# Sharp Oracle Inequalities for Low-complexity Priors

## TL;DR

This paper establishes sharp oracle inequalities for high-dimensional estimators like Lasso and nuclear norm penalties, demonstrating their theoretical performance guarantees under various data loss functions and priors.

## Contribution

It provides a unified analysis of exponential weighted aggregation and penalized estimators with general priors, highlighting their performance and differences in high-dimensional settings.

## Key findings

- Sharp oracle inequalities for Lasso, group Lasso, and nuclear norm penalties.
- Theoretical guarantees for estimators under various data loss functions.
- Efficient implementation via proximal splitting algorithms.

## Abstract

In this paper,we consider a high-dimensional statistical estimation problem in which the the number of parameters is comparable or larger than the sample size. We present a unified analysis of the performance guarantees of exponential weighted aggregation and penalized estimators with a general class of data losses and priors which encourage objects which conform to some notion of simplicity/complexity. More precisely, we show that these two estimators satisfy sharp oracle inequalities for prediction ensuring their good theoretical performances. We also highlight the differences between them. When the noise is random, we provide oracle inequalities in probability using concentration inequalities. These results are then applied to several instances including the Lasso, the group Lasso, their analysis-type counterparts, the $\ell_\infty$ and the nuclear norm penalties. All our estimators can be efficiently implemented using proximal splitting algorithms.