# Signal recovery by Stochastic Optimization

## TL;DR

This paper introduces a stochastic optimization approach for signal recovery in Generalized Linear Models, reducing the problem to solving a stochastic variational inequality with proven convergence rates.

## Contribution

It proposes a novel method that simplifies signal estimation in GLMs by linking it to stochastic VI, with weaker assumptions than traditional convexity requirements.

## Key findings

- Finite-time error bounds of $O(1/K)$ for strongly monotone cases
- Efficient computational approach for stochastic VI solutions
- Weaker structural assumptions than maximum likelihood convexity

## Abstract

We discuss an approach to signal recovery in Generalized Linear Models (GLM) in which the signal estimation problem is reduced to the problem of solving a stochastic monotone variational inequality (VI). The solution to the stochastic VI can be found in a computationally efficient way, and in the case when the VI is strongly monotone we derive finite-time upper bounds on the expected $\|\cdot\|_2^2$ error converging to 0 at the rate $O(1/K)$ as the number $K$ of observations grows. Our structural assumptions are essentially weaker than those necessary to ensure convexity of the optimization problem resulting from Maximum Likelihood estimation. In hindsight, the approach we promote can be traced back directly to the ideas behind the Rosenblatt's perceptron algorithm.