# Randomized Block Proximal Methods for Distributed Stochastic Big-Data Optimization

## TL;DR

This paper introduces a new class of distributed algorithms for large-scale stochastic convex optimization over directed graphs, capable of handling high-dimensional, nonsmooth problems with proven convergence and tested on real datasets.

## Contribution

The paper proposes a novel distributed block proximal method with convergence guarantees for stochastic big-data convex optimization over directed graphs.

## Key findings

- Convergence to optimal cost in expectation with diminishing stepsizes.
- Sublinear convergence rate with explicit bounds for constant stepsizes.
- Successful application to high-dimensional classification problems on synthetic and real data.

## Abstract

In this paper we introduce a class of novel distributed algorithms for solving stochastic big-data convex optimization problems over directed graphs. In the addressed set-up, the dimension of the decision variable can be extremely high and the objective function can be nonsmooth. The general algorithm consists of two main steps: a consensus step and an update on a single block of the optimization variable, which is then broadcast to neighbors. Three special instances of the proposed method, involving particular problem structures, are then presented. In the general case, the convergence of a dynamic consensus algorithm over random row stochastic matrices is shown. Then, the convergence of the proposed algorithm to the optimal cost is proven in expected value. Exact convergence is achieved when using diminishing (local) stepsizes, while approximate convergence is attained when constant stepsizes are employed. The convergence rate is shown to be sublinear and an explicit rate is provided in the case of constant stepsizes. Finally, the algorithm is tested on a distributed classification problem, first on synthetic data and, then, on a real, high-dimensional, text dataset.