Long scan times remain a limiting factor in MRI. Accelerated imaging is commonly required, with parallel imaging being the most clinically used approach. Recently, machine learning has also been applied to accelerated MRI reconstruction, where the focus has been on training regularizers on large datasets. In this work, we develop a scan-specific deep learning k-space method for reconstruction of undersampled data. The proposed method, Robust Artificial-neural-networks for k-space Interpolation (RAKI) learns a non-linear convolutional neural network from limited autocalibration signal. Phantom, cardiac and brain data show that RAKI improves upon the reconstruction quality of linear k-space interpolation-based parallel imaging methods.