[Paper Review] Horseshoe Regularization for Machine Learning in Complex and Deep Models

Anindya Bhadra, Jyotishka Datta+2 more|arXiv (Cornell University)|Apr 24, 2019

Gaussian Processes and Bayesian Inference90 references4 citations

TL;DR

This paper extends horseshoe regularization beyond linear Gaussian models to nonlinear, non-Gaussian, multivariate, and deep neural network settings, demonstrating its theoretical and computational viability for complex machine learning tasks. It presents methodological advances and software implementations, establishing horseshoe priors as a scalable, theoretically grounded tool for high-dimensional Bayesian inference in modern ML applications.

ABSTRACT

Since the advent of the horseshoe priors for regularization, global-local shrinkage methods have proved to be a fertile ground for the development of Bayesian methodology in machine learning, specifically for high-dimensional regression and classification problems. They have achieved remarkable success in computation, and enjoy strong theoretical support. Most of the existing literature has focused on the linear Gaussian case; see Bhadra et al. (2019b) for a systematic survey. The purpose of the current article is to demonstrate that the horseshoe regularization is useful far more broadly, by reviewing both methodological and computational developments in complex models that are more relevant to machine learning applications. Specifically, we focus on methodological challenges in horseshoe regularization in nonlinear and non-Gaussian models; multivariate models; and deep neural networks. We also outline the recent computational developments in horseshoe shrinkage for complex models along with a list of available software implementations that allows one to venture out beyond the comfort zone of the canonical linear regression problems.

Motivation & Objective

• Address the gap in horseshoe regularization beyond linear Gaussian models, which dominate existing literature.
• Investigate the theoretical and computational feasibility of global-local shrinkage priors in complex, high-dimensional machine learning models.
• Provide methodological frameworks for applying horseshoe priors in nonlinear, non-Gaussian, multivariate, and deep neural network settings.
• Develop and compile accessible software implementations to support practical adoption beyond standard linear regression.
• Bridge theoretical optimality and empirical performance in deep and complex models through scalable computational methods.

Proposed method

• Adapt horseshoe priors—characterized by global and local shrinkage hyperparameters—for use in nonlinear and non-Gaussian likelihood models via hierarchical Bayesian modeling.
• Employ variational Bayes and stochastic gradient MCMC methods to enable scalable posterior computation in high-dimensional and deep models.
• Integrate horseshoe priors into deep learning architectures such as Bayesian neural networks and deep sigmoid belief networks using probabilistic backpropagation.
• Utilize GPU-accelerated Gibbs sampling (e.g., via CUDA) to improve sampling efficiency in high-dimensional posterior inference.
• Implement software tools in Python, R, MATLAB, and Scala, with support for TensorFlow and probabilistic programming frameworks.
• Apply projection predictive methods and dynamic shrinkage processes to extend horseshoe utility in graphical models and time-series settings.

Experimental results

Research questions

• RQ1 Can horseshoe regularization maintain theoretical optimality and computational scalability in nonlinear and non-Gaussian models?
• RQ2 How can global-local shrinkage priors be effectively extended to multivariate and high-dimensional structured models?
• RQ3 What are the computational trade-offs and performance gains of using horseshoe priors in deep neural networks compared to standard regularization?
• RQ4 To what extent do variational Bayes and stochastic gradient MCMC methods enable scalable posterior inference with horseshoe priors in complex models?
• RQ5 What software infrastructure is needed to support widespread adoption of horseshoe regularization in modern machine learning pipelines?

Key findings

• Horseshoe regularization demonstrates strong empirical success in nonlinear and non-Gaussian models, including deep neural networks, despite limited theoretical optimality frameworks in these settings.
• The horseshoe prior enables effective sparsity induction and uncertainty quantification in high-dimensional problems, outperforming classical methods like lasso in certain regimes.
• GPU-accelerated implementations (e.g., GPUHorseshoe) significantly improve sampling speed in probit models, enabling practical inference in large-scale settings.
• Software packages such as HS-BNN, Bayesian Compression, and DeepGLM provide ready-to-use tools for applying horseshoe priors in Bayesian neural networks and deep learning.
• Stochastic gradient MCMC methods show promise for scaling Bayesian inference in deep models, suggesting a viable path toward fully Bayesian deep learning with horseshoe priors.
• A native TensorFlow implementation of the horseshoe distribution is available, enabling integration with large-scale machine learning workflows.