Expected Worst Case Regret via Stochastic Sequential Covering

Published: 05 Aug 2023, Last Modified: 05 Aug 2023Accepted by TMLREveryoneRevisionsBibTeX
Abstract: We study the problem of sequential prediction and online minimax regret with stochastically generated features under a general loss function. In an online learning setting, Nature selects features and associates a true label with these features. A learner uses features to predict a label, which is compared to the true label, and a loss is incurred. The total loss over $T$ rounds, when compared to a loss incurred by a set of experts, is known as a regret. We introduce the notion of *expected worst case minimax regret* that generalizes and encompasses prior known minimax regrets. For such minimax regrets, we establish tight upper bounds via a novel concept of *stochastic global sequential covering*. We show that for a hypothesis class of VC-dimension $\mathsf{VC}$ and $i.i.d.$ generated features over $T$ rounds, the cardinality of stochastic global sequential covering can be upper bounded with high probability (w.h.p.) by $e^{O(\mathsf{VC} \cdot \log^2 T)}$. We then improve this bound by introducing a new complexity measure called the *Star-Littlestone* dimension, and show that classes with Star-Littlestone dimension $\mathsf{SL}$ admit a stochastic global sequential covering of order $e^{O(\mathsf{SL} \cdot \log T)}$. We further establish upper bounds for real valued classes with finite fat-shattering numbers. Finally, by applying information-theoretic tools for the fixed design minimax regrets, we provide lower bounds for expected worst case minimax regret. We demonstrate the effectiveness of our approach by establishing tight bounds on the expected worst case minimax regrets for logarithmic loss and general mixable losses.
Submission Length: Long submission (more than 12 pages of main content)
Changes Since Last Submission: This is the camera-ready version of the paper. It has been de-anonymized, the citations have been updated, and some typographical errors have been corrected. In addition, added the acknowledgements section.
Assigned Action Editor: ~Shinichi_Nakajima2
License: Creative Commons Attribution 4.0 International (CC BY 4.0)
Submission Number: 890