Freedman's Inequality for Matrix Martingales
Freedman's inequality is a martingale counterpart to Bernstein's inequality. This result shows that the large-deviation behavior of a martingale is controlled by the predictable quadratic variation and a uniform upper bound for the martingale difference sequence. Oliveira has recently established a natural extension of Freedman's inequality that provides tail bounds for the maximum singular value of a matrix-valued martingale. This note describes a different proof of the matrix Freedman inequality that depends on a deep theorem of Lieb from matrix analysis. This argument delivers sharp constants in the matrix Freedman inequality, and it also yields tail bounds for other types of matrix martingales. The new techniques are adapted from recent work by the present author.
© 2011 Institute of Mathematical Statistics. Submitted 15 January 2011, accepted in final form 25 March 2011. Published on: May 23, 2011. Research supported by ONR Award N00014-08-1-0883, DARPA Award N66001-08-1-2065, and AFOSR Award FA9550-09-1-0643. Roberto Oliveira introduced me to Freedman's inequality and encouraged me to apply the methods from [Tro10b] to study the matrix extension of Freedman's result. I would also like to thank Yao- Liang Yu, who pointed out an inconsistency in the proof of Theorem 2.3 and who proposed the argument in Lemma 3.2. Richard Chen and Alex Gittens have helped me root out (numerous) typographic errors.
Published - Tropp2011p14039Electron_Commun_Prob.pdf