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### Abstract

This paper focuses on an Adaptive Cubic Regularisation (ARC) method for

approximating a second-order critical point of a finite sum minimisation problem.

The variant presented belongs to the framework of Bellavia, Gurioli, Morinin

and Toint (2020): it employs random models with accuracy guaranteed with a

sufficiently large prefixed probability and deterministic inexact function

evaluations within a prescribed level of accuracy. Without assuming unbiased

estimators, the expected number of iterations is O( epsilon_1^{-3/2} ) or O( \max[ epsilon_1^{-3/2},epsilon_2^{-3} ] ) when searching for a first- or second-order critical point, respectively, where epsilon_j is the jth-order tolerance. These results match the worst-case optimal complexity for the deterministic counterpart of the method.

approximating a second-order critical point of a finite sum minimisation problem.

The variant presented belongs to the framework of Bellavia, Gurioli, Morinin

and Toint (2020): it employs random models with accuracy guaranteed with a

sufficiently large prefixed probability and deterministic inexact function

evaluations within a prescribed level of accuracy. Without assuming unbiased

estimators, the expected number of iterations is O( epsilon_1^{-3/2} ) or O( \max[ epsilon_1^{-3/2},epsilon_2^{-3} ] ) when searching for a first- or second-order critical point, respectively, where epsilon_j is the jth-order tolerance. These results match the worst-case optimal complexity for the deterministic counterpart of the method.

Original language | English |
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Publication status | Accepted/In press - 2020 |

Event | Thirty-seventh International Conference on Machine Learning: ICML2020 - Duration: 13 Jul 2020 → 18 Jul 2020 |

### Conference

Conference | Thirty-seventh International Conference on Machine Learning |
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Period | 13/07/20 → 18/07/20 |

### Keywords

- Stochastic optimization
- Complexity theory

## Fingerprint Dive into the research topics of 'A stochastic ARC method with inexact function and random derivatives evaluations'. Together they form a unique fingerprint.

## Projects

- 2 Active

## Complexity in nonlinear optimization

TOINT, P., Gould, N. I. M. & Cartis, C.

1/11/08 → …

Project: Research

## Cite this

Toint, P. L. (Accepted/In press).

*A stochastic ARC method with inexact function and random derivatives evaluations*. Paper presented at Thirty-seventh International Conference on Machine Learning, .