We propose a distributed learning algorithm for the resource allocation problem in Device-to-Device (D2D) wireless networks that takes into account the throughput estimation noise. We first formulate a stochastic optimization problem with the objective of maximizing the generalized alpha fair function of the network. In order to solve it distributively, we then define and use the framework of noisy-potential games. In this context, we propose a distributed Binary Log-linear Learning Algorithm (BLLA) that converges to a Nash Equilibrium of the resource allocation game, which is also an optimal resource allocation for the optimization problem. A key enabler for the analysis of the convergence are the proposed rules for computation of resistance of trees of perturbed Markov chains. The convergence of BLLA is proved for bounded and unbounded noise, with fixed and decreasing temperature parameter. A sufficient number of estimation samples is also provided that guarantees the convergence to an optimal state. At last, we assess the performance of BLLA by extensive simulations by considering both bounded and unbounded noise cases and we show that BLLA achieves higher sum data rate compared to the state-of-the-art.