# Generic Error Bounds for the Generalized Lasso with Sub-Exponential Data

arXiv: https://arxiv.org/abs/2004.05361 · HTML: https://mlsanity.com/paper/2004.05361/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 11 Apr 2020.

## Built on by (most cited first, 6 of 6)

- [Quantizing Heavy-tailed Data in Statistical Estimation: (Near) Minimax Rates, Covariate Quantization, and Uniform Recovery](https://mlsanity.com/paper/2212.14562/) (2022)
- [Outlier Robust and Sparse Estimation of Linear Regression Coefficients](https://mlsanity.com/paper/2208.11592/) (2022)
- [Estimation of sparse linear regression coefficients under L-subexponential covariates](https://mlsanity.com/paper/2304.11958/) (2023)
- [Sparsified Simultaneous Confidence Intervals for High-Dimensional Linear Models](https://mlsanity.com/paper/2307.07574/) (2023)
- [Sparse Linear Regression when Noises and Covariates are Heavy-Tailed and Contaminated by Outliers](https://mlsanity.com/paper/2408.01336/) (2024)
- [Restricted Eigenvalues Beyond Gaussian Width: Threshold Occupancy under Heavy Tails](https://mlsanity.com/paper/2609.03504/) (2026)

## Abstract (arXiv)

This work performs a non-asymptotic analysis of the generalized Lasso under the assumption of sub-exponential data. Our main results continue recent research on the benchmark case of (sub-)Gaussian sample distributions and thereby explore what conclusions are still valid when going beyond. While many statistical features remain unaffected (e.g., consistency and error decay rates), the key difference becomes manifested in how the complexity of the hypothesis set is measured. It turns out that the estimation error can be controlled by means of two complexity parameters that arise naturally from a generic-chaining-based proof strategy. The output model can be non-realizable, while the only requirement for the input vector is a generic concentration inequality of Bernstein-type, which can be implemented for a variety of sub-exponential distributions. This abstract approach allows us to reproduce, unify, and extend previously known guarantees for the generalized Lasso. In particular, we present applications to semi-parametric output models and phase retrieval via the lifted Lasso. Moreover, our findings are discussed in the context of sparse recovery and high-dimensional estimation problems.