JournalofMachineLearningResearch6(2005)995–1018 Submitted 11/04; Revised 4/05; Published 6/05 Matrix Exponentiated Gradient Updates for On-line Learning and Bregman Projection Koji Tsuda
[email protected] Max Planck Institute for Biological Cybernetics Spemannstrasse 38 72076 Tubingen,¨ Germany, and Computational Biology Research Center National Institute of Advanced Science and Technology (AIST) 2-42 Aomi, Koto-ku, Tokyo 135-0064, Japan Gunnar Ratsch¨
[email protected] Friedrich Miescher Laboratory of the Max Planck Society Spemannstrasse 35 72076 Tubingen,¨ Germany Manfred K. Warmuth
[email protected] Computer Science Department University of California Santa Cruz, CA 95064, USA Editor: Yoram Singer Abstract We address the problem of learning a symmetric positive definite matrix. The central issue is to de- sign parameter updates that preserve positive definiteness. Our updates are motivated with the von Neumann divergence. Rather than treating the most general case, we focus on two key applications that exemplify our methods: on-line learning with a simple square loss, and finding a symmetric positive definite matrix subject to linear constraints. The updates generalize the exponentiated gra- dient (EG) update and AdaBoost, respectively: the parameter is now a symmetric positive definite matrix of trace one instead of a probability vector (which in this context is a diagonal positive def- inite matrix with trace one). The generalized updates use matrix logarithms and exponentials to preserve positive definiteness. Most importantly, we show how the derivation and the analyses of the original EG update and AdaBoost generalize to the non-diagonal case. We apply the resulting matrix exponentiated gradient (MEG) update and DefiniteBoost to the problem of learning a kernel matrix from distance measurements.