  The probability that a user will click a search result depends both on its
relevance and its position on the results page. The position based model
explains this behavior by ascribing to every item an attraction probability,
and to every position an examination probability. To be clicked, a result must
be both attractive and examined. The probabilities of an item-position pair
being clicked thus form the entries of a rank-$1$ matrix. We propose the
learning problem of a Bernoulli rank-$1$ bandit where at each step, the
learning agent chooses a pair of row and column arms, and receives the product
of their Bernoulli-distributed values as a reward. This is a special case of
the stochastic rank-$1$ bandit problem considered in recent work that proposed
an elimination based algorithm Rank1Elim, and showed that Rank1Elim's regret
scales linearly with the number of rows and columns on "benign" instances.
These are the instances where the minimum of the average row and column rewards
$\mu$ is bounded away from zero. The issue with Rank1Elim is that it fails to
be competitive with straightforward bandit strategies as $\mu \rightarrow 0$.
In this paper we propose Rank1ElimKL which simply replaces the (crude)
confidence intervals of Rank1Elim with confidence intervals based on
Kullback-Leibler (KL) divergences, and with the help of a novel result
concerning the scaling of KL divergences we prove that with this change, our
algorithm will be competitive no matter the value of $\mu$. Experiments with
synthetic data confirm that on benign instances the performance of Rank1ElimKL
is significantly better than that of even Rank1Elim, while experiments with
models derived from real data confirm that the improvements are significant
across the board, regardless of whether the data is benign or not.
