Title: Learning Pruning-Friendly Networks via Frank-Wolfe: One-Shot, Any-Sparsity, And No Retraining. Abstract: We present a novel framework to train a large deep neural network (DNN) for only $\textit{once}$, which can then be pruned to $\textit{any sparsity ratio}$ to preserve competitive accuracy $\textit{without any re-training}$. Conventional methods often require (iterative) pruning followed by re-training, which not only incurs large overhead beyond the original DNN training but also can be sensitive to retraining hyperparameters. Our core idea is to re-cast the DNN training as an explicit $\textit{pruning-aware}$ process: that is formulated with an auxiliary $K$-sparse polytope constraint, to encourage network weights to lie in a convex hull spanned by $K$-sparse vectors, potentially resulting in more sparse weight matrices. We then leverage a stochastic Frank-Wolfe (SFW) algorithm to solve this new constrained optimization, which naturally leads to sparse weight updates each time. We further note an overlooked fact that existing DNN initializations were derived to enhance SGD training (e.g., avoid gradient explosion or collapse), but was unaligned with the challenges of training with SFW. We hence also present the first learning-based initialization scheme specifically for boosting SFW-based DNN training. Experiments on CIFAR-10 and Tiny-ImageNet datasets demonstrate that our new framework named $\textbf{SFW-pruning}$ consistently achieves the state-of-the-art performance on various benchmark DNNs over a wide range of pruning ratios. Moreover, SFW-pruning only needs to train once on the same model and dataset, for obtaining arbitrary ratios, while requiring neither iterative pruning nor retraining. All codes will be released to the public. 