# Stochastic Mirror Descent for Large-Scale Sparse Recovery

arXiv: https://arxiv.org/abs/2210.12882 · HTML: https://mlsanity.com/paper/2210.12882/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 5 of its 55 references that are in the map.
- First posted on arXiv 23 Oct 2022.

## Builds on (most cited first, 5 of 5)

- [On Iterative Hard Thresholding Methods for High-dimensional M-Estimation](https://mlsanity.com/paper/1410.5137/) (2014)
- [Outlier-robust estimation of a sparse linear model using ℓ1-penalized Huber's M-estimator](https://mlsanity.com/paper/1904.06288/) (2019)
- [Between hard and soft thresholding: optimal iterative thresholding algorithms](https://mlsanity.com/paper/1804.08841/) (2018)
- [Gradient descent with nonconvex constraints: local concavity determines convergence](https://mlsanity.com/paper/1703.07755/) (2017)
- [Sparse recovery by reduced variance stochastic approximation](https://mlsanity.com/paper/2006.06365/) (2020)

## Built on by (most cited first, 1 of 1)

- [Accelerated stochastic approximation with state-dependent noise](https://mlsanity.com/paper/2307.01497/) (2023)

## Abstract (arXiv)

In this paper we discuss an application of Stochastic Approximation to statistical estimation of high-dimensional sparse parameters. The proposed solution reduces to resolving a penalized stochastic optimization problem on each stage of a multistage algorithm; each problem being solved to a prescribed accuracy by the non-Euclidean Composite Stochastic Mirror Descent (CSMD) algorithm. Assuming that the problem objective is smooth and quadratically minorated and stochastic perturbations are sub-Gaussian, our analysis prescribes the method parameters which ensure fast convergence of the estimation error (the radius of a confidence ball of a given norm around the approximate solution). This convergence is linear during the first "preliminary" phase of the routine and is sublinear during the second "asymptotic" phase. We consider an application of the proposed approach to sparse Generalized Linear Regression problem. In this setting, we show that the proposed algorithm attains the optimal convergence of the estimation error under weak assumptions on the regressor distribution. We also present a numerical study illustrating the performance of the algorithm on high-dimensional simulation data.