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

Ulysse Marteau-Ferey 1, 2, 3 Francis Bach 1, 2, 3 Alessandro Rudi 1, 2, 3 
1 SIERRA - Statistical Machine Learning and Parsimony
DI-ENS - Département d'informatique - ENS Paris, CNRS - Centre National de la Recherche Scientifique, Inria de Paris
Abstract : 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 Nyström projection techniques, and prove that it achieves optimal generalization bounds with a time complexity of order O(ndf λ), a memory complexity of order O(df 2 λ) and no dependence on the condition number, generalizing the results known for least-squares regression. Here n is the number of observations and df λ 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.
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Contributor : Ulysse Marteau-Ferey Connect in order to contact the contributor
Submitted on : Tuesday, November 19, 2019 - 6:23:58 PM
Last modification on : Wednesday, June 8, 2022 - 12:50:06 PM


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  • HAL Id : hal-02169626, version 2
  • ARXIV : 1907.01771



Ulysse Marteau-Ferey, Francis Bach, Alessandro Rudi. Globally Convergent Newton Methods for Ill-conditioned Generalized Self-concordant Losses. NeurIPS 2019 - Conference on Neural Information Processing Systems, Dec 2019, Vancouver, Canada. ⟨hal-02169626v2⟩



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