Stochastic variance-reduced prox-linear algorithms for nonconvex composite optimization

We consider the problem of minimizing composite functions of the form f ( g ( x ) ) + h ( x ) , where f and h are convex functions (which can be nonsmooth) and g is a smooth vector mapping. In addition, we assume that g is the average of finite number of component mappings or the expectation over...

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Published in:	Mathematical programming Vol. 195; no. 1-2; pp. 649 - 691
Main Authors:	Zhang, Junyu, Xiao, Lin
Format:	Journal Article
Language:	English
Published:	Berlin/Heidelberg Springer Berlin Heidelberg 01.09.2022 Springer Springer Nature B.V
Subjects:	Algorithms Calculus of Variations and Optimal Control; Optimization Combinatorics Composite functions Full Length Paper Jacobians Mapping Mathematical and Computational Physics Mathematical Methods in Physics Mathematics Mathematics and Statistics Mathematics of Computing Numerical Analysis Optimization Theoretical Variance 68W20 68Q25 Variance reduction Stochastic composite optimization Nonsmooth optimization 90C26 Sample complexity Prox-linear algorithm
ISSN:	0025-5610, 1436-4646
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Abstract	We consider the problem of minimizing composite functions of the form f ( g ( x ) ) + h ( x ) , where f and h are convex functions (which can be nonsmooth) and g is a smooth vector mapping. In addition, we assume that g is the average of finite number of component mappings or the expectation over a family of random component mappings. We propose a class of stochastic variance-reduced prox-linear algorithms for solving such problems and bound their sample complexities for finding an ϵ -stationary point in terms of the total number of evaluations of the component mappings and their Jacobians. When g is a finite average of N components, we obtain sample complexity O ( N + N 4 / 5 ϵ - 1 ) for both mapping and Jacobian evaluations. When g is a general expectation, we obtain sample complexities of O ( ϵ - 5 / 2 ) and O ( ϵ - 3 / 2 ) for component mappings and their Jacobians respectively. If in addition f is smooth, then improved sample complexities of O ( N + N 1 / 2 ϵ - 1 ) and O ( ϵ - 3 / 2 ) are derived for g being a finite average and a general expectation respectively, for both component mapping and Jacobian evaluations.
AbstractList	We consider the problem of minimizing composite functions of the form [Formula omitted], where f and h are convex functions (which can be nonsmooth) and g is a smooth vector mapping. In addition, we assume that g is the average of finite number of component mappings or the expectation over a family of random component mappings. We propose a class of stochastic variance-reduced prox-linear algorithms for solving such problems and bound their sample complexities for finding an [Formula omitted]-stationary point in terms of the total number of evaluations of the component mappings and their Jacobians. When g is a finite average of N components, we obtain sample complexity [Formula omitted] for both mapping and Jacobian evaluations. When g is a general expectation, we obtain sample complexities of [Formula omitted] and [Formula omitted] for component mappings and their Jacobians respectively. If in addition f is smooth, then improved sample complexities of [Formula omitted] and [Formula omitted] are derived for g being a finite average and a general expectation respectively, for both component mapping and Jacobian evaluations. We consider the problem of minimizing composite functions of the form f ( g ( x ) ) + h ( x ) , where f and h are convex functions (which can be nonsmooth) and g is a smooth vector mapping. In addition, we assume that g is the average of finite number of component mappings or the expectation over a family of random component mappings. We propose a class of stochastic variance-reduced prox-linear algorithms for solving such problems and bound their sample complexities for finding an ϵ -stationary point in terms of the total number of evaluations of the component mappings and their Jacobians. When g is a finite average of N components, we obtain sample complexity O ( N + N 4 / 5 ϵ - 1 ) for both mapping and Jacobian evaluations. When g is a general expectation, we obtain sample complexities of O ( ϵ - 5 / 2 ) and O ( ϵ - 3 / 2 ) for component mappings and their Jacobians respectively. If in addition f is smooth, then improved sample complexities of O ( N + N 1 / 2 ϵ - 1 ) and O ( ϵ - 3 / 2 ) are derived for g being a finite average and a general expectation respectively, for both component mapping and Jacobian evaluations. We consider the problem of minimizing composite functions of the form f(g(x))+h(x), where f and h are convex functions (which can be nonsmooth) and g is a smooth vector mapping. In addition, we assume that g is the average of finite number of component mappings or the expectation over a family of random component mappings. We propose a class of stochastic variance-reduced prox-linear algorithms for solving such problems and bound their sample complexities for finding an ϵ-stationary point in terms of the total number of evaluations of the component mappings and their Jacobians. When g is a finite average of N components, we obtain sample complexity O(N+N4/5ϵ-1) for both mapping and Jacobian evaluations. When g is a general expectation, we obtain sample complexities of O(ϵ-5/2) and O(ϵ-3/2) for component mappings and their Jacobians respectively. If in addition f is smooth, then improved sample complexities of O(N+N1/2ϵ-1) and O(ϵ-3/2) are derived for g being a finite average and a general expectation respectively, for both component mapping and Jacobian evaluations.
Audience	Academic
Author	Xiao, Lin Zhang, Junyu
Author_xml	– sequence: 1 givenname: Junyu surname: Zhang fullname: Zhang, Junyu organization: Department of Electrical Engineering, Princeton University – sequence: 2 givenname: Lin orcidid: 0000-0002-9759-3898 surname: Xiao fullname: Xiao, Lin email: linx@fb.com organization: Facebook AI Research (FAIR)
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Cites_doi	10.1093/imaiai/iay015 10.1137/140961791 10.1137/17M1151031 10.21314/JOR.2000.038 10.1007/BF01584377 10.1137/19M1285457 10.1007/BF01585997 10.1137/15M1031953 10.1080/08927020600643812 10.1007/s10107-015-0943-9 10.1137/18M1178244 10.1137/11082381X 10.1007/s10107-016-1017-3 10.1073/pnas.1908018116 10.1007/s10107-018-1311-3 10.1109/TAC.2012.2215413 10.1057/palgrave.jors.2600425 10.1007/BF00933293 10.1007/s10957-018-01452-0 10.1137/18M1230323 10.1007/BF01588325 10.1137/18M1230542 10.1137/17M1135086 10.1287/educ.1073.0032 10.1287/educ.2013.0110 10.1007/s10589-020-00179-x 10.24963/ijcai.2017/470 10.1609/aaai.v32i1.11795 10.1287/moor.2017.0889 10.1007/978-3-662-39921-7_17
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Snippet	We consider the problem of minimizing composite functions of the form f ( g ( x ) ) + h ( x ) , where f and h are convex functions (which can be nonsmooth)... We consider the problem of minimizing composite functions of the form [Formula omitted], where f and h are convex functions (which can be nonsmooth) and g is a... We consider the problem of minimizing composite functions of the form f(g(x))+h(x), where f and h are convex functions (which can be nonsmooth) and g is a...
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SubjectTerms	Algorithms Calculus of Variations and Optimal Control; Optimization Combinatorics Composite functions Full Length Paper Jacobians Mapping Mathematical and Computational Physics Mathematical Methods in Physics Mathematics Mathematics and Statistics Mathematics of Computing Numerical Analysis Optimization Theoretical Variance
Title	Stochastic variance-reduced prox-linear algorithms for nonconvex composite optimization
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