Distributed Online Optimization with Stochastic Agent Availability

TL;DR

This paper studies distributed online optimization in federated learning with stochastic agent availability, proposing a variant of FTRL and providing regret bounds that account for probabilistic agent activity.

Contribution

It introduces a distributed FTRL algorithm tailored for stochastic agent availability and derives regret bounds that depend on the probability of agent activity and network properties.

Findings

Expected network regret scales as $(rac{ ext{condition number}}{p^2}) imes ext{min}ig\{ ext{sqrt}(N), N^{1/4}/ ext{sqrt}(p)igig\} ext{sqrt}(T)$.

Regret bounds hold with high probability, not just in expectation.

The average-case regret notion aligns with the worst-case, indicating bounds are tight.

Abstract

Motivated by practical federated learning settings where clients may not be always available, we investigate a variant of distributed online optimization where agents are active with a known probability $p$ at each time step, and communication between neighboring agents can only take place if they are both active. We introduce a distributed variant of the FTRL algorithm and analyze its network regret, defined through the average of the instantaneous regret of the active agents. Our analysis shows that, for any connected communication graph $G$ over $N$ agents, the expected network regret of our FTRL variant after $T$ steps is at most of order $(\kappa/p^2)\min\big\{\sqrt{N},N^{1/4}/\sqrt{p}\big\}\sqrt{T}$, where $\kappa$ is the condition number of the Laplacian of $G$. We then show that similar regret bounds also hold with high probability. Moreover, we show that our notion of regret (average-case over the agents) is essentially equivalent to the standard notion of regret (worst-case over agents), implying that our bounds are not significantly improvable when $p=1$. Our theoretical results are supported by experiments on synthetic datasets.