In this paper, we propose a learning-based low-overhead channel estimation method for coordinated beamforming in ultra-dense networks. We first show through simulation that the channel state information (CSI) of geographically separated base stations (BSs) exhibits strong non-linear correlations in terms of mutual information. This finding enables us to adopt a novel learning-based approach to remotely infer the quality of different beamforming patterns at a dense-layer BS based on the CSI of an umbrella control-layer BS. The proposed scheme can reduce channel acquisition overhead by replacing pilot-aided channel estimation with the online inference from an artificial neural network, which is fitted offline. Moreover, we propose to exploit joint learning of multiple CBSs and involve more candidate beam patterns to obtain better performance. Simulation results based on stochastic ray-tracing channel models show that the proposed scheme can reach an accuracy of 99.74% in settings with 20 beamforming patterns.