Successive Reduction of Arms in Multi-Armed Bandits

The relevance of the multi-armed bandit problem has risen in the past few years with the need for online optimization techniques in Internet systems, such as online advertisement and news article recommendation. At the same time, these applications reveal that state-of-the-art solution schemes do not scale well with the number of bandit arms. In this paper, we present two types of Successive Reduction (SR) strategies - 1) Successive Reduction Hoeffding (SRH) and 2) Successive Reduction Order Statistics (SRO). Both use an Order Statistics based Thompson Sampling method for arm selection, and then successively eliminate bandit arms from consideration based on a confidence threshold. While SRH uses Hoeffding Bounds for elimination, SRO uses the probability of an arm being superior to the currently selected arm to measure confidence. A computationally efficient scheme for pair-wise calculation of the latter probability is also presented in this paper. Using SR strategies, sampling resources and arm pulls are not wasted on arms that are unlikely to be the optimal one. To demonstrate the scalability of our proposed schemes, we compare them with two state-of-the-art approaches, namely pure Thompson Sampling and UCB-Tuned. The empirical results are truly conclusive, with the performance advantage of proposed SRO scheme increasing persistently with the number of bandit arms while the SRH scheme shows similar performance as pure Thompson Sampling. We thus believe that SR algorithms will open up for improved performance in Internet based on-line optimization, and tackling of larger problems.