Stochastic Rank-1 Bandits

We propose stochastic rank-$1$ bandits, a class of online learning problems where at each step a learning agent chooses a pair of row and column arms, and receives the product of their values as a reward. The challenge is that the values of the row and column are unobserved. These values are stochastic and drawn independently of each other. We propose an efficient algorithm for solving our problem, Rank1Elim, and derive a $O((K + L) (1 / \Delta) \log n)$ upper bound on its $n$-step regret, where $K$ is the number of rows, $L$ is the number of columns, and $\Delta$ is the minimum of the row and column gaps. This is the first bandit algorithm for finding the maximum entry of a rank-$1$ matrix whose regret is linear in $K + L$, $1 / \Delta$, and $\log n$. We evaluate our proposed algorithm on both synthetic and real-world problems, and observe that it leverages the structure of our problems and can learn near-optimal solutions even when our modeling assumptions are mildly violated.