Enabling scalable stochastic gradient-based inference for Gaussian processes by employing the Unbiased LInear System SolvEr (ULISSE)

In applications of Gaussian processes where quantification of uncertainty is of primary interest, it is necessary to accurately characterize the posterior distribution over covariance parameters. This paper proposes an adaptation of the Stochastic Gradient Langevin Dynamics algorithm to draw samples from the posterior distribution over covariance parameters with negligible bias and without the need to compute the marginal likelihood. In Gaussian process regression, this has the enormous advantage that stochastic gradients can be computed by solving linear systems only. A novel unbiased linear systems solver based on parallelizable covariance matrix-vector products is developed to accelerate the unbiased estimation of gradients. The results demonstrate the possibility to enable scalable and exact (in a Monte Carlo sense) quantification of uncertainty in Gaussian processes without imposing any special structure on the covariance or reducing the number of input vectors.Comment: 10 pages - paper accepted at ICML 201

Bayesian Inference of Log Determinants

The log-determinant of a kernel matrix appears in a variety of machine learning problems, ranging from determinantal point processes and generalized Markov random fields, through to the training of Gaussian processes. Exact calculation of this term is often intractable when the size of the kernel matrix exceeds a few thousand. In the spirit of probabilistic numerics, we reinterpret the problem of computing the log-determinant as a Bayesian inference problem. In particular, we combine prior knowledge in the form of bounds from matrix theory and evidence derived from stochastic trace estimation to obtain probabilistic estimates for the log-determinant and its associated uncertainty within a given computational budget. Beyond its novelty and theoretic appeal, the performance of our proposal is competitive with state-of-the-art approaches to approximating the log-determinant, while also quantifying the uncertainty due to budget-constrained evidence.Comment: 12 pages, 3 figure

MCMC for variationally sparse Gaussian processes

Gaussian process (GP) models form a core part of probabilistic machine learning. Considerable research effort has been made into attacking three issues with GP models: how to compute efficiently when the number of data is large; how to approximate the posterior when the likelihood is not Gaussian and how to estimate covariance function parameter posteriors. This paper simultaneously addresses these, using a variational approximation to the posterior which is sparse in support of the function but otherwise free-form. The result is a Hybrid Monte-Carlo sampling scheme which allows for a non-Gaussian approximation over the function values and covariance parameters simultaneously, with efficient computations based on inducing-point sparse GPs. Code to replicate each experiment in this paper will be available shortly.JH was funded by an MRC fellowship, AM and ZG by EPSRC grant EP/I036575/1 and a Google Focussed Research award.This is the final version of the article. It first appeared from the Neural Information Processing Systems Foundation via https://papers.nips.cc/paper/5875-mcmc-for-variationally-sparse-gaussian-processe

Adaptive Multiple Importance Sampling for Gaussian Processes

In applications of Gaussian processes where quantification of uncertainty is a strict requirement, it is necessary to accurately characterize the posterior distribution over Gaussian process covariance parameters. Normally, this is done by means of standard Markov chain Monte Carlo (MCMC) algorithms. Motivated by the issues related to the complexity of calculating the marginal likelihood that can make MCMC algorithms inefficient, this paper develops an alternative inference framework based on Adaptive Multiple Importance Sampling (AMIS). This paper studies the application of AMIS in the case of a Gaussian likelihood, and proposes the Pseudo-Marginal AMIS for non-Gaussian likelihoods, where the marginal likelihood is unbiasedly estimated. The results suggest that the proposed framework outperforms MCMC-based inference of covariance parameters in a wide range of scenarios and remains competitive for moderately large dimensional parameter spaces.Comment: 27 page

Sampling from Gaussian Process Posteriors using Stochastic Gradient Descent

Gaussian processes are a powerful framework for quantifying uncertainty and for sequential decision-making but are limited by the requirement of solving linear systems. In general, this has a cubic cost in dataset size and is sensitive to conditioning. We explore stochastic gradient algorithms as a computationally efficient method of approximately solving these linear systems: we develop low-variance optimization objectives for sampling from the posterior and extend these to inducing points. Counterintuitively, stochastic gradient descent often produces accurate predictions, even in cases where it does not converge quickly to the optimum. We explain this through a spectral characterization of the implicit bias from non-convergence. We show that stochastic gradient descent produces predictive distributions close to the true posterior both in regions with sufficient data coverage, and in regions sufficiently far away from the data. Experimentally, stochastic gradient descent achieves state-of-the-art performance on sufficiently large-scale or ill-conditioned regression tasks. Its uncertainty estimates match the performance of significantly more expensive baselines on a large-scale Bayesian optimization task

Learning Binary Data Representation for Optical Processing Units

Optical Processing Units (OPUs) are computing devices that perform random projections of input data by exploiting the physical phenomenon of scattering a light source through a diffusive medium. Random projections calculated by OPUs have been used successfully for approximating kernel ridge regression for large datasets with low power consumption and at high speed. However, OPUs require the input data to be binary. In this paper, we propose to use shallow and deep neural networks (NN) as binary encoders to perform input data binarization. The difficulty in developing a binarization strategy which is learned in an end-to-end fashion along with kernel ridge regression parameters, is due to the non-differentiability of the operation performed by the OPU. We overcome this difficulty by considering OPUs as a black-box and by employing the REINFORCE gradient estimator, which allows us to calculate the gradient of the loss function with respect to the weights of the binarization encoder and to optimize these together with the parameters of kernel ridge regression with gradient- based optimization. Through our experimental campaign on a variety of tasks and datasets, we show that our method outperforms alternative unsupervised and supervised binarization techniques

Learning Inconsistent Preferences with Kernel Methods

We propose a probabilistic kernel approach for preferential learning from pairwise duelling data using Gaussian Processes. Different from previous methods, we do not impose a total order on the item space, hence can capture more expressive latent preferential structures such as inconsistent preferences and clusters of comparable items. Furthermore, we prove the universality of the proposed kernels, i.e. that the corresponding reproducing kernel Hilbert Space (RKHS) is dense in the space of skew-symmetric preference functions. To conclude the paper, we provide an extensive set of numerical experiments on simulated and real-world datasets showcasing the competitiveness of our proposed method with state-of-the-art

Infinity Learning: Learning Markov Chains from Aggregate Steady-State Observations

We consider the task of learning a parametric Continuous Time Markov Chain (CTMC) sequence model without examples of sequences, where the training data consists entirely of aggregate steady-state statistics. Making the problem harder, we assume that the states we wish to predict are unobserved in the training data. Specifically, given a parametric model over the transition rates of a CTMC and some known transition rates, we wish to extrapolate its steady state distribution to states that are unobserved. A technical roadblock to learn a CTMC from its steady state has been that the chain rule to compute gradients will not work over the arbitrarily long sequences necessary to reach steady state ---from where the aggregate statistics are sampled. To overcome this optimization challenge, we propose $\infty$-SGD, a principled stochastic gradient descent method that uses randomly-stopped estimators to avoid infinite sums required by the steady state computation, while learning even when only a subset of the CTMC states can be observed. We apply $\infty$-SGD to a real-world testbed and synthetic experiments showcasing its accuracy, ability to extrapolate the steady state distribution to unobserved states under unobserved conditions (heavy loads, when training under light loads), and succeeding in difficult scenarios where even a tailor-made extension of existing methods fails

A Bayesian conjugate gradient method (with Discussion)

A fundamental task in numerical computation is the solution of large linear systems. The conjugate gradient method is an iterative method which offers rapid convergence to the solution, particularly when an effective preconditioner is employed. However, for more challenging systems a substantial error can be present even after many iterations have been performed. The estimates obtained in this case are of little value unless further information can be provided about the numerical error. In this paper we propose a novel statistical model for this numerical error set in a Bayesian framework. Our approach is a strict generalisation of the conjugate gradient method, which is recovered as the posterior mean for a particular choice of prior. The estimates obtained are analysed with Krylov subspace methods and a contraction result for the posterior is presented. The method is then analysed in a simulation study as well as being applied to a challenging problem in medical imaging