Adaptive subgradient methods for online learning and stochastic optimization. (English) Zbl 1280.68164

Duchi, John; Hazan, Elad; Singer, Yoram

J. Mach. Learn. Res. 12, 2121-2159 (2011).

Summary: 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. Metaphorically, the adaptation allows us to find needles in haystacks in the form of very predictive but rarely seen features. Our paradigm stems from recent advances in stochastic optimization and 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 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 give several efficient algorithms for empirical risk minimization problems with common and important regularization functions and domain constraints. We experimentally study our theoretical analysis and show that adaptive subgradient methods outperform state-of-the-art, yet non-adaptive, subgradient algorithms.

MSC:

68T05	Learning and adaptive systems in artificial intelligence
90C25	Convex programming
90C15	Stochastic programming

Keywords:

subgradient methods; adaptivity; online learning; stochastic convex optimization

Software:

AdaGrad