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Conference Paper

Matrix Exponential Gradient Updates for On-line Learning and Bregman Projection

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Tsuda,  K
Department Empirical Inference, Max Planck Institute for Biological Cybernetics, Max Planck Society;
Max Planck Institute for Biological Cybernetics, Max Planck Society;

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Rätsch,  G
Department Empirical Inference, Max Planck Institute for Biological Cybernetics, Max Planck Society;
Max Planck Institute for Biological Cybernetics, Max Planck Society;

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Citation

Tsuda, K., Rätsch, G., & Warmuth, M. (2005). Matrix Exponential Gradient Updates for On-line Learning and Bregman Projection. In L. Saul, Y. Weiss, & L. Bottou (Eds.), Advances in Neural Information Processing Systems 17 (pp. 1425-1432). Cambridge, MA, USA: MIT Press.


Cite as: https://hdl.handle.net/11858/00-001M-0000-0013-D527-4
Abstract
We address the problem of learning a symmetric positive definite matrix. The central issue is to design 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 symmetric linear constraints. The updates generalize the Exponentiated Gradient (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 definite matrix with trace one). The generalized updates use matrix logarithms and exponentials
to preserve positive definiteness. Most importantly, we show how the analysis of each algorithm generalizes to the non-diagonal case. We apply both new algorithms, called the Matrix Exponentiated Gradient (MEG) update and DefiniteBoost, to learn a kernel matrix from distance
measurements.