{"type":"ScholarlyArticle","title":"Online Combinatorial Optimization with Group Fairness Constraints","authors":[{"name":"Negin Golrezaei","slug":"negin-golrezaei","given":"Negin","family":"Golrezaei","affiliation":"Sloan School of Management , MIT"},{"name":"Rad Niazadeh","slug":"rad-niazadeh","given":"Rad","family":"Niazadeh","affiliation":"Booth School of Business , University of Chicago"},{"name":"Kumar Kshitij Patel","slug":"kumar-kshitij-patel","given":"Kumar Kshitij","family":"Patel","affiliation":"Toyota Technological Institute at Chicago (TTIC)"},{"name":"Fransisca Susan","slug":"fransisca-susan","given":"Fransisca","family":"Susan","affiliation":"Sloan School of Management , MIT"}],"abstract":"As digital marketplaces and services continue to expand, it is crucial to maintain a safe and fair environment for all users. This requires implementing fairness constraints into the sequential decision-making processes of these platforms to ensure equal treatment. However, this can be challenging as these processes often need to solve NP-complete problems with exponentially large decision spaces at each time step. To overcome this, we propose a general framework incorporating robustness and fairness into NP-complete problems, such as optimizing product ranking and maximizing sub-modular functions. Our framework casts the problem as a max-min game between a primal player aiming to maximize the platform's objective and a dual player in charge of group fairness constraints. We show that one can trace the entire Pareto fairness curve by changing the thresholds on the fairness constraints. We provide theoretical guarantees for our method and empirically evaluate it, demonstrating its effectiveness.","doi":"10.24963/ijcai.2024/44","url":"https://ijcai.org/proceedings/2024/44","pdf":"https://ijcai.org/proceedings/2024/0044.pdf","markdown":"https://ijcai.org/proceedings/2024/44.md","bibtex":"https://ijcai.org/proceedings/2024/bibtex/44","volume":{"slug":"2024","title":"Proceedings of the Thirty-Third International Joint Conference on Artificial Intelligence","short_title":"IJCAI 2024","url":"https://ijcai.org/proceedings/2024/","year":2024,"location":"Jeju, Korea","dates":"3-9 August 2024","editors":"Kate Larson","isbn":"978-1-956792-04-1","publisher":"International Joint Conferences on Artificial Intelligence Organization","doi_prefix":"10.24963/ijcai.2024"},"section":"AI Ethics, Trust, Fairness","track":"Main Track","number":44,"pages":{"first":394,"last":402},"keywords":["Machine Learning: ML: Optimization","AI Ethics, Trust, Fairness: ETF: Fairness and diversity","Machine Learning: ML: Online learning","Constraint Satisfaction and Optimization: CSO: Mixed discrete and continuous optimization"],"videos":[],"references":[{"position":1,"raw":"[Asadpour et al., 2022] Arash Asadpour, Rad Niazadeh, Amin Saberi, and Ali Shameli. Sequential submodular maximization and applications to ranking an assortment of products. Operations Research, 2022.","title":"Sequential submodular maximization and applications to ranking an assortment of products","authors":"Arash Asadpour, Rad Niazadeh, Amin Saberi, Ali Shameli","year":"2022","venue":"Operations Research","doi":null,"url":null,"internal_url":null},{"position":2,"raw":"[Balseiro et al., 2021] Santiago Balseiro, Haihao Lu, and Va- hab Mirrokni. Regularized online allocation problems: Fairness and beyond. In International Conference on Ma- chine Learning, pages 630-639. PMLR, 2021.","title":"Santiago Balseiro, Haihao Lu, and Vahab Mirrokni. Regularized online allocation problems: Fairness and beyond","authors":"Balseiro","year":"2021","venue":"International Conference on Machine Learning","doi":null,"url":null,"internal_url":null},{"position":3,"raw":"[Ben-Tal et al., 2013] Aharon Ben-Tal, Dick Den Hertog, Anja De Waegenaere, Bertrand Melenberg, and Gijs Ren- nen. Robust solutions of optimization problems af- fected by uncertain probabilities. Management Science, 59(2):341-357, 2013.","title":"Robust solutions of optimization problems affected by uncertain probabilities","authors":"Ben-Tal","year":"2013","venue":"Management Science","doi":null,"url":null,"internal_url":null},{"position":4,"raw":"[Bertsimas et al., 2011] Dimitris Bertsimas, Vivek F. Farias, and Nikolaos Trichakis. The price of fairness. Operations research, 59(1):17-31, 2011.","title":"The price of fairness","authors":"Dimitris Bertsimas, F Vivek, Nikolaos Farias, Trichakis","year":"2011","venue":"Operations research","doi":null,"url":null,"internal_url":null},{"position":5,"raw":"[Bertsimas et al., 2012] Dimitris Bertsimas, Vivek F. Farias, and Nikolaos Trichakis. On the efficiency-fairness trade- off. Management Science, 58(12):2234-2250, 2012.","title":"On the efficiency-fairness tradeoff","authors":"Dimitris Bertsimas, F Vivek, Nikolaos Farias, Trichakis","year":"2012","venue":"Management Science","doi":null,"url":null,"internal_url":null},{"position":6,"raw":"[Besbes et al., 2014] Omar Besbes, Yonatan Gur, and Assaf Zeevi. Stochastic multi-armed-bandit problem with non- stationary rewards. Advances in neural information pro- cessing systems, 27, 2014.","title":"Stochastic multi-armed-bandit problem with nonstationary rewards","authors":"Omar Besbes, Yonatan Gur, Assaf Zeevi","year":"2014","venue":"Advances in neural information processing systems","doi":null,"url":null,"internal_url":null},{"position":7,"raw":"[Biega et al., 2018] Asia J. Biega, Krishna P. Gummadi, and Gerhard Weikum. Equity of attention: Amortizing indi- vidual fairness in rankings. In The 41st international acm sigir conference on research & development in informa- tion retrieval, pages 405-414, 2018.","title":"Equity of attention: Amortizing individual fairness in rankings","authors":"Biega","year":"2018","venue":"The 41st international acm sigir conference on research & development in information retrieval","doi":null,"url":null,"internal_url":null},{"position":8,"raw":"[California Legislative Service, 2020] California Legislative Service. The california privacy rights act of 2020, 2020. Available at: https://cppa.ca.gov/regulations/.","title":"California Legislative Service. The california privacy rights act of 2020, 2020","authors":null,"year":"2020","venue":null,"doi":null,"url":"https://cppa.ca.gov/regulations/","internal_url":null},{"position":9,"raw":"[Celis et al., 2018] Elisa L. Celis, Damian Straszak, and Nisheeth K. Vishnoi. Ranking with fairness constraints. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Schloss Dagstuhl- Leibniz-Zentrum fuer Informatik, 2018.","title":"Ranking with fairness constraints","authors":"Elisa L Celis, Damian Straszak, Nisheeth K Vishnoi","year":"2018","venue":"45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)","doi":null,"url":null,"internal_url":null},{"position":10,"raw":"[Chen et al., 2018] Lin Chen, Hamed Hassani, and Amin Karbasi. Online continuous submodular maximization. In International Conference on Artificial Intelligence and Statistics, pages 1896-1905. PMLR, 2018.","title":"Online continuous submodular maximization","authors":"Lin Chen, Hamed Hassani, Amin Karbasi","year":"2018","venue":"International Conference on Artificial Intelligence and Statistics","doi":null,"url":null,"internal_url":null},{"position":11,"raw":"[Chen et al., 2022] Qinyi Chen, Negin Golrezaei, Fransisca Susan, and Edy Baskoro. Fair assortment planning. arXiv preprint arXiv:2208.07341, 2022.","title":"Fair assortment planning","authors":"Chen","year":"2022","venue":null,"doi":null,"url":null,"internal_url":null},{"position":12,"raw":"[Chierichetti et al., 2019] Flavio Chierichetti, Ravi Kumar, Silvio Lattanzi, and Sergei Vassilvtiskii. Matroids, match- ings, and fairness. In The 22nd International Conference on Artificial Intelligence and Statistics, pages 2212-2220.","title":"Matroids, matchings, and fairness","authors":"Flavio Chierichetti, Ravi Kumar, Silvio Lattanzi, Sergei Vassilvtiskii","year":null,"venue":"The 22nd International Conference on Artificial Intelligence and Statistics","doi":null,"url":null,"internal_url":null},{"position":13,"raw":"PMLR, 2019.","title":null,"authors":"Pmlr","year":"2019","venue":null,"doi":null,"url":null,"internal_url":null},{"position":14,"raw":"[Chouldechova and Roth, 2018] Alexandra Chouldechova and Aaron Roth. The frontiers of fairness in machine learning. arXiv preprint arXiv:1810.08810, 2018.","title":"The frontiers of fairness in machine learning","authors":"Roth Chouldechova","year":"2018","venue":null,"doi":null,"url":null,"internal_url":null},{"position":15,"raw":"[Dekel, 2012] Ofer Dekel. Combinatorial optimization. Lec- ture Notes, 2012. [Online; accessed 2024-01-17]. Avail- able: https://courses.cs.washington.edu/courses/cse599s/ 12sp/scribes/lecture16.pdf.","title":"Combinatorial optimization","authors":"Ofer Dekel, Dekel","year":"2012","venue":null,"doi":null,"url":"https://courses.cs.washington.edu/courses/cse599s/12sp/scribes/lecture16.pdf","internal_url":null},{"position":16,"raw":"[Deng et al., 2022] Yuan Deng, Negin Golrezaei, Patrick Jaillet, Jason Cheuk Nam Liang, and Vahab Mirrokni. Fairness in the autobidding world with machine-learned advice. arXiv preprint arXiv:2209.04748, 2022.","title":"Fairness in the autobidding world with machine-learned advice","authors":"Yuan Deng, Negin Golrezaei, Patrick Jaillet, Jason Cheuk, Nam Liang, Vahab Mirrokni","year":"2022","venue":null,"doi":null,"url":null,"internal_url":null},{"position":17,"raw":"[Dobzinski and Schapira, 2006] Shahar Dobzinski and Michael Schapira. An improved approximation algorithm for combinatorial auctions with submodular bidders. In Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm, pages 1064-1073, 2006.","title":"An improved approximation algorithm for combinatorial auctions with submodular bidders","authors":"Schapira Dobzinski, Shahar Dobzinski, Michael Schapira","year":"2006","venue":"Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm","doi":null,"url":null,"internal_url":null},{"position":18,"raw":"[Duchi et al., 2021] John C. Duchi, Peter W. Glynn, and Hongseok Namkoong. Statistics of robust optimization: A generalized empirical likelihood approach. Mathemat- ics of Operations Research, 46(3):946-969, 2021.","title":"Statistics of robust optimization: A generalized empirical likelihood approach","authors":"John C Duchi, Peter W Glynn, Hongseok Namkoong","year":"2021","venue":"Mathematics of Operations Research","doi":null,"url":null,"internal_url":null},{"position":19,"raw":"[Dudik et al., 2020] Miroslav Dudik, Nika Haghtalab, Haipeng Luo, Robert E. Schapire, Vasilis Syrgkanis, and Jennifer Wortman Vaughan. Oracle-efficient online learning and auction design. Journal of the ACM (JACM), 67(5):1-57, 2020.","title":"Oracle-efficient online learning and auction design","authors":"Dudik","year":"2020","venue":"Journal of the ACM (JACM)","doi":null,"url":null,"internal_url":null},{"position":20,"raw":"[Dwork et al., 2012] Cynthia Dwork, Moritz Hardt, Toniann Pitassi, Omer Reingold, and Richard Zemel. Fairness through awareness. In Proceedings of the 3rd innovations in theoretical computer science conference, pages 214- 226, 2012.","title":"Fairness through awareness","authors":"Cynthia Dwork, Moritz Hardt, Toniann Pitassi, Omer Reingold, Richard Zemel","year":"2012","venue":"Proceedings of the 3rd innovations in theoretical computer science conference","doi":null,"url":null,"internal_url":null},{"position":21,"raw":"[Garcia-Soriano and Bonchi, 2021] David Garcia-Soriano and Francesco Bonchi. Maxmin-fair ranking: individual fairness under group-fairness constraints. In Proceedings of the 27th ACM SIGKDD Conference on Knowledge Discovery & Data Mining, pages 436-446, 2021.","title":"Maxmin-fair ranking: individual fairness under group-fairness constraints","authors":"Garcia-Soriano, Bonchi, David Garcia-Soriano, Francesco Bonchi","year":"2021","venue":"Proceedings of the 27th ACM SIGKDD Conference on Knowledge Discovery & Data Mining","doi":null,"url":null,"internal_url":null},{"position":22,"raw":"[Goyal et al., 2016] Vineet Goyal, Retsef Levi, and Danny Segev. Near-optimal algorithms for the assortment plan- ning problem under dynamic substitution and stochastic demand. Operations Research, 64(1):219-235, 2016.","title":"Near-optimal algorithms for the assortment planning problem under dynamic substitution and stochastic demand","authors":"Vineet Goyal, Retsef Levi, Danny Segev","year":"2016","venue":"Operations Research","doi":null,"url":null,"internal_url":null},{"position":23,"raw":"[Harper and Konstan, 2015] Maxwell Harper and Joseph Konstan. The movielens datasets: History and context. Acm transactions on interactive intelligent systems (tiis), 5(4):1-19, 2015.","title":"The movielens datasets: History and context. Acm transactions on interactive intelligent systems","authors":"Konstan Harper, Maxwell Harper, Joseph Konstan","year":"2015","venue":"tiis)","doi":null,"url":null,"internal_url":null},{"position":24,"raw":"[Hazan, 2016] Elad Hazan. Introduction to online convex optimization. Foundations and Trends® in Optimization, 2(3-4):157-325, 2016.","title":"Introduction to online convex optimization","authors":"Elad Hazan","year":"2016","venue":"Foundations and Trends® in Optimization","doi":null,"url":null,"internal_url":null},{"position":25,"raw":"[Hitsch et al., 2010] Gunter J. Hitsch, Ali Hortacsu, and Dan Ariely. What makes you click?-mate preferences in online dating. Quantitative marketing and Economics, 8:393-427, 2010.","title":"What makes you click?-mate preferences in online dating","authors":"Gunter J Hitsch, Ali Hortacsu, Dan Ariely","year":"2010","venue":"Quantitative marketing and Economics","doi":null,"url":null,"internal_url":null},{"position":26,"raw":"[Jafarnia-Jahromi et al., 2021] Mehdi Jafarnia-Jahromi, Liyu Chen, Rahul Jain, and Haipeng Luo. Online learning for stochastic shortest path model via posterior sampling. arXiv preprint arXiv:2106.05335, 2021.","title":"Online learning for stochastic shortest path model via posterior sampling","authors":"Mehdi Jafarnia-Jahromi, Liyu Chen, Rahul Jain, Haipeng Luo","year":"2021","venue":null,"doi":null,"url":null,"internal_url":null},{"position":27,"raw":"[Joseph et al., 2016] Matthew Joseph, Michael Kearns, Jamie H. Morgenstern, and Aaron Roth. Fairness in learning: Classic and contextual bandits. Advances in neural information processing systems, 29, 2016.","title":"Fairness in learning: Classic and contextual bandits","authors":"Matthew Joseph, Michael Kearns, Jamie H Morgenstern, Aaron Roth","year":"2016","venue":"Advances in neural information processing systems","doi":null,"url":null,"internal_url":null},{"position":28,"raw":"[Kakade et al., 2007] Sham M. Kakade, Adam Tauman Kalai, and Katrina Ligett. Playing games with approxima- tion algorithms. In Proceedings of the thirty-ninth annual ACM symposium on Theory of computing, pages 546-555, 2007.","title":"Playing games with approximation algorithms","authors":"M Sham, Adam Kakade, Katrina Tauman Kalai, Ligett","year":"2007","venue":"Proceedings of the thirty-ninth annual ACM symposium on Theory of computing","doi":null,"url":null,"internal_url":null},{"position":29,"raw":"[Kalai and Vempala, 2005] Adam Kalai and Santosh Vem- pala. Efficient algorithms for online decision problems. Journal of Computer and System Sciences, 71(3):291-307, 2005.","title":"Adam Kalai and Santosh Vempala. Efficient algorithms for online decision problems","authors":"Vempala Kalai","year":"2005","venue":"Journal of Computer and System Sciences","doi":null,"url":null,"internal_url":null},{"position":30,"raw":"[Kearns et al., 2018] Michael Kearns, Seth Neel, Aaron Roth, and Zhiwei Steven Wu. Preventing fairness gerry- mandering: Auditing and learning for subgroup fairness. In International conference on machine learning, pages 2564-2572. PMLR, 2018.","title":"Preventing fairness gerrymandering: Auditing and learning for subgroup fairness","authors":"Michael Kearns, Seth Neel, Aaron Roth, Zhiwei Steven Wu","year":"2018","venue":"International conference on machine learning","doi":null,"url":null,"internal_url":null},{"position":31,"raw":"[Kearns et al., 2019] Michael Kearns, Seth Neel, Aaron Roth, and Zhiwei Steven Wu. An empirical study of rich subgroup fairness for machine learning. In Proceedings of the conference on fairness, accountability, and trans- parency, pages 100-109, 2019.","title":"An empirical study of rich subgroup fairness for machine learning","authors":"Michael Kearns, Seth Neel, Aaron Roth, Zhiwei Steven Wu","year":"2019","venue":"Proceedings of the conference on fairness, accountability, and transparency","doi":null,"url":null,"internal_url":null},{"position":32,"raw":"[Kempe et al., 2003] David Kempe, Jon Kleinberg, and Éva Tardos. Maximizing the spread of influence through a so- cial network. In Proceedings of the ninth ACM SIGKDD international conference on Knowledge discovery and data mining, pages 137-146, 2003.","title":"Maximizing the spread of influence through a social network","authors":"David Kempe, Jon Kleinberg, Éva Tardos","year":"2003","venue":"Proceedings of the ninth ACM SIGKDD international conference on Knowledge discovery and data mining","doi":null,"url":null,"internal_url":null},{"position":33,"raw":"[Kleinberg et al., 2018] Jon Kleinberg, Jens Ludwig, Send- hil Mullainathan, and Ashesh Rambachan. Algorithmic fairness. In Aea papers and proceedings, volume 108, pages 22-27, 2018.","title":"Algorithmic fairness","authors":"Kleinberg","year":"2018","venue":"Aea papers and proceedings","doi":null,"url":null,"internal_url":null},{"position":34,"raw":"[Liu et al., 2022] Qingsong Liu, Weihang Xu, Siwei Wang, and Zhixuan Fang. Combinatorial bandits with linear con- straints: Beyond knapsacks and fairness. Advances in Neu- ral Information Processing Systems, 35:2997-3010, 2022.","title":"Combinatorial bandits with linear constraints: Beyond knapsacks and fairness","authors":"Qingsong Liu, Weihang Xu, Siwei Wang, Zhixuan Fang","year":"2022","venue":"Advances in Neural Information Processing Systems","doi":null,"url":null,"internal_url":null},{"position":35,"raw":"[Loi et al., 2019] Michele Loi, Anders Herlitz, and Hoda Heidari. A philosophical theory of fairness for prediction- based decisions. Available at SSRN 3450300, 2019.","title":"A philosophical theory of fairness for predictionbased decisions","authors":"Michele Loi, Anders Herlitz, Hoda Heidari","year":"2019","venue":null,"doi":null,"url":null,"internal_url":null},{"position":36,"raw":"[Manshadi et al., 2021] Vahideh Manshadi, Rad Niazadeh, and Scott Rodilitz. Fair dynamic rationing. In Proceedings of the 22nd ACM Conference on Economics and Compu- tation, pages 694-695, 2021.","title":"Vahideh Manshadi, Rad Niazadeh, and Scott Rodilitz. Fair dynamic rationing","authors":"Manshadi","year":"2021","venue":"Proceedings of the 22nd ACM Conference on Economics and Computation","doi":null,"url":null,"internal_url":null},{"position":37,"raw":"[Miettinen, 1999] Kaisa Miettinen. Nonlinear multiobjective optimization, volume 12. Springer Science & Business Media, 1999.","title":"Nonlinear multiobjective optimization","authors":"Kaisa Miettinen","year":"1999","venue":null,"doi":null,"url":null,"internal_url":null},{"position":38,"raw":"[Nemhauser et al., 1978] George L. Nemhauser, Lau- rence A. Wolsey, and Marshall L. Fisher. An analysis of approximations for maximizing submodular set func- tions-i. Mathematical programming, 14(1):265-294, 1978.","title":"An analysis of approximations for maximizing submodular set functions-i","authors":"George L Nemhauser, Laurence A Wolsey, Marshall L Fisher","year":"1978","venue":"Mathematical programming","doi":null,"url":null,"internal_url":null},{"position":39,"raw":"[Niazadeh et al., 2021] Rad Niazadeh, Negin Golrezaei, Joshua R. Wang, Fransisca Susan, and Ashwinkumar Badanidiyuru. Online learning via offline greedy algo- rithms: Applications in market design and optimization. In Proceedings of the 22nd ACM Conference on Economics and Computation, pages 737-738, 2021.","title":"Online learning via offline greedy algorithms: Applications in market design and optimization","authors":"Niazadeh","year":"2021","venue":"Proceedings of the 22nd ACM Conference on Economics and Computation","doi":null,"url":null,"internal_url":null},{"position":40,"raw":"[Park and Kwan, 2020] Yoo Min Park and Mei-Po Kwan. Understanding racial disparities in exposure to traffic- related air pollution: Considering the spatiotemporal dy- namics of population distribution. International journal of environmental research and public health, 17(3):908, 2020.","title":"Understanding racial disparities in exposure to trafficrelated air pollution: Considering the spatiotemporal dynamics of population distribution","authors":"Kwan ; ] Park, Min Yoo, Mei-Po Park, Kwan","year":"2020","venue":"International journal of environmental research and public health","doi":null,"url":null,"internal_url":null},{"position":41,"raw":"[Rahmattalabi et al., 2019] Aida Rahmattalabi, Phebe Vayanos, Anthony Fulginiti, Eric Rice, Bryan Wilder, Amulya Yadav, and Milind Tambe. Exploring algorithmic fairness in robust graph covering problems. Advances in neural information processing systems, 32, 2019.","title":"Exploring algorithmic fairness in robust graph covering problems","authors":"Rahmattalabi","year":"2019","venue":"Advances in neural information processing systems","doi":null,"url":null,"internal_url":null},{"position":42,"raw":"[Ribeiro, 2002] Carlos H. C. Ribeiro. Reinforcement learn- ing agents. Artificial intelligence review, 17:223-250, 2002.","title":"Reinforcement learning agents","authors":"H C Carlos, Ribeiro","year":"2002","venue":"Artificial intelligence review","doi":null,"url":null,"internal_url":null},{"position":43,"raw":"[Roughgarden and Wang, 2018] Tim Roughgarden and Joshua R. Wang. An optimal learning algorithm for online unconstrained submodular maximization. In Conference On Learning Theory, pages 1307-1325. PMLR, 2018.","title":"An optimal learning algorithm for online unconstrained submodular maximization","authors":"Wang ; Roughgarden, Tim Roughgarden, Joshua R Wang","year":"2018","venue":"Conference On Learning Theory","doi":null,"url":null,"internal_url":null},{"position":44,"raw":"[Sagawa et al., 2019] Shiori Sagawa, Pang Wei Koh, Tat- sunori B. Hashimoto, and Percy Liang. Distribution- ally robust neural networks for group shifts: On the im- portance of regularization for worst-case generalization. arXiv preprint arXiv:1911.08731, 2019.","title":"Distributionally robust neural networks for group shifts: On the importance of regularization for worst-case generalization","authors":"Shiori Sagawa, Pang Wei Koh, B Tatsunori, Percy Hashimoto, Liang","year":"2019","venue":null,"doi":null,"url":null,"internal_url":null},{"position":45,"raw":"[Shalev-Shwartz and Singer, 2007] Shai Shalev-Shwartz and Yoram Singer. A primal-dual perspective of online learn- ing algorithms. Machine Learning, 69:115-142, 2007.","title":"A primal-dual perspective of online learning algorithms","authors":"Shalev-Shwartz Singer, Shai Shalev-Shwartz, Yoram Singer","year":"2007","venue":"Machine Learning","doi":null,"url":null,"internal_url":null},{"position":46,"raw":"[Singh and Joachims, 2018] Ashudeep Singh and Thorsten Joachims. Fairness of exposure in rankings. In Proceed- ings of the 24th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining, pages 2219- 2228, 2018.","title":"Ashudeep Singh and Thorsten Joachims. Fairness of exposure in rankings","authors":"Joachims Singh","year":"2018","venue":"Proceedings of the 24th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining","doi":null,"url":null,"internal_url":null},{"position":47,"raw":"[Soma et al., 2022] Tasuku Soma, Khashayar Gatmiry, and Stefanie Jegelka. Optimal algorithms for group distribu- tionally robust optimization and beyond. arXiv preprint arXiv:2212.13669, 2022.","title":"Optimal algorithms for group distributionally robust optimization and beyond","authors":"Soma","year":"2022","venue":null,"doi":null,"url":null,"internal_url":null},{"position":48,"raw":"[Streeter and Golovin, 2008] Matthew Streeter and Daniel Golovin. An online algorithm for maximizing submodu- lar functions. Advances in Neural Information Processing Systems, 21, 2008.","title":"An online algorithm for maximizing submodular functions","authors":"Golovin Streeter, Matthew Streeter, Daniel Golovin","year":"2008","venue":"Advances in Neural Information Processing Systems","doi":null,"url":null,"internal_url":null},{"position":49,"raw":"[Tang and Yuan, 2023] Shaojie Tang and Jing Yuan. Beyond submodularity: a unified framework of randomized set se- lection with group fairness constraints. Journal of Combi- natorial Optimization, 45(4):102, 2023.","title":"Beyond submodularity: a unified framework of randomized set selection with group fairness constraints","authors":"Yuan ; ] Shaojie Tang, Jing Tang, Yuan","year":"2023","venue":"Journal of Combinatorial Optimization","doi":null,"url":null,"internal_url":null},{"position":50,"raw":"[Zinkevich, 2003] Martin Zinkevich. Online convex pro- gramming and generalized infinitesimal gradient ascent. In Proceedings of the 20th international conference on ma- chine learning (icml-03), pages 928-936, 2003.","title":"Online convex programming and generalized infinitesimal gradient ascent","authors":"Martin Zinkevich","year":"2003","venue":"Proceedings of the 20th international conference on machine learning (icml-03)","doi":null,"url":null,"internal_url":null}],"license_note":"Copyright International Joint Conferences on Artificial Intelligence."}