Approximation guarantees for parallelized maximization of monotone non-submodular function with a cardinality constraint
Emerging applications in machine learning have imposed the problem of monotone non-submodular maximization subject to a cardinality constraint. Meanwhile, parallelism is prevalent for large-scale optimization problems in bigdata scenario while adaptive complexity is an important measurement of paral...
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| Veröffentlicht in: | Journal of combinatorial optimization Jg. 43; H. 5; S. 1671 - 1690 |
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01.07.2022
Springer Nature B.V |
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| Abstract | Emerging applications in machine learning have imposed the problem of monotone non-submodular maximization subject to a cardinality constraint. Meanwhile, parallelism is prevalent for large-scale optimization problems in bigdata scenario while adaptive complexity is an important measurement of parallelism since it quantifies the number of sequential rounds by which the multiple independent functions can be evaluated in parallel. For a monotone non-submodular function and a cardinality constraint, this paper devises an adaptive algorithm for maximizing the function value with the cardinality constraint through employing the generic submodularity ratio
γ
to connect the monotone set function with submodularity. The algorithm achieves an approximation ratio of
1
-
e
-
γ
2
-
ε
and consumes
O
(
log
(
n
/
η
)
/
ε
2
)
adaptive rounds and
O
(
n
log
log
(
k
)
/
ε
3
)
oracle queries in expectation. Furthermore, when
γ
=
1
, the algorithm achieves an approximation guarantee
1
-
1
/
e
-
ε
, achieving the same ratio as the state-of-art result for the submodular version of the problem. |
|---|---|
| AbstractList | Emerging applications in machine learning have imposed the problem of monotone non-submodular maximization subject to a cardinality constraint. Meanwhile, parallelism is prevalent for large-scale optimization problems in bigdata scenario while adaptive complexity is an important measurement of parallelism since it quantifies the number of sequential rounds by which the multiple independent functions can be evaluated in parallel. For a monotone non-submodular function and a cardinality constraint, this paper devises an adaptive algorithm for maximizing the function value with the cardinality constraint through employing the generic submodularity ratio γ to connect the monotone set function with submodularity. The algorithm achieves an approximation ratio of 1-e-γ2-ε and consumes O(log(n/η)/ε2) adaptive rounds and O(nloglog(k)/ε3) oracle queries in expectation. Furthermore, when γ=1, the algorithm achieves an approximation guarantee 1-1/e-ε, achieving the same ratio as the state-of-art result for the submodular version of the problem. Emerging applications in machine learning have imposed the problem of monotone non-submodular maximization subject to a cardinality constraint. Meanwhile, parallelism is prevalent for large-scale optimization problems in bigdata scenario while adaptive complexity is an important measurement of parallelism since it quantifies the number of sequential rounds by which the multiple independent functions can be evaluated in parallel. For a monotone non-submodular function and a cardinality constraint, this paper devises an adaptive algorithm for maximizing the function value with the cardinality constraint through employing the generic submodularity ratio γ to connect the monotone set function with submodularity. The algorithm achieves an approximation ratio of 1 - e - γ 2 - ε and consumes O ( log ( n / η ) / ε 2 ) adaptive rounds and O ( n log log ( k ) / ε 3 ) oracle queries in expectation. Furthermore, when γ = 1 , the algorithm achieves an approximation guarantee 1 - 1 / e - ε , achieving the same ratio as the state-of-art result for the submodular version of the problem. |
| Author | Guo, Longkun Cui, Min Xu, Dachuan Wu, Dan |
| Author_xml | – sequence: 1 givenname: Min surname: Cui fullname: Cui, Min organization: Department of Operations Research and Information Engineering, Beijing University of Technology – sequence: 2 givenname: Dachuan surname: Xu fullname: Xu, Dachuan organization: Department of Operations Research and Information Engineering, Beijing University of Technology – sequence: 3 givenname: Longkun orcidid: 0000-0003-2891-4253 surname: Guo fullname: Guo, Longkun email: longkun.guo@gmail.com organization: Shandong Key Laboratory of Computer Networks, School of Computer Science and Technology, Shandong Computer Science Center, Qilu University of Technology (Shandong Academy of Sciences) – sequence: 4 givenname: Dan surname: Wu fullname: Wu, Dan organization: School of Mathematics and Statistics, Henan University of Science and Technology |
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| Keywords | Non-submodular optimization Submodularity ratio Parallel algorithm Cardinality constraint |
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| References | AttigeriGManoharaPMMRadhikaMPFeature selection using submodular approach for financial big dataJ Inf Process Syst201915613061325 Chekuri C, Quanrud K (2019b) Parallelizing greedy for submodular set function maximization in matroids and beyond. In: Proceedings of the 51st annual ACM SIGACT symposium on theory of computing, pp 78–89 Ene A, Nguyn HL (2020) Parallel algorithm for non-monotone DR-submodular maximization. In: Proceedings of the 37th international conference on machine learning, pp 2902–2911 Lin Y, Chen W, Lui JC (2017) Boosting information spread: an algorithmic approach. In: Proceedings of the international conference on data engineering, pp 883–894 Agarwal A, Assadi S, Khanna S (2019) Stochastic submodular cover with limited adaptivity. In: Proceedings of the 30th annual ACM-SIAM symposium on discrete algorithms, pp 323–342 Balkanski E, Singer Y (2018) The adaptive complexity of maximizing a submodular function. In: Proceedings of the 50th annual ACM SIGACT symposium on theory of computing, pp 1138–1151 Pan X, Jegelka S, Gonzalez JE, Bradley JK, Jordan MI (2014) Parallel double greedy submodular maximization. In: Proceedings of the 27th international conference on neural information processing systems, vol 1, pp 118–126 ChernoffHA measure of asymptotic efficiency for tests of a hypothesis based on the sum of observationsAnn Math Stat19522344935075751810.1214/aoms/1177729330 Mirrokni V, Zadimoghaddam M (2015) Randomized composable core-sets for distributed submodular maximization. In: Proceedings of the 47th annual ACM symposium on theory of computing, pp 153–162 Ene A, Nguyn HL, Vladu A (2019) Submodular maximization with matroid and packing constraints in parallel. In: Proceedings of the 51st annual ACM SIGACT symposium on theory of computing, pp 90–101 Nong Q, Sun T, Gong S, Fang Q, Du D, Shao X (2019) Maximize a monotone function with a generic submodularity ratio. In: Proceedings of the international proceedings on algorithmic applications in management, pp 249–260 TangSBeyond pointwise submodularity: non-monotone adaptive submodular maximization in linear timeTheor Comput Sci2021850249261418303510.1016/j.tcs.2020.11.007 KempeDKleinbergJTardosEMaximizing the spread of influence through a social networkTheory Comput2015111105147333984710.4086/toc.2015.v011a004 NemhauserGLWolseyLABest algorithms for approximating the maximum of a submodular set functionMath Oper Res19783317718850665610.1287/moor.3.3.177 BuchbinderNFeldmanMNaorJSchwartzRA tight linear time (1/2)-approximation for unconstrained submodular maximizationSIAM J Comput201544513841402341614010.1137/130929205 Balkanski E, Singer Y (2020) A lower bound for parallel submodular minimization. In: Proceedings of the 52nd annual ACM SIGACT symposium on theory of computing, pp 130–139 GolovinDKrauseAAdaptive submodularity: theory and applications in active learning and stochastic optimizationJ Artif Intell Res201042142748628748071230.90141 Kuhnle A (2021) Nearly linear-time, parallelizable algorithms for non-monotone submodular maximization. In: Proceedings of the 35th AAAI conference on artificial intelligence Zhang Z, Liu B, Wang Y, Xu D, Zhang D (2019) Greedy algorithm for maximization of non-submodular functions subject to knapsack constraint. In: Proceedings of the international proceedings on computing and combinatorics conference, pp 651–662 Breuer A, Balkanski E, Singer Y (2020) The FAST algorithm for submodular maximization. In: Proceedings of the 37th international conference on machine learning, pp 1134–1143 Chekuri C, Quanrud K (2019a) Submodular function maximization in parallel via the multilinear relaxation. In: Proceedings of the 30th annual ACM-SIAM symposium on discrete algorithms, pp 303–322 FeigeUMirrokniVVondrakJMaximizing non-monotone submodular functionsSIAM J Comput201140411331153282531210.1137/090779346 NemhauserGLWolseyLAFisherMLAn analysis of approximations for maximizing submodular set functions IMath Program197814126529450386610.1007/BF01588971 Fahrbach M, Miller GL, Peng R, Sawlani S, Wang J, Xu SC (2018) Graph sketching against adaptive adversaries applied to the minimum degree algorithm. In: Proceedings of the 59th annual symposium on foundations of computer science, pp 101–112 KawaharaYNaganoKOkamotoYSubmodular fractional programming for balanced clusteringPattern Recogn Lett201142223524310.1016/j.patrec.2010.08.008 AlonNSpencerJThe probabilistic method2000New YorkWiley10.1002/0471722154 Kuhnle A, Smith J, Crawford VG, Thai MT (2018) Fast maximization of non-submodular, monotonic functions on the integer lattice. In: Proceedings of the 35th international proceedings on international conference on machine learning, pp 2786–2795 Lin H, Bilmes JA (2011) A class of submodular functions for document summarization. In: Proceedings of the 49th annual meeting of the association for computational linguistics: human language technologies, vol 1, pp 510–520 Balkanski E, Rubinstein A, Singer Y (2019) An exponential speedup in parallel running time for submodular maximization without loss in approximation. In: Proceedings of the 30th annual ACM-SIAM symposium on discrete algorithms, pp 283–302 Bian AA, Buhmann JM, Krause A, Tschiatschek S (2017) Guarantees for greedy maximization of non-submodular functions with applications. In: Proceedings of the 34th international proceedings on international conference on machine learning, vol 70, pp 498–507 Fahrbach M, Mirrokni V, Zadimoghaddam M (2019) Submodular maximization with nearly optimal approximation, adaptivity and query complexity. In: Proceedings of the 30th annual ACM-SIAM symposium on discrete algorithms, pp 255–273 Parambath SA, Chawla S, Vijayakumar N (2018) SAGA: a submodular greedy algorithm for group recommendation. In: Proceedings of international conference on international conference on artificial intelligence, pp 3900–3908 Ene A, Nguyen HL (2019) Submodular maximization with nearly-optimal approximation and adaptivity in nearly-linear time. In: Proceedings of the 30th annual ACM-SIAM symposium on discrete algorithms, pp 274–282 GongSNongQLiuWFangQParametric monotone function maximization with matroid constraintsJ Global Optim2019753833849401996910.1007/s10898-019-00800-2 Wei K, Iyer R, Bilmes JA (2015) Submodularity in data subset selection and active learning. 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| References_xml | – reference: Fahrbach M, Miller GL, Peng R, Sawlani S, Wang J, Xu SC (2018) Graph sketching against adaptive adversaries applied to the minimum degree algorithm. In: Proceedings of the 59th annual symposium on foundations of computer science, pp 101–112 – reference: Zhang Z, Liu B, Wang Y, Xu D, Zhang D (2019) Greedy algorithm for maximization of non-submodular functions subject to knapsack constraint. In: Proceedings of the international proceedings on computing and combinatorics conference, pp 651–662 – reference: Fahrbach M, Mirrokni V, Zadimoghaddam M (2019) Submodular maximization with nearly optimal approximation, adaptivity and query complexity. In: Proceedings of the 30th annual ACM-SIAM symposium on discrete algorithms, pp 255–273 – reference: NemhauserGLWolseyLAFisherMLAn analysis of approximations for maximizing submodular set functions IMath Program197814126529450386610.1007/BF01588971 – reference: Wei K, Iyer R, Bilmes JA (2015) Submodularity in data subset selection and active learning. In: Proceedings of the 32nd international conference on international conference on machine learning, vol 37, pp 1954–1963 – reference: Kuhnle A, Smith J, Crawford VG, Thai MT (2018) Fast maximization of non-submodular, monotonic functions on the integer lattice. In: Proceedings of the 35th international proceedings on international conference on machine learning, pp 2786–2795 – reference: Ene A, Nguyen HL (2019) Submodular maximization with nearly-optimal approximation and adaptivity in nearly-linear time. In: Proceedings of the 30th annual ACM-SIAM symposium on discrete algorithms, pp 274–282 – reference: Ene A, Nguyn HL (2020) Parallel algorithm for non-monotone DR-submodular maximization. In: Proceedings of the 37th international conference on machine learning, pp 2902–2911 – reference: GongSNongQLiuWFangQParametric monotone function maximization with matroid constraintsJ Global Optim2019753833849401996910.1007/s10898-019-00800-2 – reference: Balkanski E, Rubinstein A, Singer Y (2019) An exponential speedup in parallel running time for submodular maximization without loss in approximation. In: Proceedings of the 30th annual ACM-SIAM symposium on discrete algorithms, pp 283–302 – reference: Breuer A, Balkanski E, Singer Y (2020) The FAST algorithm for submodular maximization. In: Proceedings of the 37th international conference on machine learning, pp 1134–1143 – reference: Mirrokni V, Zadimoghaddam M (2015) Randomized composable core-sets for distributed submodular maximization. In: Proceedings of the 47th annual ACM symposium on theory of computing, pp 153–162 – reference: Lin H, Bilmes JA (2011) A class of submodular functions for document summarization. In: Proceedings of the 49th annual meeting of the association for computational linguistics: human language technologies, vol 1, pp 510–520 – reference: Parambath SA, Chawla S, Vijayakumar N (2018) SAGA: a submodular greedy algorithm for group recommendation. In: Proceedings of international conference on international conference on artificial intelligence, pp 3900–3908 – reference: Agarwal A, Assadi S, Khanna S (2019) Stochastic submodular cover with limited adaptivity. In: Proceedings of the 30th annual ACM-SIAM symposium on discrete algorithms, pp 323–342 – reference: Bian AA, Buhmann JM, Krause A, Tschiatschek S (2017) Guarantees for greedy maximization of non-submodular functions with applications. In: Proceedings of the 34th international proceedings on international conference on machine learning, vol 70, pp 498–507 – reference: Ene A, Nguyn HL, Vladu A (2019) Submodular maximization with matroid and packing constraints in parallel. In: Proceedings of the 51st annual ACM SIGACT symposium on theory of computing, pp 90–101 – reference: GolovinDKrauseAAdaptive submodularity: theory and applications in active learning and stochastic optimizationJ Artif Intell Res201042142748628748071230.90141 – reference: AttigeriGManoharaPMMRadhikaMPFeature selection using submodular approach for financial big dataJ Inf Process Syst201915613061325 – reference: KawaharaYNaganoKOkamotoYSubmodular fractional programming for balanced clusteringPattern Recogn Lett201142223524310.1016/j.patrec.2010.08.008 – reference: FeigeUMirrokniVVondrakJMaximizing non-monotone submodular functionsSIAM J Comput201140411331153282531210.1137/090779346 – reference: AlonNSpencerJThe probabilistic method2000New YorkWiley10.1002/0471722154 – reference: Chekuri C, Quanrud K (2019b) Parallelizing greedy for submodular set function maximization in matroids and beyond. In: Proceedings of the 51st annual ACM SIGACT symposium on theory of computing, pp 78–89 – reference: Balkanski E, Singer Y (2020) A lower bound for parallel submodular minimization. In: Proceedings of the 52nd annual ACM SIGACT symposium on theory of computing, pp 130–139 – reference: Chekuri C, Quanrud K (2019a) Submodular function maximization in parallel via the multilinear relaxation. In: Proceedings of the 30th annual ACM-SIAM symposium on discrete algorithms, pp 303–322 – reference: NemhauserGLWolseyLABest algorithms for approximating the maximum of a submodular set functionMath Oper Res19783317718850665610.1287/moor.3.3.177 – reference: Lin Y, Chen W, Lui JC (2017) Boosting information spread: an algorithmic approach. 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| SubjectTerms | Adaptive algorithms Approximation Combinatorics Convex and Discrete Geometry Machine learning Mathematical analysis Mathematical Modeling and Industrial Mathematics Mathematics Mathematics and Statistics Maximization Operations Research/Decision Theory Optimization Theory of Computation |
| Title | Approximation guarantees for parallelized maximization of monotone non-submodular function with a cardinality constraint |
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