Homogeneous Second-Order Descent Framework: A Fast Alternative to Newton-Type Methods Article Swipe

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Chang He , Yuntian Jiang , Chuwen Zhang , Dongdong Ge , Bo Jiang , Yinyu Ye ·

YOU? · · 2023 · Open Access · · DOI: https://doi.org/10.48550/arxiv.2306.17516

This paper proposes a homogeneous second-order descent framework (HSODF) for nonconvex and convex optimization based on the generalized homogeneous model (GHM). In comparison to the Newton steps, the GHM can be solved by extremal symmetric eigenvalue procedures and thus grant an advantage in ill-conditioned problems. Moreover, GHM extends the ordinary homogeneous model (OHM) (Zhang et al. 2022) to allow adaptiveness in the construction of the aggregated matrix. Consequently, HSODF is able to recover some well-known second-order methods, such as trust-region methods and gradient regularized methods, while maintaining comparable iteration complexity bounds. We also study two specific realizations of HSODF. One is adaptive HSODM, which has a parameter-free $O(ε^{-3/2})$ global complexity bound for nonconvex second-order Lipschitz continuous objective functions. The other one is homotopy HSODM, which is proven to have a global linear rate of convergence without strong convexity. The efficiency of our approach to ill-conditioned and high-dimensional problems is justified by some preliminary numerical results.

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Mathematics Mathematical optimization Lipschitz continuity Applied mathematics Eigenvalues and eigenvectors Convexity Hessian matrix Homotopy Convergence (economics) Homogeneous Rate of convergence Matrix (chemical analysis) Upper and lower bounds Computer science Mathematical analysis Combinatorics Pure mathematics Economics Channel (broadcasting) Materials science Physics Economic growth Financial economics Computer network Composite material Quantum mechanics

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Type: preprint
Language: en
Landing Page: http://arxiv.org/abs/2306.17516
PDF: https://arxiv.org/pdf/2306.17516
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DOI: https://doi.org/10.48550/arxiv.2306.17516

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Title: Homogeneous Second-Order Descent Framework: A Fast Alternative to Newton-Type Methods

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Language: en

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Publication year: 2023

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Publication date: 2023-06-30

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Authors: Chang He, Yuntian Jiang, Chuwen Zhang, Dongdong Ge, Bo Jiang, Yinyu Ye

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Concepts: Mathematics, Mathematical optimization, Lipschitz continuity, Applied mathematics, Eigenvalues and eigenvectors, Convexity, Hessian matrix, Homotopy, Convergence (economics), Homogeneous, Rate of convergence, Matrix (chemical analysis), Upper and lower bounds, Computer science, Mathematical analysis, Combinatorics, Pure mathematics, Economics, Channel (broadcasting), Materials science, Physics, Economic growth, Financial economics, Computer network, Composite material, Quantum mechanics

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abstract_inverted_index.approach	142
abstract_inverted_index.extremal	33
abstract_inverted_index.gradient	82
abstract_inverted_index.homotopy	122
abstract_inverted_index.methods,	76, 84
abstract_inverted_index.ordinary	49
abstract_inverted_index.problems	147
abstract_inverted_index.proposes	2
abstract_inverted_index.results.	154
abstract_inverted_index.specific	95
abstract_inverted_index.Lipschitz	114
abstract_inverted_index.Moreover,	45
abstract_inverted_index.advantage	41
abstract_inverted_index.framework	7
abstract_inverted_index.iteration	88
abstract_inverted_index.justified	149
abstract_inverted_index.nonconvex	10, 112
abstract_inverted_index.numerical	153
abstract_inverted_index.objective	116
abstract_inverted_index.problems.	44
abstract_inverted_index.symmetric	34
abstract_inverted_index.aggregated	65
abstract_inverted_index.comparable	87
abstract_inverted_index.comparison	22
abstract_inverted_index.complexity	89, 109
abstract_inverted_index.continuous	115
abstract_inverted_index.convexity.	137
abstract_inverted_index.efficiency	139
abstract_inverted_index.eigenvalue	35
abstract_inverted_index.functions.	117
abstract_inverted_index.procedures	36
abstract_inverted_index.well-known	74
abstract_inverted_index.convergence	134
abstract_inverted_index.generalized	17
abstract_inverted_index.homogeneous	4, 18, 50
abstract_inverted_index.maintaining	86
abstract_inverted_index.preliminary	152
abstract_inverted_index.regularized	83
abstract_inverted_index.adaptiveness	59
abstract_inverted_index.construction	62
abstract_inverted_index.optimization	13
abstract_inverted_index.realizations	96
abstract_inverted_index.second-order	5, 75, 113
abstract_inverted_index.trust-region	79
abstract_inverted_index.Consequently,	67
abstract_inverted_index.$O(ε^{-3/2})$	107
abstract_inverted_index.parameter-free	106
abstract_inverted_index.ill-conditioned	43, 144
abstract_inverted_index.high-dimensional	146
cited_by_percentile_year
countries_distinct_count	0
institutions_distinct_count	6
citation_normalized_percentile