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Deep learning models with convolutional and recurrent networks are now ubiquitous and analyze massive amounts of audio, image, video, text and graph data, with applications in automatic translation, speech-to-text, scene understanding, ranking user preferences, ad placement, etc. Competing frameworks for building these networks such as TensorFlow, Chainer, CNTK, Torch/PyTorch, Caffe1/2, MXNet and Theano, explore different tradeoffs between usability and expressiveness, research or production orientation and supported hardware. They operate on a DAG of computational operators, wrapping high-performance libraries such as CUDNN (for NVIDIA GPUs) or NNPACK (for various CPUs), and automate memory allocation, synchronization, distribution. Custom operators are needed where the computation does not fit existing high-performance library calls, usually at a high engineering cost. This is frequently required when new operators are invented by researchers: such operators suffer a severe performance penalty, which limits the pace of innovation. Furthermore, even if there is an existing runtime call these frameworks can use, it often doesn't offer optimal performance for a user's particular network architecture and dataset, missing optimizations between operators as well as optimizations that can be done knowing the size and shape of data. Our contributions include (1) a language close to the mathematics of deep learning called Tensor Comprehensions, (2) a polyhedral Just-In-Time compiler to convert a mathematical description of a deep learning DAG into a CUDA kernel with delegated memory management and synchronization, also providing optimizations such as operator fusion and specialization for specific sizes, (3) a compilation cache populated by an autotuner. [Abstract cutoff]