# 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} }

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…

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## References

SHOWING 1-10 OF 41 REFERENCES

### Newton Sketch: A Near Linear-Time Optimization Algorithm with Linear-Quadratic Convergence

- Computer Science, MathematicsSIAM J. Optim.
- 2017

A randomized second-order method for optimization known as the Newton Sketch, based on performing an approximate Newton step using a randomly projected or sub-sampled Hessian, is proposed, which has super-linear convergence with exponentially high probability and convergence and complexity guarantees that are independent of condition numbers and related problem-dependent quantities.

### Convergence rates of sub-sampled Newton methods

- Computer ScienceNIPS
- 2015

This paper uses sub-sampling techniques together with low-rank approximation to design a new randomized batch algorithm which possesses comparable convergence rate to Newton's method, yet has much smaller per-iteration cost.

### Adaptivity of averaged stochastic gradient descent to local strong convexity for logistic regression

- Mathematics, Computer ScienceJ. Mach. Learn. Res.
- 2014

After N iterations, with a constant step-size proportional to 1/R2√N where N is the number of observations and R is the maximum norm of the observations, the convergence rate is always of order O(1/ √N), and improves to O(R2/µN), which shows that averaged stochastic gradient is adaptive to unknown local strong convexity of the objective function.

### An optimal randomized incremental gradient method

- Computer ScienceMath. Program.
- 2018

It is shown that the total number of gradient evaluations performed by RPDG can be several times smaller, both in expectation and with high probability, than those performed by deterministic optimal first-order methods under favorable situations.

### Sub-sampled Newton methods

- Computer Science, MathematicsMath. Program.
- 2019

For large-scale finite-sum minimization problems, we study non-asymptotic and high-probability global as well as local convergence properties of variants of Newton’s method where the Hessian and/or…

### Global linear convergence of Newton's method without strong-convexity or Lipschitz gradients

- MathematicsArXiv
- 2018

It is shown that Newton's method converges globally at a linear rate for objective functions whose Hessians are stable, and holds even if an approximate Hessian is used, and if the subproblems are only solved approximately.

### Linear Convergence with Condition Number Independent Access of Full Gradients

- Computer ScienceNIPS
- 2013

This paper proposes to remove the dependence on the condition number by allowing the algorithm to access stochastic gradients of the objective function, and presents a novel algorithm named Epoch Mixed Gradient Descent (EMGD) that is able to utilize two kinds of gradients.

### Accelerated Stochastic Matrix Inversion: General Theory and Speeding up BFGS Rules for Faster Second-Order Optimization

- Computer ScienceNeurIPS
- 2018

This work develops the first accelerated (deterministic and stochastic) quasi-Newton updates, which lead to provably more aggressive approximations of the inverse Hessian, and lead to speed-ups over classical non-accelerated rules in numerical experiments.

### Optimal Rates for the Regularized Least-Squares Algorithm

- Mathematics, Computer ScienceFound. Comput. Math.
- 2007

A complete minimax analysis of the problem is described, showing that the convergence rates obtained by regularized least-squares estimators are indeed optimal over a suitable class of priors defined by the considered kernel.

### Exact and Inexact Subsampled Newton Methods for Optimization

- Computer Science
- 2016

This paper analyzes an inexact Newton method that solves linear systems approximately using the conjugate gradient (CG) method, and that samples the Hessian and not the gradient (the gradient is assumed to be exact).