Dimension-free PAC-Bayesian bounds for matrices, vectors, and linear least squares regression

Abstract

This paper is focused on dimension-free PAC-Bayesian bounds, under weak polynomial moment assumptions, allowing for heavy tailed sample distributions. It covers the estimation of the mean of a vector or a matrix, with applications to least squares linear regression. Special efforts are devoted to the estimation of Gram matrices, due to their prominent role in high-dimension data analysis.

arXiv commentVersion 1 needed some further proofreading

References in corpus (3)

• Simultaneous analysis of Lasso and Dantzig selector
• Geometric median and robust estimation in Banach spaces
• Dimension-free PAC-Bayesian bounds for the estimation of the mean of a random vector

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• PAC-Bayes under potentially heavy tails
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• A Unified Approach to Robust Mean Estimation
• MONK -- Outlier-Robust Mean Embedding Estimation by Median-of-Means
• Distribution-Free Robust Linear Regression
• Robust Modifications of U-statistics and Applications to Covariance Estimation Problems
• Differentially Private (Gradient) Expectation Maximization Algorithm with Statistical Guarantees
• Distribution-robust mean estimation via smoothed random perturbations
• PAC-Bayes unleashed: generalisation bounds with unbounded losses
• Optimal Mean Estimation without a Variance
• New bounds for $k$-means and information $k$-means
• Learning via Wasserstein-Based High Probability Generalisation Bounds
• Robust learning with anytime-guaranteed feedback
• Iteratively Reweighted ${\ell }_1$-Penalized Robust Regression