Online Learning in Dynamically Changing Environments

• URL: http://arxiv.org/abs/2302.00103v1
• Date: Tue, 31 Jan 2023 21:10:03 GMT
• Title: Online Learning in Dynamically Changing Environments
• Authors: Changlong Wu, Ananth Grama, Wojciech Szpankowski
• Abstract summary: We study the problem of online learning and online regret when samples are drawn from a general unknown non-stationary process. We prove a tight (upto $sqrtlog T$ factor) bound $O(sqrtKTcdotmathsfVC(mathcalH)log T)$ for the expected worst case regret of any finite VC-dimensional class. We extend these results to general smooth adversary processes with unknown reference measure.
• Score: 11.731001328350983
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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