Title: Statistical Adaptive Stochastic Optimization. Abstract: We investigate statistical methods for automatically scheduling the learning rate (step size) in stochastic optimization. First, we consider a broad family of stochastic optimization methods with constant hyperparameters (including the learning rate and various forms of momentum) and derive a general necessary condition for the resulting dynamics to be stationary. Based on this condition, we develop a simple online statistical test to detect (non-)stationarity and use it to automatically drop the learning rate by a constant factor whenever stationarity is detected. Unlike in prior work, our stationarity condition and our statistical test applies to different algorithms without modification. Finally, we propose a smoothed stochastic line-search method that can be used to warm up the optimization process before the statistical test can be applied effectively. This removes the expensive trial and error for setting a good initial learning rate. The combined method is highly autonomous and it attains state-of-the-art training and testing performance in our experiments on several deep learning tasks.