Title: Searching for Convolutions and a More Ambitious NAS. Abstract: An important goal of neural architecture search (NAS) is to automate-away the design of neural networks on new tasks in under-explored domains, thus helping to democratize machine learning. However, current NAS research largely focuses on search spaces consisting of existing operations---such as different types of convolution---that are already known to work well on well-studied problems---often in computer vision. Our work is motivated by the following question: can we enable users to build their own search spaces and discover the right neural operations given data from their specific domain? We make progress towards this broader vision for NAS by introducing a space of operations generalizing the convolution that enables search over a large family of parameterizable linear-time matrix-vector functions. Our flexible construction allows users to design their own search spaces adapted to the nature and shape of their data, to warm-start search methods using convolutions when they are known to perform well, or to discover new operations from scratch when they do not. We evaluate our approach on several novel search spaces over vision and text data, on all of which simple NAS search algorithms can find operations that perform better than baseline layers.