Prediction under Uncertainty in Sparse Spectrum Gaussian Processes with Applications to Filtering and Control

In many sequential prediction and decision-making problems such as Bayesian filtering and probabilistic model-based planning and control, we need to cope with the challenge of prediction under uncertainty, where the goal is to compute the predictive distribution p(y) given a input distribution p(x) and a probabilistic model p(y|x). Computing the exact predictive distribution is generally intractable. In this work, we consider a special class of problems in which the input distribution p(x) is a multivariate Gaussian, and the probabilistic model p(y|x) is learned from data and specified by a sparse spectral representation of Gaussian processes (SSGPs). SSGPs are a powerful tool for scaling Gaussian processes (GPs) to large datasets by approximating the covariance function using finite-dimensional random Fourier features. Existing SSGP algorithms for regression assume deterministic inputs, precluding their use in many sequential prediction and decision-making applications where accounting for input uncertainty is crucial. To address this prediction under uncertainty problem, we propose an exact moment-matching approach with closed-form expressions for predictive distributions. Our method is more general and scalable than its standard GP counterpart, and is naturally applicable to multi-step prediction or uncertainty propagation. We show that our method can be used to develop new algorithms for Bayesian filtering and stochastic model predictive control, and we evaluate the applicability of our method with both simulated and real-world experiments.