Adaptive subgradient methods for online learning and stochastic optimization

We present a new family of subgradient methods that dynamically incorporate knowledge of the geometry of the data observed in earlier iterations to perform more informative gradient-based learning. The adaptation, in essence, allows us to find needles in haystacks in the form of very predictive yet rarely observed features. Our paradigm stems from recent advances in online learning which employ proximal functions to control the gradient steps of the algorithm. We describe and analyze an apparatus for adaptively modifying the proximal function, which significantly simplifies the task of setting a learning rate and results in regret guarantees that are provably as good as the best proximal function that can be chosen in hindsight. We corroborate our theoretical results with experiments on a text classification task, showing substantial improvements for classification with sparse datasets.