Title: Training Neural Networks for and by Interpolation. Abstract: In modern supervised learning, many deep neural networks are able to interpolate the data: the empirical loss can be driven to near zero on all samples simultaneously. In this work, we explicitly exploit this interpolation property for the design of a new optimization algorithm for deep learning. Specifically, we use it to compute an adaptive learning-rate in closed form at each iteration. This results in the Adaptive Learning-rates for Interpolation with Gradients (ALI-G) algorithm. ALI-G retains the main advantage of SGD which is a low computational cost per iteration. But unlike SGD, the learning-rate of ALI-G uses a single constant hyper-parameter and does not require a decay schedule, which makes it considerably easier to tune. We provide convergence guarantees of ALI-G in the stochastic convex setting. Notably, all our convergence results tackle the realistic case where the interpolation property is satisfied up to some tolerance. We provide experiments on a variety of architectures and tasks: (i) learning a differentiable neural computer; (ii) training a wide residual network on the SVHN data set; (iii) training a Bi-LSTM on the SNLI data set; and (iv) training wide residual networks and densely connected networks on the CIFAR data sets. ALI-G produces state-of-the-art results among adaptive methods, and even yields comparable performance with SGD, which requires manually tuned learning-rate schedules. Furthermore, ALI-G is simple to implement in any standard deep learning framework and can be used as a drop-in replacement in existing code.