Deep Horseshoe Gaussian Processes

• URL: http://arxiv.org/abs/2403.01737v1
• Date: Mon, 4 Mar 2024 05:30:43 GMT
• Title: Deep Horseshoe Gaussian Processes
• Authors: Isma\"el Castillo and Thibault Randrianarisoa
• Abstract summary: We introduce the deep Horseshoe Gaussian process Deep-HGP, a new simple prior based on deep Gaussian processes with a squared-exponential kernel. We show that the associated tempered posterior distribution recovers the unknown true regression curve optimally in terms of quadratic loss, up to a logarithmic factor.
• Score: 1.0742675209112622
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Deep Gaussian processes have recently been proposed as natural objects to fit, similarly to deep neural networks, possibly complex features present in modern data samples, such as compositional structures. Adopting a Bayesian nonparametric approach, it is natural to use deep Gaussian processes as prior distributions, and use the corresponding posterior distributions for statistical inference. We introduce the deep Horseshoe Gaussian process Deep-HGP, a new simple prior based on deep Gaussian processes with a squared-exponential kernel, that in particular enables data-driven choices of the key lengthscale parameters. For nonparametric regression with random design, we show that the associated tempered posterior distribution recovers the unknown true regression curve optimally in terms of quadratic loss, up to a logarithmic factor, in an adaptive way. The convergence rates are simultaneously adaptive to both the smoothness of the regression function and to its structure in terms of compositions. The dependence of the rates in terms of dimension are explicit, allowing in particular for input spaces of dimension increasing with the number of observations.

Related papers

• von Mises Quasi-Processes for Bayesian Circular Regression [57.88921637944379]
  We explore a family of expressive and interpretable distributions over circle-valued random functions. The resulting probability model has connections with continuous spin models in statistical physics. For posterior inference, we introduce a new Stratonovich-like augmentation that lends itself to fast Markov Chain Monte Carlo sampling.
  arXiv  Detail & Related papers  (2024-06-19T01:57:21Z)
• Sparse Variational Contaminated Noise Gaussian Process Regression with Applications in Geomagnetic Perturbations Forecasting [4.675221539472143]
  We propose a scalable inference algorithm for fitting sparse Gaussian process regression models with contaminated normal noise on large datasets. We show that our approach yields shorter prediction intervals for similar coverage and accuracy when compared to an artificial dense neural network baseline.
  arXiv  Detail & Related papers  (2024-02-27T15:08:57Z)
• Stochastic Gradient Descent for Gaussian Processes Done Right [86.83678041846971]
  We show that when emphdone right -- by which we mean using specific insights from optimisation and kernel communities -- gradient descent is highly effective. We introduce a emphstochastic dual descent algorithm, explain its design in an intuitive manner and illustrate the design choices. Our method places Gaussian process regression on par with state-of-the-art graph neural networks for molecular binding affinity prediction.
  arXiv  Detail & Related papers  (2023-10-31T16:15:13Z)
• Instance-Dependent Generalization Bounds via Optimal Transport [51.71650746285469]
  Existing generalization bounds fail to explain crucial factors that drive the generalization of modern neural networks. We derive instance-dependent generalization bounds that depend on the local Lipschitz regularity of the learned prediction function in the data space. We empirically analyze our generalization bounds for neural networks, showing that the bound values are meaningful and capture the effect of popular regularization methods during training.
  arXiv  Detail & Related papers  (2022-11-02T16:39:42Z)
• Density Estimation with Autoregressive Bayesian Predictives [1.5771347525430772]
  In the context of density estimation, the standard Bayesian approach is to target the posterior predictive. We develop a novel parameterization of the bandwidth using an autoregressive neural network that maps the data into a latent space.
  arXiv  Detail & Related papers  (2022-06-13T20:43:39Z)
• Distributed Sketching for Randomized Optimization: Exact Characterization, Concentration and Lower Bounds [54.51566432934556]
  We consider distributed optimization methods for problems where forming the Hessian is computationally challenging. We leverage randomized sketches for reducing the problem dimensions as well as preserving privacy and improving straggler resilience in asynchronous distributed systems.
  arXiv  Detail & Related papers  (2022-03-18T05:49:13Z)
• Amortised inference of fractional Brownian motion with linear computational complexity [0.0]
  We introduce a simulation-based, amortised Bayesian inference scheme to infer the parameters of random walks. Our approach learns the posterior distribution of the walks' parameters with a likelihood-free method. We adapt this scheme to show that a finite decorrelation time in the environment can furthermore be inferred from individual trajectories.
  arXiv  Detail & Related papers  (2022-03-15T14:43:16Z)
• Non-Gaussian Gaussian Processes for Few-Shot Regression [71.33730039795921]
  We propose an invertible ODE-based mapping that operates on each component of the random variable vectors and shares the parameters across all of them. NGGPs outperform the competing state-of-the-art approaches on a diversified set of benchmarks and applications.
  arXiv  Detail & Related papers  (2021-10-26T10:45:25Z)
• Wasserstein-Splitting Gaussian Process Regression for Heterogeneous Online Bayesian Inference [9.7471390457395]
  We employ variational free energy approximations of GPs operating in tandem with online expectation propagation steps. We introduce a local splitting step which instantiates a new GP whenever the posterior distribution changes significantly. Over time, this yields an ensemble of sparse GPs which may be updated incrementally.
  arXiv  Detail & Related papers  (2021-07-26T17:52:46Z)
• Sparse Algorithms for Markovian Gaussian Processes [18.999495374836584]
  Sparse Markovian processes combine the use of inducing variables with efficient Kalman filter-likes recursion. We derive a general site-based approach to approximate the non-Gaussian likelihood with local Gaussian terms, called sites. Our approach results in a suite of novel sparse extensions to algorithms from both the machine learning and signal processing, including variational inference, expectation propagation, and the classical nonlinear Kalman smoothers. The derived methods are suited to literature-temporal data, where the model has separate inducing points in both time and space.
  arXiv  Detail & Related papers  (2021-03-19T09:50:53Z)
• Efficiently Sampling Functions from Gaussian Process Posteriors [76.94808614373609]
  We propose an easy-to-use and general-purpose approach for fast posterior sampling. We demonstrate how decoupled sample paths accurately represent Gaussian process posteriors at a fraction of the usual cost.
  arXiv  Detail & Related papers  (2020-02-21T14:03:16Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.