This paper addresses the guided depth completion task in which the goal is to predict a dense depth map given a guidance RGB image and sparse depth measurements. Recent advances on this problem nurture hopes that one day we can acquire accurate and dense depth at a very low cost. A major challenge of guided depth completion is to effectively make use of extremely sparse measurements, e.g., measurements covering less than 1% of the image pixels. In this paper, we propose a fully differentiable model that avoids convolving on sparse tensors by jointly learning depth interpolation and refinement. More specifically, we propose a differentiable kernel regression layer that interpolates the sparse depth measurements via learned kernels. We further refine the interpolated depth map using a residual depth refinement layer which leads to improved performance compared to learning absolute depth prediction using a vanilla network. We provide experimental evidence that our differentiable kernel regression layer not only enables end-to-end training from very sparse measurements using standard convolutional network architectures, but also leads to better depth interpolation results compared to existing heuristically motivated methods. We demonstrate that our method outperforms many state-of-the-art guided depth completion techniques on both NYUv2 and KITTI. We further show the generalization ability of our method with respect to the density and spatial statistics of the sparse depth measurements.