Maximal-Capacity Discrete Memoryless Channel Identification

The problem of identifying the channel with the highest capacity among several discrete memoryless channels (DMCs) is considered. The problem is cast as a pure-exploration multi-armed bandit problem, which follows the practical use of training sequences to sense the communication channel statistics. A gap-elimination algorithm termed <monospace xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">BestChanID</monospace> is proposed, which is oblivious to the capacity-achieving input distributions, and is guaranteed to output the DMC with the largest capacity, with a desired confidence. Furthermore, two additional algorithms <monospace xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">NaiveChanSel</monospace> and <monospace xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">MedianChanEl</monospace>, which output with certain confidence a DMC with capacity close to the maximal, are also presented. Each of these algorithms is shown to be beneficial in a different regime and can be used as a subroutine of <monospace xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">BestChanID</monospace>. To analyze the algorithms’ guarantees, a capacity estimator is proposed and tight confidence bounds on the estimator error are derived. Based on this estimator, the sample complexity of all the proposed algorithms is analyzed as a function of the desired confidence parameter, the number of channels, and the channels’ input and output alphabet sizes. The cost of best channel identification is shown to scale quadratically with the alphabet size, and a fundamental lower bound is derived on the number of channel senses required to identify the best channel with a certain confidence.