A Regret-Variance Trade-Off in Online Learning

Dirk van der Hoeven, Nikita Zhivotovskiy, Nicolò Cesa-Bianchi

Keywords: Online learning, statistical learning, corrupted feedback, selective sampling, abstention

TL;DR: We state a regret-variance trade-off in online learning and provide multiple applications.

Abstract:

We consider prediction with expert advice for strongly convex and bounded losses, and investigate trade-offs between regret and ``variance'' (i.e., squared difference of learner's predictions and best expert predictions).
With $K$ experts, the Exponentially Weighted Average (EWA) algorithm is known to achieve $O(\log K)$ regret.
We prove that a variant of EWA either achieves a \textsl{negative} regret (i.e., the algorithm outperforms the best expert), or guarantees a $O(\log K)$ bound on \textsl{both} variance and regret.
Building on this result, we show several examples of how variance of predictions can be exploited in learning.
In the online to batch analysis, we show that a large empirical variance allows to stop the online to batch conversion early and outperform the risk of the best predictor in the class. We also recover the optimal rate of model selection aggregation when we do not consider early stopping.
In online prediction with corrupted losses, we show that the effect of corruption on the regret can be compensated by a large variance.
In online selective sampling, we design an algorithm that samples less when the variance is large, while guaranteeing the optimal regret bound in expectation.
In online learning with abstention, we use a similar term as the variance to derive the first high-probability $O(\log K)$ regret bound in this setting.
Finally, we extend our results to the setting of online linear regression.