Variational Inference for Sparse Spectrum Gaussian Process Regression" presents a crucial step towards addressing the scalability issues inherent in Gaussian Processes. Through adept algorithm design, the paper sets the foundation for further exploration into hybrid inferential techniques that merge the paradigms of variational inference and sparse representations in complex data modeling scenarios.