Streaming sparse Gaussian process approximations

Sparse pseudo-point approximations for Gaussian process (GP) models provide a suite of methods that support deployment of GPs in the large data regime and enable analytic intractabilities to be sidestepped. However, the field lacks a principled method to handle streaming data in which both the posterior distribution over function values and the hyperparameter estimates are updated in an online fashion. The small number of existing approaches either use suboptimal hand-crafted heuristics for hyperparameter learning, or suffer from catastrophic forgetting or slow updating when new data arrive. This paper develops a new principled framework for deploying Gaussian process probabilistic models in the streaming setting, providing methods for learning hyperparameters and optimising pseudo-input locations. The proposed framework is assessed using synthetic and real-world datasets

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

Sign Stable Projections, Sign Cauchy Projections and Chi-Square Kernels

The method of stable random projections is popular for efficiently computing the Lp distances in high dimension (where 0<p<=2), using small space. Because it adopts nonadaptive linear projections, this method is naturally suitable when the data are collected in a dynamic streaming fashion (i.e., turnstile data streams). In this paper, we propose to use only the signs of the projected data and analyze the probability of collision (i.e., when the two signs differ). We derive a bound of the collision probability which is exact when p=2 and becomes less sharp when p moves away from 2. Interestingly, when p=1 (i.e., Cauchy random projections), we show that the probability of collision can be accurately approximated as functions of the chi-square similarity. For example, when the (un-normalized) data are binary, the maximum approximation error of the collision probability is smaller than 0.0192. In text and vision applications, the chi-square similarity is a popular measure for nonnegative data when the features are generated from histograms. Our experiments confirm that the proposed method is promising for large-scale learning applications