Stochastic Rank1 Bandits
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Proceedings of the 20th International Conference on Artificial Intelligence and Statistics, PMLR 54:392401, 2017.
Abstract
We propose stochastic rank1 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 main challenge of the problem is that the individual values of the row and column are unobserved. We assume that these values are stochastic and drawn independently. We propose a computationallyefficient algorithm for solving our problem, which we call Rank1Elim. We derive a O((K + L) (1 / Delta) log n) upper bound on its nstep 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; under the assumption that the mean row and column rewards are bounded away from zero. To the best of our knowledge, we present the first bandit algorithm that finds the maximum entry of a rank1 matrix whose regret is linear in K + L, 1 / Delta, and log n. We also derive a nearly matching lower bound. Finally, we evaluate Rank1Elim empirically on multiple problems. We observe that it leverages the structure of our problems and can learn nearoptimal solutions even if our modeling assumptions are mildly violated.
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