  Online learning to rank (ONL2R) is a foundational problem for recommender
systems and has received increasing attention in recent years. Among the
existing approaches for ONL2R, a natural modeling architecture is the
multi-armed bandit framework coupled with the position-based click model.
However, developing efficient online learning policies for MAB-based ONL2R with
position-based click models is highly challenging due to the combinatorial
nature of the problem, and partial observability in the position-based click
model. To date, results in MAB-based ONL2R with position-based click models
remain rather limited, which motivates us to fill this gap in this work. Our
main contributions in this work are threefold: i) We propose the first general
MAB framework that captures all key ingredients of ONL2R with position-based
click models. Our model considers personalized and equal treatments in ONL2R
ranking recommendations, both of which are widely used in practice; ii) Based
on the above analytical framework, we develop two unified greed- and UCB-based
policies called GreedyRank and UCBRank, each of which can be applied to
personalized and equal ranking treatments; and iii) We show that both
GreedyRank and UCBRank enjoy $O(\sqrt{t}\ln t)$ and $O(\sqrt{t\ln t})$ anytime
sublinear regret for personalized and equal treatment, respectively. For the
fundamentally hard equal ranking treatment, we identify classes of collective
utility functions and their associated sufficient conditions under which
$O(\sqrt{t}\ln t)$ and $O(\sqrt{t\ln t})$ anytime sublinear regrets are still
achievable for GreedyRank and UCBRank, respectively. Our numerical experiments
also verify our theoretical results and demonstrate the efficiency of
GreedyRank and UCBRank in seeking the optimal action under various problem
settings.
