# A Hybrid Stochastic Gradient Tracking Method for Distributed Online Optimization Over Time-Varying Directed Networks

## TL;DR

This paper introduces TV-HSGT, a novel distributed online optimization algorithm that effectively handles stochastic gradients over time-varying directed networks, improving convergence and regret bounds without requiring gradient boundedness.

## Contribution

The paper presents a hybrid stochastic gradient tracking algorithm that operates over directed networks without Perron vector estimation, enhancing dynamic regret bounds in online optimization.

## Key findings

- TV-HSGT outperforms existing methods in dynamic environments.
- It reduces gradient variance through recursive stochastic gradient integration.
- Experimental results validate its effectiveness in logistic regression tasks.

## Abstract

With the increasing scale and dynamics of data, distributed online optimization has become essential for real-time decision-making in various applications. However, existing algorithms often rely on bounded gradient assumptions and overlook the impact of stochastic gradients, especially in time-varying directed networks. This study proposes a novel Time-Varying Hybrid Stochastic Gradient Tracking algorithm named TV-HSGT, based on hybrid stochastic gradient tracking and variance reduction mechanisms. Specifically, TV-HSGT integrates row-stochastic and column-stochastic communication schemes over time-varying digraphs, eliminating the need for Perron vector estimation or out-degree information. By combining current and recursive stochastic gradients, it effectively reduces gradient variance while accurately tracking global descent directions. Theoretical analysis demonstrates that TV-HSGT can achieve improved bounds on dynamic regret without assuming gradient boundedness. Experimental results on logistic regression tasks confirm the effectiveness of TV-HSGT in dynamic and resource-constrained environments.