# Newton-type methods for non-convex optimization under inexact Hessian information

@article{Xu2017NewtontypeMF, title={Newton-type methods for non-convex optimization under inexact Hessian information}, author={Peng Xu and Farbod Roosta-Khorasani and Michael W. Mahoney}, journal={Mathematical Programming}, year={2017}, pages={1-36} }

We consider variants of trust-region and adaptive cubic regularization methods for non-convex optimization, in which the Hessian matrix is approximated. Under certain condition on the inexact Hessian, and using approximate solution of the corresponding sub-problems, we provide iteration complexity to achieve $$\varepsilon $$ε-approximate second-order optimality which have been shown to be tight. Our Hessian approximation condition offers a range of advantages as compared with the prior works…

## 151 Citations

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- 2020

This work proposes a new trust-region method which employs suitable approximations of the objective function, gradient and Hessian built via random subsampling techniques and shows that the new procedure is more efficient, in terms of overall computational cost, than the standard trust- region scheme with subsampled Hessians.

### Convergence of Newton-MR under Inexact Hessian Information

- Computer Science, MathematicsSIAM J. Optim.
- 2021

This work draws from matrix perturbation theory to estimate the distance between the subspaces underlying the exact and approximate Hessian matrices in Newton-MR, which extends the application range of the classical Newton-CG beyond convexity to invex problems.

### Stochastic Second-order Methods for Non-convex Optimization with Inexact Hessian and Gradient

- Computer ScienceArXiv
- 2018

This paper studies a family of stochastic trust region and cubic regularization methods when gradient, Hessian and function values are computed inexactly, and shows the iteration complexity to achieve $\epsilon$-approximate second-order optimality is in the same order with previous work for which gradient and functionvalues are computed exactly.

### Adaptively Accelerating Cubic Regularized Newton's Methods for Convex Optimization via Random Sampling

- Computer Science
- 2018

This paper proposes to compute an approximated Hessian matrix by either uniform or non-uniformly sub-sampling the components of the objective, and develops accelerated adaptive cubic regularization approaches that provide theoretical guarantees on global iteration complexity of O(\epsilon^{-1/3}) with high probability.

### Accelerating Adaptive Cubic Regularization of Newton's Method via Random Sampling

- Computer Science, Mathematics
- 2018

This paper proposes to compute an approximated Hessian matrix by either uniformly or non-uniformly sub-sampling the components of the objective and develops accelerated adaptive cubic regularization approaches, which provide theoretical guarantees on global iteration complexity of O (cid:15) − 1 / 3 ) with high probability.

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- Computer ScienceIMA Journal of Numerical Analysis
- 2022

This approach is a first attempt to introduce inexact Hessian and/or gradient information into the Newton-CG algorithm of Royer & Wright, and derives iteration complexity bounds for achieving $\epsilon $-approximate second-order optimality that match best-known lower bounds.

### First-Order Methods for Nonconvex Quadratic Minimization

- Mathematics, Computer ScienceSIAM Rev.
- 2020

When the authors use Krylov subspace solutions to approximate the cubic-regularized Newton step, the results recover the strongest known convergence guarantees to approximate second-order stationary points of general smooth nonconvex functions.

### Stochastic analysis of an adaptive cubic regularization method under inexact gradient evaluations and dynamic Hessian accuracy

- Computer ScienceOptimization
- 2020

An extended version of the adaptive cubic regularization method with dynamic inexact Hessian information for nonconvex optimization inherits the innovative use of adaptive accuracy requirements for Hessian approximations introduced in the just quoted paper and additionally employs inexact computations of the gradient.

### Cubic Regularization Methods with Second-Order Complexity Guarantee Based on a New Subproblem Reformulation

- Computer Science, MathematicsJournal of the Operations Research Society of China
- 2022

A new reformulation of the cubic regularization subproblem is proposed, an unconstrained convex problem that requires computing the minimum eigenvalue of the Hessian and a variant of adaptive regularization with cubics (ARC) is derived.

### WORST-CASE EVALUATION COMPLEXITY AND OPTIMALITY OF SECOND-ORDER METHODS FOR NONCONVEX SMOOTH OPTIMIZATION

- Computer ScienceProceedings of the International Congress of Mathematicians (ICM 2018)
- 2019

A new general class of inexact second-order algorithms for unconstrained optimization that includes regularization and trust-region variations of Newton's method as well as of their linesearch variants is considered, implying that these methods have optimal worst-case evaluation complexity within a wider class of second- order methods, and that Newton'smethod is suboptimal.

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