Adversarial Delays in Online Strongly-Convex Optimization

TL;DR

This paper analyzes the impact of adversarial delays on strongly-convex online optimization, providing a simple regret bound for online-gradient-descent that generalizes existing results and applies to various delay scenarios.

Contribution

It introduces a unified regret bound for online-gradient-descent under adversarial delays, extending and generalizing previous results without distributional assumptions.

Findings

Regret bound of O(∑_{t=1}^T log(1 + d_t/t)) for adversarial delays

Special cases recover known bounds like O(log T) and O(τ log T)

The analysis applies to arbitrary delay sequences without distributional assumptions.

Abstract

We consider the problem of strongly-convex online optimization in presence of adversarial delays; in a T-iteration online game, the feedback of the player's query at time t is arbitrarily delayed by an adversary for d_t rounds and delivered before the game ends, at iteration t+d_t-1. Specifically for \algo{online-gradient-descent} algorithm we show it has a simple regret bound of \Oh{\sum_{t=1}^T \log (1+ \frac{d_t}{t})}. This gives a clear and simple bound without resorting any distributional and limiting assumptions on the delays. We further show how this result encompasses and generalizes several of the existing known results in the literature. Specifically it matches the celebrated logarithmic regret \Oh{\log T} when there are no delays (i.e. d_t = 1) and regret bound of \Oh{\tau \log T} for constant delays d_t = \tau.