• Corpus ID: 195791848

Globally Convergent Newton Methods for Ill-conditioned Generalized Self-concordant Losses

@inproceedings{MarteauFerey2019GloballyCN,
  title={Globally Convergent Newton Methods for Ill-conditioned Generalized Self-concordant Losses},
  author={Ulysse Marteau-Ferey and Francis R. Bach and Alessandro Rudi},
  booktitle={Neural Information Processing Systems},
  year={2019}
}

• Ulysse Marteau-Ferey, F. Bach, Alessandro Rudi
• Published in
  Neural Information Processing…
  1 July 2019
• Computer Science, Mathematics

In this paper, we study large-scale convex optimization algorithms based on the Newton method applied to regularized generalized self-concordant losses, which include logistic regression and softmax regression. We first prove that our new simple scheme based on a sequence of problems with decreasing regularization parameters is provably globally convergent, that this convergence is linear with a constant factor which scales only logarithmically with the condition number. In the parametric setting, we obtain an algorithm with the same scaling than regular first-order methods but with an improved behavior, in particular in ill-conditioned problems. Second, in the non parametric machine learning setting, we provide an explicit algorithm combining the previous scheme with Nystr{o}m projection techniques, and prove that it achieves optimal generalization bounds with a time complexity of order O(ndf $\lambda$), a memory complexity of order O(df 2 $\lambda$) and no dependence on the condition number, generalizing the results known for least-squares regression. Here n is the number of observations and df $\lambda$ is the associated degrees of freedom. In particular, this is the first large-scale algorithm to solve logistic and softmax regressions in the non-parametric setting with large condition numbers and theoretical guarantees.