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We study the problem of sequential prediction and online minimax regret withstochastically generated features under a general loss function. We introduce anotion of expected worst case minimax regret that generalizes and encompassesprior known minimax regrets. For such minimax regrets we establish tight upperbounds via a novel concept of stochastic global sequential covering. We showthat for a hypothesis class of VC-dimension $\mathsf{VC}$ and $i.i.d.$generated features of length $T$, the cardinality of the stochastic globalsequential covering can be upper bounded with high probability (whp) by$e^{O(\mathsf{VC} \cdot \log^2 T)}$. We then improve this bound by introducinga new complexity measure called the Star-Littlestone dimension, and show thatclasses with Star-Littlestone dimension $\mathsf{SL}$ admit a stochastic globalsequential covering of order $e^{O(\mathsf{SL} \cdot \log T)}$. We furtherestablish upper bounds for real valued classes with finite fat-shatteringnumbers. Finally, by applying information-theoretic tools of the fixed designminimax regrets, we provide lower bounds for the expected worst case minimaxregret. We demonstrate the effectiveness of our approach by establishing tightbounds on the expected worst case minimax regrets for logarithmic loss andgeneral mixable losses.

Expected Worst Case Regret via Stochastic Sequential Covering