Temporal Variability in Implicit Online Learning
This work addresses a theoretical bottleneck in online learning for researchers, providing incremental improvements in regret analysis for Implicit algorithms.
The paper tackles the gap between practical success and theoretical regret analysis of Implicit algorithms in online learning by proving a novel static regret bound that depends on the temporal variability of loss functions, showing constant regret under constant variability with proper tuning, and presenting an adaptive algorithm that achieves this bound without prior knowledge, validated on classification and regression datasets.
In the setting of online learning, Implicit algorithms turn out to be highly successful from a practical standpoint. However, the tightest regret analyses only show marginal improvements over Online Mirror Descent. In this work, we shed light on this behavior carrying out a careful regret analysis. We prove a novel static regret bound that depends on the temporal variability of the sequence of loss functions, a quantity which is often encountered when considering dynamic competitors. We show, for example, that the regret can be constant if the temporal variability is constant and the learning rate is tuned appropriately, without the need of smooth losses. Moreover, we present an adaptive algorithm that achieves this regret bound without prior knowledge of the temporal variability and prove a matching lower bound. Finally, we validate our theoretical findings on classification and regression datasets.