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When fitting Bayesian machine learning models on scarce data, the main challenge is to obtain suitable prior knowledge and encode it into the model. Recent advances in meta-learning offer powerful methods for extracting such prior knowledge from data acquired in related tasks. When it comes to meta-learning in Gaussian process models, approaches in this setting have mostly focused on learning the kernel function of the prior, but not on learning its mean function. In this work, we explore meta-learning the mean function of a Gaussian process prior. We present analytical and empirical evidence that mean function learning can be useful in the meta-learning setting, discuss the risk of overfitting, and draw connections to other meta-learning approaches, such as model agnostic meta-learning and functional PCA.
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We introduce a framework for Continual Learning (CL) based on Bayesian inference over the function space rather than the parameters of a deep neural network. This method, referred to as functional regularisation for Continual Learning, avoids forgetting a previous task by constructing and memorising an approximate posterior belief over the underlying task-specific function. To achieve this we rely on a Gaussian process obtained by treating the weights of the last layer of a neural network as random and Gaussian distributed. Then, the training algorithm sequentially encounters tasks and constructs posterior beliefs over the task-specific functions by using inducing point sparse Gaussian process methods. At each step a new task is first learnt and then a summary is constructed consisting of (i) inducing inputs -- a fixed-size subset of the task inputs selected such that it optimally represents the task -- and (ii) a posterior distribution over the function values at these inputs. This summary then regularises learning of future tasks, through Kullback-Leibler regularisation terms. Our method thus unites approaches focused on (pseudo-)rehearsal with those derived from a sequential Bayesian inference perspective in a principled way, leading to strong results on accepted benchmarks.
For a learning task, Gaussian process (GP) is interested in learning the statistical relationship between inputs and outputs, since it offers not only the prediction mean but also the associated variability. The vanilla GP however struggles to learn complicated distribution with the property of, e.g., heteroscedastic noise, multi-modality and non-stationarity, from massive data due to the Gaussian marginal and the cubic complexity. To this end, this article studies new scalable GP paradigms including the non-stationary heteroscedastic GP, the mixture of GPs and the latent GP, which introduce additional latent variables to modulate the outputs or inputs in order to learn richer, non-Gaussian statistical representation. We further resort to different variational inference strategies to arrive at analytical or tighter evidence lower bounds (ELBOs) of the marginal likelihood for efficient and effective model training. Extensive numerical experiments against state-of-the-art GP and neural network (NN) counterparts on various tasks verify the superiority of these scalable modulated GPs, especially the scalable latent GP, for learning diverse data distributions.
Spectral approximation and variational inducing learning for the Gaussian process are two popular methods to reduce computational complexity. However, in previous research, those methods always tend to adopt the orthonormal basis functions, such as eigenvectors in the Hilbert space, in the spectrum method, or decoupled orthogonal components in the variational framework. In this paper, inspired by quantum physics, we introduce a novel basis function, which is tunable, local and bounded, to approximate the kernel function in the Gaussian process. There are two adjustable parameters in these functions, which control their orthogonality to each other and limit their boundedness. And we conduct extensive experiments on open-source datasets to testify its performance. Compared to several state-of-the-art methods, it turns out that the proposed method can obtain satisfactory or even better results, especially with poorly chosen kernel functions.
We present a multi-task learning formulation for Deep Gaussian processes (DGPs), through non-linear mixtures of latent processes. The latent space is composed of private processes that capture within-task information and shared processes that capture across-task dependencies. We propose two different methods for segmenting the latent space: through hard coding shared and task-specific processes or through soft sharing with Automatic Relevance Determination kernels. We show that our formulation is able to improve the learning performance and transfer information between the tasks, outperforming other probabilistic multi-task learning models across real-world and benchmarking settings.
Deep Gaussian Processes learn probabilistic data representations for supervised learning by cascading multiple Gaussian Processes. While this model family promises flexible predictive distributions, exact inference is not tractable. Approximate inference techniques trade off the ability to closely resemble the posterior distribution against speed of convergence and computational efficiency. We propose a novel Gaussian variational family that allows for retaining covariances between latent processes while achieving fast convergence by marginalising out all global latent variables. After providing a proof of how this marginalisation can be done for general covariances, we restrict them to the ones we empirically found to be most important in order to also achieve computational efficiency. We provide an efficient implementation of our new approach and apply it to several benchmark datasets. It yields excellent results and strikes a better balance between accuracy and calibrated uncertainty estimates than its state-of-the-art alternatives.
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