===== SGD-QN: Careful Quasi-Newton Stochastic Gradient Descent =====
//Abstract//:
The SGDQN algorithm is a stochastic gradient descent
algorithm that makes careful use of second-order information
and splits the parameter update into independently scheduled
components. Thanks to this design, SGDQN iterates nearly as
fast as a first-order stochastic gradient
descent but requires less iterations
to achieve the same accuracy.
This algorithm won the "Wild Track" of the first
PASCAL Large Scale Learning Challenge.
//Errata//:
Please see section [[#Errata]] below.
Antoine Bordes, Léon Bottou and Patrick Gallinari: **SGD-QN: Careful Quasi-Newton Stochastic Gradient Descent**, //Journal of Machine Learning Research//, 10:1737--1754, July 2009.
[[http://jmlr.csail.mit.edu/papers/v10/bordes09a.html|JMLR Link]]
[[http://jmlr.csail.mit.edu/papers/v11/bordes10a.html|JMLR Erratum]]
[[http://leon.bottou.org/publications/djvu/jmlr-2009.djvu|jmlr-2009.djvu]]
[[http://leon.bottou.org/publications/pdf/jmlr-2009.pdf|jmlr-2009.pdf]]
[[http://leon.bottou.org/publications/psgz/jmlr-2009.ps.gz|jmlr-2009.ps.gz]]
@article{bordes-bottou-gallinari-2009,
author = {Bordes, Antoine and Bottou, L\'{e}on and Gallinari, Patrick},
title = {SGD-QN: Careful Quasi-Newton Stochastic Gradient Descent},
journal = {Journal of Machine Learning Research},
year = {2009},
volume = {10},
pages = {1737--1754},
month = {July},
url = {http://leon.bottou.org/papers/bordes-bottou-gallinari-2009},
}
==== Implementation ====
The complete source code of
[[http://webia.lip6.fr/~bordes/mywiki/doku.php?id=sgdqn|LibSGDQN]]
is available on
[[http://webia.lip6.fr/~bordes/mywiki/doku.php|Antoine's]] web site.
This source code comes with a script that replicates the
experiments discussed in this paper.
==== Appendix ====
The appendix contains a derivation of upper and lower bounds
on the asymptotic convergence speed of stochastic gradient algorithm.
The constants are exact in the case of second order stochastic gradient.
==== Errata ====
The SGDQN algorithm as described in this paper contains a subtle flaw
described in a subsequent [[:papers:bordes-2010|erratum]].
There is a missing 1/2 factor in the bounds of theorem 1.
\[
\def\w{\mathbf{w}}
{\frac{1}{2}} \frac{{\mathrm tr}(\mathbf{HBGB})}{2\lambda_{\max}-1}\,t^{-1} + {\mathrm o}(t^{-1})
~\leq~ \mathbb{E}_{\sigma}\big[\:{\cal P}_n(\w_t)-{\cal P}_n(\w^*_n)\:\big] ~\leq~
{\frac{1}{2}} \frac{{\mathrm tr}(\mathbf{HBGB})}{2\lambda_{\min}-1}\,t^{-1} + {\mathrm o}(t^{-1})
\]
The version of the paper found on this site contains the correct theorem and proof.