Learning Multiuser Channel Allocations in Cognitive Radio Networks: A Combinatorial Multi-Armed Bandit Formulation

We consider the following fundamental problem in the context of channelized dynamic spectrum access. There are $M$ secondary users and $N \ge M$ orthogonal channels. Each secondary user requires a single channel for operation that does not conflict with the channels assigned to the other users. Due to geographic dispersion, each secondary user can potentially see different primary user occupancy behavior on each channel. Time is divided into discrete decision rounds. The throughput obtainable from spectrum opportunities on each user-channel combination over a decision period is modeled as an arbitrarily-distributed non-negative random variable with bounded support but unknown mean, i.i.d. over time. The objective is to search for an allocation of channels for all users that maximizes the expected sum throughput. We formulate this problem as a combinatorial multi-armed bandit (MAB), in which each arm corresponds to a matching of the users to channels. Unlike most prior work on multi-armed bandits, this combinatorial formulation results in dependent arms. Moreover, the number of arms grows super-exponentially as the permutation $P(N,M)$. We present a novel matching-learning algorithm with polynomial storage and polynomial computation per decision period for this problem, and prove that it results in a regret (the gap between the expected sum-throughput obtained by a genie-aided perfect allocation and that obtained by this algorithm) that is uniformly upper-bounded for all time $n$ by a function that grows as $O(M^4 N log n)$, i.e. polynomial in the number of unknown parameters and logarithmic in time. We also discuss how our results provide a non-trivial generalization of known theoretical results on multi-armed bandits.