A Stochastic Quasi-Newton Method for Online Convex Optimization

We develop stochastic variants of the wellknown BFGS quasi-Newton optimization method, in both full and memory-limited (LBFGS) forms, for online optimization of convex functions. The resulting algorithm performs comparably to a well-tuned natural gradient descent but is scalable to very high-dimensional problems. On standard benchmarks in natural language processing, it asymptotically outperforms previous stochastic gradient methods for parameter estimation in conditional random fields. We are working on analyzing the convergence of online (L)BFGS, and extending it to nonconvex optimization problems.

[1]  Ken Brodlie An assessment of two approaches to variable metric methods , 1977, Math. Program..

[2]  H. Robbins,et al.  A Convergence Theorem for Non Negative Almost Supermartingales and Some Applications , 1985 .

[3]  Lee A. Feldkamp,et al.  Decoupled extended Kalman filter training of feedforward layered networks , 1991, IJCNN-91-Seattle International Joint Conference on Neural Networks.

[4]  Martin Fodslette Møller A scaled conjugate gradient algorithm for fast supervised learning , 1993, Neural Networks.

[5]  Ralph Neuneier,et al.  How to Train Neural Networks , 1996, Neural Networks: Tricks of the Trade.

[6]  Nicol N. Schraudolph Local Gain Adaptation in Stochastic Gradient Descent , 1999 .

[7]  Kenji Fukumizu,et al.  Adaptive Method of Realizing Natural Gradient Learning for Multilayer Perceptrons , 2000, Neural Computation.

[8]  Sabine Buchholz,et al.  Introduction to the CoNLL-2000 Shared Task Chunking , 2000, CoNLL/LLL.

[9]  D K Smith Numerical Optimization , 2001, J. Oper. Res. Soc..

[10]  Andrew McCallum,et al.  Conditional Random Fields: Probabilistic Models for Segmenting and Labeling Sequence Data , 2001, ICML.

[11]  Nicol N. Schraudolph,et al.  Fast Curvature Matrix-Vector Products for Second-Order Gradient Descent , 2002, Neural Computation.

[12]  Martial Hebert,et al.  Discriminative Fields for Modeling Spatial Dependencies in Natural Images , 2003, NIPS.

[13]  Nicol N. Schraudolph,et al.  Combining Conjugate Direction Methods with Stochastic Approximation of Gradients , 2003, AISTATS.

[14]  Fernando Pereira,et al.  Shallow Parsing with Conditional Random Fields , 2003, NAACL.

[15]  Nigel Collier,et al.  Introduction to the Bio-entity Recognition Task at JNLPBA , 2004, NLPBA/BioNLP.

[16]  Léon Bottou,et al.  On-line learning for very large data sets , 2005 .

[17]  Mark W. Schmidt,et al.  Accelerated training of conditional random fields with stochastic gradient methods , 2006, ICML.

[18]  H. Robbins A Stochastic Approximation Method , 2007 .