Sequential Gaussian Processes for Online Learning of Nonstationary Functions

Many machine learning problems can be framed in the context of estimating functions, and often these are time-dependent functions that are estimated in real-time as observations arrive. Gaussian processes (GPs) are an attractive choice for modeling real-valued nonlinear functions due to their flexibility and uncertainty quantification. However, the typical GP regression model suffers from several drawbacks: i) Conventional GP inference scales $O(N^{3})$ with respect to the number of observations; ii) updating a GP model sequentially is not trivial; and iii) covariance kernels often enforce stationarity constraints on the function, while GPs with non-stationary covariance kernels are often intractable to use in practice. To overcome these issues, we propose an online sequential Monte Carlo algorithm to fit mixtures of GPs that capture non-stationary behavior while allowing for fast, distributed inference. By formulating hyperparameter optimization as a multi-armed bandit problem, we accelerate mixing for real time inference. Our approach empirically improves performance over state-of-the-art methods for online GP estimation in the context of prediction for simulated non-stationary data and hospital time series data

Continual Multi-task Gaussian Processes

We address the problem of continual learning in multi-task Gaussian process (GP) models for handling sequential input-output observations. Our approach extends the existing prior-posterior recursion of online Bayesian inference, i.e.\ past posterior discoveries become future prior beliefs, to the infinite functional space setting of GP. For a reason of scalability, we introduce variational inference together with an sparse approximation based on inducing inputs. As a consequence, we obtain tractable continual lower-bounds where two novel Kullback-Leibler (KL) divergences intervene in a natural way. The key technical property of our method is the recursive reconstruction of conditional GP priors conditioned on the variational parameters learned so far. To achieve this goal, we introduce a novel factorization of past variational distributions, where the predictive GP equation propagates the posterior uncertainty forward. We then demonstrate that it is possible to derive GP models over many types of sequential observations, either discrete or continuous and amenable to stochastic optimization. The continual inference approach is also applicable to scenarios where potential multi-channel or heterogeneous observations might appear. Extensive experiments demonstrate that the method is fully scalable, shows a reliable performance and is robust to uncertainty error propagation over a plenty of synthetic and real-world datasets