$k\texttt{-experts}$ -- Online Policies and Fundamental Limits
This work addresses an online learning problem for scenarios requiring multi-expert selection, offering theoretical and practical advancements, though it is incremental as it builds on the classic experts framework.
The paper tackles the $k$-experts problem, a generalization of Prediction with Expert's Advice where the learner selects $k$ experts per round, and proposes the SAGE framework to achieve sublinear regret or improve existing guarantees for a wide class of reward functions, with experiments on standard datasets.
We introduce the $\texttt{$k$-experts}$ problem - a generalization of the classic Prediction with Expert's Advice framework. Unlike the classic version, where the learner selects exactly one expert from a pool of $N$ experts at each round, in this problem, the learner can select a subset of $k$ experts at each round $(1\leq k\leq N)$. The reward obtained by the learner at each round is assumed to be a function of the $k$ selected experts. The primary objective is to design an online learning policy with a small regret. In this pursuit, we propose $\texttt{SAGE}$ ($\textbf{Sa}$mpled Hed$\textbf{ge}$) - a framework for designing efficient online learning policies by leveraging statistical sampling techniques. For a wide class of reward functions, we show that $\texttt{SAGE}$ either achieves the first sublinear regret guarantee or improves upon the existing ones. Furthermore, going beyond the notion of regret, we fully characterize the mistake bounds achievable by online learning policies for stable loss functions. We conclude the paper by establishing a tight regret lower bound for a variant of the $\texttt{$k$-experts}$ problem and carrying out experiments with standard datasets.