{"type":"ScholarlyArticle","title":"Non-monotone DR-submodular Maximization over General Convex Sets","authors":[{"name":"Christoph Dürr","slug":"christoph-durr","given":"Christoph","family":"Dürr","affiliation":"LIP6 , Sorbonne University , France"},{"name":"Nguyen Kim Thang","slug":"nguyen-kim-thang","given":"Nguyen","family":"Kim Thang","affiliation":"IBISC , Univ Évry , University Paris-Saclay , France"},{"name":"Abhinav Srivastav","slug":"abhinav-srivastav","given":"Abhinav","family":"Srivastav","affiliation":"IBISC , Univ Évry , University Paris-Saclay , France"},{"name":"Léo Tible","slug":"leo-tible","given":"Léo","family":"Tible","affiliation":"IBISC , Univ Évry , University Paris-Saclay , France"}],"abstract":"Many real-world problems can often be cast as the optimization of DR-submodular functions defined over a convex domain. These functions play an important role with applications in many areas of applied mathematics, such as machine learning, computer vision, operation research, communication systems or economics. In addition, they capture a subclass of non-convex optimization that provides both practical and theoretical guarantees.\r\nIn this paper, we show that for maximizing non-monotone DR-submodular functions over a general convex set (such as up-closed convex sets, conic convex set, etc) the Frank-Wolfe algorithm achieves an approximation guarantee which depends on the convex set. To the best of our knowledge, this is the first approximation guarantee. Finally we benchmark our algorithm on problems arising in machine learning domain with the real-world datasets.","doi":"10.24963/ijcai.2020/297","url":"https://ijcai.org/proceedings/2020/297","pdf":"https://ijcai.org/proceedings/2020/0297.pdf","markdown":"https://ijcai.org/proceedings/2020/297.md","bibtex":"https://ijcai.org/proceedings/2020/bibtex/297","volume":{"slug":"2020","title":"Proceedings of the Twenty-Ninth International Joint Conference on Artificial Intelligence","short_title":"IJCAI 2020","url":"https://ijcai.org/proceedings/2020/","year":2020,"location":"Yokohama","dates":"January 2021 (postponed from 11-17 July 2020)","editors":"Christian Bessiere","isbn":"978-0-9992411-6-5","publisher":"International Joint Conferences on Artificial Intelligence Organization","doi_prefix":"10.24963/ijcai.2020"},"section":"Machine Learning","track":"Main track","number":297,"pages":{"first":2148,"last":2154},"keywords":["Machine Learning: Online Learning","Machine Learning: Learning Theory"],"videos":[{"label":"Video","url":"https://videos.ijcai20.org/24384/file_24384.m3u8","length_sec":263,"provider":"hls"}],"references":[{"position":1,"raw":"[Bach and others, 2013] Francis Bach et al. Learning with submodular functions: A convex optimization perspec- tive. Foundations and Trends® in Machine Learning, 6(2- 3):145-373, 2013.","title":"Learning with submodular functions: A convex optimization perspective","authors":"Francis Bach","year":"2013","venue":"Foundations and Trends® in Machine Learning","doi":null,"url":null,"internal_url":null},{"position":2,"raw":"[Bach, 2016] Francis Bach. Submodular functions: from dis- crete to continuous domains. Mathematical Programming, pages 1-41, 2016.","title":"Submodular functions: from discrete to continuous domains","authors":"Francis Bach","year":"2016","venue":"Mathematical Programming","doi":null,"url":null,"internal_url":null},{"position":3,"raw":"[Bian et al., 2017a] Andrew An Bian, Kfir Levy, Andreas Krause, and Joachim M. Buhmann. Continuous DR- submodular maximization: Structure and algorithms. In NeurIPS, 2017.","title":"Continuous DRsubmodular maximization: Structure and algorithms","authors":"Bian","year":"2017","venue":"NeurIPS","doi":null,"url":null,"internal_url":null},{"position":4,"raw":"[Bian et al., 2017b] Andrew An Bian, Baharan Mirza- soleiman, Joachim Buhmann, and Andreas Krause. Guar- anteed non-convex optimization: Submodular maximiza- tion over continuous domains. In AISTATS, pages 111- 120, 2017.","title":"Guaranteed non-convex optimization: Submodular maximiza-tion over continuous domains","authors":"Bian","year":"2017","venue":"AISTATS","doi":null,"url":null,"internal_url":null},{"position":5,"raw":"[Bian et al., 2018] An Bian, Joachim M Buhmann, and An- dreas Krause. Optimal DR-submodular maximization and applications to provable mean field inference. In NeurIPS, 2018.","title":"Optimal DR-submodular maximization and applications to provable mean field inference","authors":"Bian","year":"2018","venue":"NeurIPS","doi":null,"url":null,"internal_url":null},{"position":6,"raw":"[Buchbinder and Feldman, 2018] Niv Buchbinder and Moran Feldman. Deterministic algorithms for submodular maximization problems. ACM TALG, 14(3):32, 2018.","title":"Deterministic algorithms for submodular maximization problems","authors":"Feldman Buchbinder, Niv Buchbinder, Moran Feldman","year":"2018","venue":"ACM TALG","doi":null,"url":null,"internal_url":null},{"position":7,"raw":"[Buchbinder et al., 2015] Niv Buchbinder, Moran Feldman, Joseph Seffi, and Roy Schwartz. A tight linear time (1/2)- approximation for unconstrained submodular maximiza- tion. SIAM Computing, 44(5):1384-1402, 2015.","title":"A tight linear time (1/2)approximation for unconstrained submodular maximization","authors":"Buchbinder","year":"2015","venue":"SIAM Computing","doi":null,"url":null,"internal_url":null},{"position":8,"raw":"[Calinescu et al., 2011] Gruia Calinescu, Chandra Chekuri, Martin Pál, and Jan Vondrák. Maximizing a monotone submodular function subject to a matroid constraint. SIAM J. Computing, 40(6):1740-1766, 2011.","title":"Maximizing a monotone submodular function subject to a matroid constraint","authors":"Gruia Calinescu, Chandra Chekuri, Martin Pál, Jan Vondrák","year":"2011","venue":"SIAM J. Computing","doi":null,"url":null,"internal_url":null},{"position":9,"raw":"[Chen et al., 2018] Lin Chen, Hamed Hassani, and Amin Karbasi. Online continuous submodular maximization. In AISTAT, 2018. [Djolonga and Krause, 2014] Josip Djolonga and Andreas Krause. From map to marginals: Variational inference in bayesian submodular models. In NeurIPS, pages 244-252, 2014.","title":"From map to marginals: Variational inference in bayesian submodular models","authors":"Lin Chen, Hamed Hassani, Amin Karbasi","year":"2014","venue":"Josip Djolonga and Andreas Krause","doi":null,"url":null,"internal_url":null},{"position":10,"raw":"[Djolonga et al., 2016] Josip Djolonga, Sebastian Tschi- atschek, and Andreas Krause. Variational inference in mixed probabilistic submodular models. In NeurIPS, pages 1759-1767, 2016.","title":"Josip Djolonga, Sebastian Tschiatschek, and Andreas Krause. Variational inference in mixed probabilistic submodular models","authors":"Djolonga","year":"2016","venue":"NeurIPS","doi":null,"url":null,"internal_url":null},{"position":11,"raw":"[Elenberg et al., 2017] Ethan Elenberg, Alexandros G Di- makis, Moran Feldman, and Amin Karbasi. Streaming weak submodularity: Interpreting neural networks on the fly. In NeurIPS, pages 4044-4054, 2017.","title":"Streaming weak submodularity: Interpreting neural networks on the fly","authors":"Ethan Elenberg, Alexandros G Dimakis, Moran Feldman, Amin Karbasi","year":"2017","venue":"NeurIPS","doi":null,"url":null,"internal_url":null},{"position":12,"raw":"[Elenberg et al., 2018] Ethan R Elenberg, Rajiv Khanna, Alexandros G Dimakis, Sahand Negahban, et al. Re- stricted strong convexity implies weak submodularity. The Annals of Statistics, 46(6B):3539-3568, 2018.","title":"Restricted strong convexity implies weak submodularity","authors":"Rajiv Ethan R Elenberg, Alexandros G Khanna, Sahand Dimakis, Negahban","year":"2018","venue":"The Annals of Statistics","doi":null,"url":null,"internal_url":null},{"position":13,"raw":"[Feldman et al., 2011] Moran Feldman, Joseph Naor, and Roy Schwartz. A unified continuous greedy algorithm for submodular maximization. In FOCS, pages 570-579, 2011.","title":"A unified continuous greedy algorithm for submodular maximization","authors":"Moran Feldman, Joseph Naor, Roy Schwartz","year":"2011","venue":"FOCS","doi":null,"url":null,"internal_url":null},{"position":14,"raw":"[Golovin and Krause, 2011] Daniel Golovin and Andreas Krause. Adaptive submodularity: Theory and applications in active learning and stochastic optimization. J. of AI Re- search, 42:427-486, 2011.","title":"Adaptive submodularity: Theory and applications in active learning and stochastic optimization","authors":"Golovin, Krause, Daniel Golovin, Andreas Krause","year":"2011","venue":"J. of AI Research","doi":null,"url":null,"internal_url":null},{"position":15,"raw":"[Gomez Rodriguez et al., 2010] Manuel Gomez Rodriguez, Jure Leskovec, and Andreas Krause. Inferring networks of diffusion and influence. In SIGKDD, pages 1019-1028, 2010.","title":"Inferring networks of diffusion and influence","authors":"Manuel Gomez Rodriguez, Jure Leskovec, Andreas Krause","year":"2010","venue":"SIGKDD","doi":null,"url":null,"internal_url":null},{"position":16,"raw":"[Guillory and Bilmes, 2011] Andrew Guillory and Jeff A Bilmes. Simultaneous learning and covering with adver- sarial noise. In ICML, volume 11, pages 369-376, 2011.","title":"Simultaneous learning and covering with adversarial noise","authors":"Bilmes Guillory, Andrew Guillory, Jeff A Bilmes","year":"2011","venue":"ICML","doi":null,"url":null,"internal_url":null},{"position":17,"raw":"[Hassani et al., 2017] Hamed Hassani, Mahdi Soltanolkotabi, and Amin Karbasi. Gradient meth- ods for submodular maximization. In NeurIPS, pages 5841-5851, 2017.","title":"Gradient methods for submodular maximization","authors":"Hamed Hassani, Mahdi Soltanolkotabi, Amin Karbasi","year":"2017","venue":"NeurIPS","doi":null,"url":null,"internal_url":null},{"position":18,"raw":"[Ito and Fujimaki, 2016] Shinji Ito and Ryohei Fujimaki. Large-scale price optimization via network flow. In NeurIPS, pages 3855-3863, 2016.","title":"Large-scale price optimization via network flow","authors":"Fujimaki ; ] Shinji Ito, Ryohei Ito, Fujimaki","year":"2016","venue":"NeurIPS","doi":null,"url":null,"internal_url":null},{"position":19,"raw":"[Iwata et al., 2001] Satoru Iwata, Lisa Fleischer, and Satoru Fujishige. A combinatorial strongly polynomial algorithm for minimizing submodular functions. J. ACM, 48(4):761- 777, 2001.","title":"A combinatorial strongly polynomial algorithm for minimizing submodular functions","authors":"Satoru Iwata, Lisa Fleischer, Satoru Fujishige","year":"2001","venue":"J. ACM","doi":null,"url":null,"internal_url":null},{"position":20,"raw":"[Kempe et al., 2003] David Kempe, Jon Kleinberg, and Éva Tardos. Maximizing the spread of influence through a so- cial network. In SIGKDD, pages 137-146, 2003.","title":"Maximizing the spread of influence through a social network","authors":"David Kempe, Jon Kleinberg, Éva Tardos","year":"2003","venue":"SIGKDD","doi":null,"url":null,"internal_url":null},{"position":21,"raw":"[Kulesza et al., 2012] Alex Kulesza, Ben Taskar, et al. De- terminantal point processes for machine learning. Founda- tions and Trends® in Machine Learning, 5(2-3):123-286, 2012.","title":"Determinantal point processes for machine learning","authors":"Alex Kulesza, Ben Taskar","year":"2012","venue":"Foundations and Trends® in Machine Learning","doi":null,"url":null,"internal_url":null},{"position":22,"raw":"[Lin and Bilmes, 2011] Hui Lin and Jeff Bilmes. A class of submodular functions for document summarization. In ACL, pages 510-520, 2011.","title":"A class of submodular functions for document summarization","authors":"Hui Lin, Jeff Bilmes","year":"2011","venue":"ACL","doi":null,"url":null,"internal_url":null},{"position":23,"raw":"[Niazadeh et al., 2018] Rad Niazadeh, Tim Roughgarden, and Joshua R Wang. Optimal algorithms for continuous non-monotone submodular and dr-submodular maximiza- tion. In NeurIPS, 2018.","title":"Optimal algorithms for continuous non-monotone submodular and dr-submodular maximization","authors":"Rad Niazadeh, Tim Roughgarden, Joshua R Wang","year":"2018","venue":"NeurIPS","doi":null,"url":null,"internal_url":null},{"position":24,"raw":"[Singla et al., 2014] Adish Singla, Ilija Bogunovic, Gábor Bartók, Amin Karbasi, and Andreas Krause. Near- optimally teaching the crowd to classify. In ICML, pages 154-162, 2014.","title":"Nearoptimally teaching the crowd to classify","authors":"Adish Singla, Ilija Bogunovic, Gábor Bartók, Amin Karbasi, Andreas Krause","year":"2014","venue":"ICML","doi":null,"url":null,"internal_url":null},{"position":25,"raw":"[Soma and Yoshida, 2017] Tasuku Soma and Yuichi Yoshida. Non-monotone DR-submodular function maximization. In AAAI, pages 898-904, 2017.","title":"Non-monotone DR-submodular function maximization","authors":"Yoshida ; ] Tasuku Soma, Yuichi Soma, Yoshida","year":"2017","venue":"AAAI","doi":null,"url":null,"internal_url":null},{"position":26,"raw":"[Vondrák, 2013] Jan Vondrák. Symmetry and approximabil- ity of submodular maximization problems. SIAM J. Com- puting, 42(1):265-304, 2013.","title":"Symmetry and approximability of submodular maximization problems","authors":"Jan Vondrák","year":"2013","venue":"SIAM J. Computing","doi":null,"url":null,"internal_url":null}],"license_note":"Copyright International Joint Conferences on Artificial Intelligence."}