## 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.
@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 LibSGDQN is available on 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 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.