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\title{Online Learning with Stochastically Partitioning Experts}

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\author[1]{\href{puranjaydatta@gmail.com}{Puranjay~Datta}{}}
\author[1]{Sharayu~Moharir}
\author[2]{Jaya~Prakash~Champati}
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    Electrical Engineering Department\\
    IIT Bombay\\
    India
}
\affil[2]{%
    Computer Science Department\\
    University of Victoria\\
    Canada
}

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\begin{document}

\maketitle
 

\begin{abstract}
We study a variant of the experts problem in which new experts are revealed over time according to a stochastic process. The experts are represented by partitions of a hypercube $\mathbb{B}$ in $d$-dimensional Euclidean space. In each round, a point is drawn from $\mathbb{B}$ in an independent and identically distributed manner using an unknown distribution. For each chosen point, we draw $d$ orthogonal hyperplanes parallel to the $d$ faces of $\mathbb{B}$ passing through the point. The set of experts available in a round is the set of partitions of $\mathbb{B}$ created by 
all the hyperplanes drawn up to that point. Losses are adversarial, and 
the performance metrics of interest include expected regret and high probability bounds on the sample-path regret. 
We propose a suitably adapted version of the Hedge algorithm called \algo, which uses a constant learning rate and has $O(\sqrt{2^d T \log T})$ expected regret, which is order-optimal. 
Further, we show that for \algo, there exists a trade-off between choosing a learning rate that has optimal expected regret and a learning rate that leads to a high probability sample-path regret bound. We address this limitation by proposing Ada\algo, a variant of \algo~that uses an adaptive learning rate by tracking the loss of the experts revealed up to that round. Ada\algo~simultaneously achieves $O(\log(\log T)\sqrt{ T \log T})$ expected regret and $O(\log T \sqrt{T \log T})$ sample-path regret, with probability at least $1-T^{-c}$, where $c > 0$ is a constant dependent on $d$.

%\color{red}Keywords \color{black}
%We study the decision-theoretic online learning problem with branching experts, where new experts may be revealed to the learner as the game progresses, and each new expert has a cumulative loss `close' to one of the existing experts and from there on it evolves independently. If the experts are revealed by an adversary and the losses are adversary, Gofer et al. \cite{Gofer13} proposed the Hedge with partial restarts algorithm and obtained the optimal $O(\sqrt{TN_T})$ regret bound, where $N_T$ is the number of experts revealed till time $T$. We study the problem under the stochastic setting where in each round an expert is drawn i.i.d. using a fixed (and unknown) distribution from a continuous and bounded expert-set and is revealed to the learner. We show that \algo, a vanilla extension of the Hedge algorithm, has $\sqrt{T \ln T} $ regret bound. If the expert-set is finite with size $N$, we show a simple modification to \algo~has $O(\sqrt{T \ln N})$ regret, without requiring the losses of the new experts be related to revealed experts, with high probability for `well-behaved' distributions. 
\end{abstract}

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% Acknowledgments---Will not appear in anonymized version
%\acks{This research was supported by NSERC Discovery Grant (54196-52150) and SCPP grant from IIT Bombay, India.}We thank a bunch of people and funding agency
\section*{Acknowledgments}
This research was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) Discovery Grant (54196-52150) and an SCPP grant from IIT Bombay, India.

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