Online Learning in Dynamically Changing Environments
This work addresses the problem of adapting to dynamically changing data distributions for online learning algorithms, representing an incremental step in regret analysis for non-stationary settings.
The paper tackles online learning in non-stationary environments by introducing a dynamic process model with bounded changes, proving tight regret bounds for VC-dimensional classes under absolute loss and general mixable losses, and extending results to smooth adversary processes.
We study the problem of online learning and online regret minimization when samples are drawn from a general unknown non-stationary process. We introduce the concept of a dynamic changing process with cost $K$, where the conditional marginals of the process can vary arbitrarily, but that the number of different conditional marginals is bounded by $K$ over $T$ rounds. For such processes we prove a tight (upto $\sqrt{\log T}$ factor) bound $O(\sqrt{KT\cdot\mathsf{VC}(\mathcal{H})\log T})$ for the expected worst case regret of any finite VC-dimensional class $\mathcal{H}$ under absolute loss (i.e., the expected miss-classification loss). We then improve this bound for general mixable losses, by establishing a tight (up to $\log^3 T$ factor) regret bound $O(K\cdot\mathsf{VC}(\mathcal{H})\log^3 T)$. We extend these results to general smooth adversary processes with unknown reference measure by showing a sub-linear regret bound for $1$-dimensional threshold functions under a general bounded convex loss. Our results can be viewed as a first step towards regret analysis with non-stationary samples in the distribution blind (universal) regime. This also brings a new viewpoint that shifts the study of complexity of the hypothesis classes to the study of the complexity of processes generating data.