Adversarial Delays in Online Strongly-Convex Optimization

We consider the problem of strongly-convex online optimization in presence of adversarial delays [1]; in a T -iteration online game, the feedback of the player’s query at time t is arbitrarily delayed by an adversary for dt rounds and delivered before the game ends, at iteration t+ dt − 1. Specifically for online-gradient-descent algorithm we show it has a simple regret bound of O (∑T t=1 log(1 + dt 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 O (log T ) [2] when there are no delays (i.e. dt = 1) and the regret bound of O (τ log T ) [3] for constant delays dt = τ .