Online Learning Algorithms for Stochastic Water-Filling

Water-filling is the term for the classic solution to the problem of allocating constrained power to a set of parallel channels to maximize the total data-rate. It is used widely in practice, for example, for power allocation to sub-carriers in multi-user OFDM systems such as WiMax. The classic water-filling algorithm is deterministic and requires perfect knowledge of the channel gain to noise ratios. In this paper we consider how to do power allocation over stochastically time-varying (i.i.d.) channels with unknown gain to noise ratio distributions. We adopt an online learning framework based on stochastic multi-armed bandits. We consider two variations of the problem, one in which the goal is to find a power allocation to maximize $\sum\limits_i \mathbb{E}[\log(1 + SNR_i)]$, and another in which the goal is to find a power allocation to maximize $\sum\limits_i \log(1 + \mathbb{E}[SNR_i])$. For the first problem, we propose a \emph{cognitive water-filling} algorithm that we call CWF1. We show that CWF1 obtains a regret (defined as the cumulative gap over time between the sum-rate obtained by a distribution-aware genie and this policy) that grows polynomially in the number of channels and logarithmically in time, implying that it asymptotically achieves the optimal time-averaged rate that can be obtained when the gain distributions are known. For the second problem, we present an algorithm called CWF2, which is, to our knowledge, the first algorithm in the literature on stochastic multi-armed bandits to exploit non-linear dependencies between the arms. We prove that the number of times CWF2 picks the incorrect power allocation is bounded by a function that is polynomial in the number of channels and logarithmic in time, implying that its frequency of incorrect allocation tends to zero.