On the Value of Stochastic Side Information in Online Learning

We study the effectiveness of stochastic side information in deterministic online learning scenarios. We propose a forecaster to predict a deterministic sequence where its performance is evaluated against an expert class. We assume that certain stochastic side information is available to the forecaster but not the experts. We define the minimax expected regret for evaluating the forecaster's performance, for which we obtain both upper and lower bounds. Consequently, our results characterize the improvement in the regret due to the stochastic side information. Compared with the classical online learning problem with regret scales with $O(\sqrt{n})$, the regret can be negative when the stochastic side information is more powerful than the experts. To illustrate, we apply the proposed bounds to two concrete examples of different types of side information.