{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,28]],"date-time":"2026-04-28T00:49:17Z","timestamp":1777337357441,"version":"3.51.4"},"reference-count":38,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2026,4,1]],"date-time":"2026-04-01T00:00:00Z","timestamp":1775001600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2026,4,1]],"date-time":"2026-04-01T00:00:00Z","timestamp":1775001600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Syst Sci Complex"],"published-print":{"date-parts":[[2026,4]]},"DOI":"10.1007\/s11424-026-5503-2","type":"journal-article","created":{"date-parts":[[2026,4,28]],"date-time":"2026-04-28T00:03:31Z","timestamp":1777334611000},"page":"831-856","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["No-Regret Learning in Network Stochastic Zero-Sum Games"],"prefix":"10.1007","volume":"39","author":[{"given":"Shijie","family":"Huang","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jinlong","family":"Lei","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yiguang","family":"Hong","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2026,4,28]]},"reference":[{"key":"5503_CR1","volume-title":"Theory of Games and Economic Behavior","author":"O Morgenstern","year":"1953","unstructured":"Morgenstern O and Von Neumann J, Theory of Games and Economic Behavior, Princeton University Press, Princeton, 1953."},{"key":"5503_CR2","doi-asserted-by":"publisher","DOI":"10.1515\/9781400831050","volume-title":"Robust Optimization","author":"A Ben-Tal","year":"2009","unstructured":"Ben-Tal A, El Ghaoui L, and Nemirovski A, Robust Optimization, Princeton University Press, Princeton, 2009."},{"key":"5503_CR3","unstructured":"Goodfellow I, Pouget-Abadie J, Mirza M, et al., Generative adversarial nets, Advances in Neural Information Processing Systems, 2014, 2672\u20132680."},{"issue":"6218","key":"5503_CR4","doi-asserted-by":"publisher","first-page":"145","DOI":"10.1126\/science.1259433","volume":"347","author":"M Bowling","year":"2015","unstructured":"Bowling M, Burch N, Johanson M, et al., Heads-up limit hold\u2019em poker is solved, Science, 2015, 347(6218): 145\u2013149.","journal-title":"Science"},{"issue":"2","key":"5503_CR5","doi-asserted-by":"publisher","first-page":"286","DOI":"10.2307\/1969529","volume":"54","author":"J Nash","year":"1951","unstructured":"Nash J, Non-cooperative games, Annals of Mathematics, 1951, 54(2): 286\u2013295.","journal-title":"Annals of Mathematics"},{"key":"5503_CR6","first-page":"993","volume-title":"Conference on Learning Theory","author":"A Rakhlin","year":"2013","unstructured":"Rakhlin A and Sridharan K, Online learning with predictable sequences, Conference on Learning Theory, 2013, 993\u20131019."},{"key":"5503_CR7","unstructured":"Rakhlin S and Sridharan K, Optimization, learning, and games with predictable sequences, Advances in Neural Information Processing Systems, 2013, 3066\u20133074."},{"key":"5503_CR8","first-page":"2488","volume-title":"International Conference on Machine Learning","author":"E A Kangarshahi","year":"2018","unstructured":"Kangarshahi E A, Hsieh Y P, Sahin M F, et al., Let\u2019s be honest: An optimal no-regret framework for zero-sum games, International Conference on Machine Learning, 2018, 2488\u20132496."},{"issue":"6","key":"5503_CR9","doi-asserted-by":"publisher","first-page":"1683","DOI":"10.1016\/j.automatica.2013.02.062","volume":"49","author":"B Gharesifard","year":"2013","unstructured":"Gharesifard B and Cortes J, Distributed convergence to Nash equilibria in two-network zero-sum games, Automatica, 2013, 49(6): 1683\u20131692.","journal-title":"Automatica"},{"issue":"10","key":"5503_CR10","doi-asserted-by":"publisher","first-page":"2920","DOI":"10.1109\/TAC.2015.2504962","volume":"61","author":"Y Lou","year":"2015","unstructured":"Lou Y, Hong Y, Xie L, et al., Nash equilibrium computation in subnetwork zero-sum games with switching communications, IEEE Transactions on Automatic Control, 2015, 61(10): 2920\u20132935.","journal-title":"IEEE Transactions on Automatic Control"},{"issue":"10","key":"5503_CR11","doi-asserted-by":"publisher","first-page":"6620","DOI":"10.1109\/TAC.2024.3379253","volume":"69","author":"S Huang","year":"2024","unstructured":"Huang S, Lei J, Hong Y, et al., No-regret distributed learning in subnetwork zero-sum games, IEEE Transactions on Automatic Control, 2024, 69(10): 6620\u20136635.","journal-title":"IEEE Transactions on Automatic Control"},{"issue":"2","key":"5503_CR12","doi-asserted-by":"publisher","first-page":"243","DOI":"10.1287\/opre.43.2.243","volume":"43","author":"A Washburn","year":"1995","unstructured":"Washburn A and Wood K, Two-person zero-sum games for network interdiction, Operations Research, 1995, 43(2): 243\u2013251.","journal-title":"Operations Research"},{"key":"5503_CR13","first-page":"7953","volume-title":"International Conference on Machine Learning","author":"A Rajeswaran","year":"2020","unstructured":"Rajeswaran A, Mordatch I, and Kumar V, A game theoretic framework for model based reinforcement learning, International Conference on Machine Learning, 2020, 7953\u20137963."},{"issue":"2","key":"5503_CR14","doi-asserted-by":"publisher","first-page":"107","DOI":"10.1561\/2200000018","volume":"4","author":"S Shalev-Shwartz","year":"2011","unstructured":"Shalev-Shwartz S, Online learning and online convex optimization, Foundations and Trends in Machine Learning, 2011, 4(2): 107\u2013194.","journal-title":"Foundations and Trends in Machine Learning"},{"key":"5503_CR15","volume-title":"Logarithmic regret algorithms for strongly convex repeated games","author":"S Shalev-Shwartz","year":"2007","unstructured":"Shalev-Shwartz S and Singer Y, Logarithmic regret algorithms for strongly convex repeated games, The Hebrew University, Jerusalem, Israel, 2007."},{"issue":"1","key":"5503_CR16","doi-asserted-by":"publisher","first-page":"1","DOI":"10.2140\/pjm.1956.6.1","volume":"6","author":"D Blackwell","year":"1956","unstructured":"Blackwell D, An analog of the minimax theorem for vector payoffs, Pacific Journal of Mathematics, 1956, 6(1): 1\u20138.","journal-title":"Pacific Journal of Mathematics"},{"issue":"1\u20132","key":"5503_CR17","doi-asserted-by":"publisher","first-page":"79","DOI":"10.1006\/game.1999.0738","volume":"29","author":"Y Freund","year":"1999","unstructured":"Freund Y and Schapire R E, Adaptive game playing using multiplicative weights, Games and Economic Behavior, 1999, 29(1\u20132): 79\u2013103.","journal-title":"Games and Economic Behavior"},{"key":"5503_CR18","unstructured":"Zinkevich M, Johanson M, Bowling M, et al., Regret minimization in games with incomplete information, Advances in Neural Information Processing Systems, 2007, 1729\u20131736."},{"key":"5503_CR19","first-page":"3507","volume-title":"International Conference on Machine Learning","author":"Y Cai","year":"2023","unstructured":"Cai Y and Zheng W, Doubly optimal no-regret learning in monotone games, International Conference on Machine Learning, PMLR, 2023, 3507\u20133524."},{"key":"5503_CR20","first-page":"27604","volume":"34","author":"C Daskalakis","year":"2021","unstructured":"Daskalakis C, Fishelson M, and Golowich N, Near-optimal no-regret learning in general games, Advances in Neural Information Processing Systems, 2021, 34: 27604\u201327616.","journal-title":"Advances in Neural Information Processing Systems"},{"issue":"1","key":"5503_CR21","doi-asserted-by":"publisher","first-page":"465","DOI":"10.1007\/s10107-018-1254-8","volume":"173","author":"P Mertikopoulos","year":"2019","unstructured":"Mertikopoulos P and Zhou Z, Learning in games with continuous action sets and unknown payoff functions, Mathematical Programming, 2019, 173(1): 465\u2013507.","journal-title":"Mathematical Programming"},{"issue":"1","key":"5503_CR22","doi-asserted-by":"publisher","first-page":"183","DOI":"10.1109\/TAC.2025.3586980","volume":"71","author":"L Liao","year":"2026","unstructured":"Liao L, Yuan D, Ho D W C, et al., Distributed dynamic no-regret learning in two-network zero-sum games, IEEE Transactions on Automatic Control, 2026, 71(1): 183\u2013197.","journal-title":"IEEE Transactions on Automatic Control"},{"issue":"4","key":"5503_CR23","doi-asserted-by":"publisher","first-page":"1574","DOI":"10.1137\/070704277","volume":"19","author":"A Nemirovski","year":"2009","unstructured":"Nemirovski A, Juditsky A, Lan G, et al., Robust stochastic approximation approach to stochastic programming, SIAM Journal on Optimization, 2009, 19(4): 1574\u20131609.","journal-title":"SIAM Journal on Optimization"},{"issue":"4","key":"5503_CR24","doi-asserted-by":"publisher","first-page":"1779","DOI":"10.1137\/130919362","volume":"24","author":"Y Chen","year":"2014","unstructured":"Chen Y, Lan G, and Ouyang Y, Optimal primal-dual methods for a class of saddle point problems, SIAM Journal on Optimization, 2014, 24(4): 1779\u20131814.","journal-title":"SIAM Journal on Optimization"},{"key":"5503_CR25","first-page":"1153","volume-title":"Proceedings of the 34th International Conference on Neural Information Processing Systems","author":"J Yang","year":"2020","unstructured":"Yang J, Kiyavash N, and He N, Global convergence and variance reduction for a class of nonconvex-nonconcave minimax problems, Proceedings of the 34th International Conference on Neural Information Processing Systems, 2020, 1153\u20131165."},{"issue":"1","key":"5503_CR26","doi-asserted-by":"publisher","first-page":"157","DOI":"10.1287\/moor.2018.0986","volume":"45","author":"J Lei","year":"2020","unstructured":"Lei J, Shanbhag U V, Pang J S, et al., On synchronous, asynchronous, and randomized best-response schemes for stochastic Nash games, Mathematics of Operations Research, 2020, 45(1): 157\u2013190.","journal-title":"Mathematics of Operations Research"},{"key":"5503_CR27","doi-asserted-by":"publisher","first-page":"143","DOI":"10.1007\/978-3-642-34097-0_7","volume-title":"Agent-Based Optimization","author":"K Srivastava","year":"2013","unstructured":"Srivastava K, Nedi\u0107 A, and Stipanovi\u0107 D, Distributed Bregman-distance algorithms for min-max optimization, Agent-Based Optimization, Springer Berlin, Heidelberg, 2013, 143\u2013174."},{"issue":"4","key":"5503_CR28","doi-asserted-by":"publisher","first-page":"922","DOI":"10.1109\/TAC.2010.2041686","volume":"55","author":"A Nedi\u0107","year":"2010","unstructured":"Nedi\u0107 A, Ozdaglar A, and Parrilo P A, Constrained consensus and optimization in multi-agent networks, IEEE Transactions on Automatic Control, 2010, 55(4): 922\u2013938.","journal-title":"IEEE Transactions on Automatic Control"},{"key":"5503_CR29","first-page":"928","volume-title":"International Conference on Machine Learning","author":"M Zinkevich","year":"2003","unstructured":"Zinkevich M, Online convex programming and generalized infinitesimal gradient ascent, International Conference on Machine Learning, 2003, 928\u2013936."},{"issue":"12","key":"5503_CR30","doi-asserted-by":"publisher","first-page":"6409","DOI":"10.1109\/TAC.2017.2650563","volume":"62","author":"S Lee","year":"2017","unstructured":"Lee S, Nedi\u0107 A, and Raginsky M, Stochastic dual averaging for decentralized online optimization on time-varying communication graphs, IEEE Transactions on Automatic Control, 2017, 62(12): 6409\u20136416.","journal-title":"IEEE Transactions on Automatic Control"},{"key":"5503_CR31","first-page":"1","volume-title":"International Conference on Learning Representations","author":"P Mertikopoulos","year":"2019","unstructured":"Mertikopoulos P, Lecouat B, Zenati H, et al., Optimistic mirror descent in saddle-point problems: Going the extra (gradient) mile, International Conference on Learning Representations, 2019, 1\u201323."},{"key":"5503_CR32","first-page":"233","volume-title":"Optimizing Methods in Statistics","author":"H Robbins","year":"1971","unstructured":"Robbins H and Siegmund D, A convergence theorem for non negative almost supermartingales and some applications, Optimizing Methods in Statistics, 1971, 233\u2013257."},{"issue":"3\u20134","key":"5503_CR33","doi-asserted-by":"publisher","first-page":"157","DOI":"10.1561\/2400000013","volume":"2","author":"E Hazan","year":"2016","unstructured":"Hazan E, Introduction to online convex optimization, Foundations and Trends in Optimization, 2016, 2(3\u20134): 157\u2013325.","journal-title":"Foundations and Trends in Optimization"},{"key":"5503_CR34","first-page":"373","volume-title":"Proceedings of the Fortieth Annual ACM Symposium on Theory of Computing","author":"A Blum","year":"2008","unstructured":"Blum A, Hajiaghayi M T, Ligett K, et al., Regret minimization and the price of total anarchy, Proceedings of the Fortieth Annual ACM Symposium on Theory of Computing, 2008, 373\u2013382."},{"key":"5503_CR35","first-page":"6175","volume-title":"International Conference on Machine Learning","author":"Y Lei","year":"2021","unstructured":"Lei Y, Yang Z, Yang T, et al., Stability and generalization of stochastic gradient methods for minimax problems, International Conference on Machine Learning, 2021, 6175\u20136186."},{"key":"5503_CR36","first-page":"568","volume-title":"International Conference on Artificial Intelligence and Statistics","author":"J Zhang","year":"2021","unstructured":"Zhang J, Hong M, Wang M, et al., Generalization bounds for stochastic saddle point problems, International Conference on Artificial Intelligence and Statistics, 2021, 568\u2013576."},{"key":"5503_CR37","volume-title":"Proc. of AAAI-21","author":"L Adam","year":"2021","unstructured":"Adam L, Hor\u010d\u00edk R, Kasl T, et al., Double oracle algorithm for computing equilibria in continuous games, Proc. of AAAI-21, 2021."},{"key":"5503_CR38","first-page":"536","volume-title":"International Conference on Machine Learning","author":"I Anagnostides","year":"2022","unstructured":"Anagnostides I, Panageas I, Farina G, et al., On last-iterate convergence beyond zero-sum games, International Conference on Machine Learning, PMLR, 2022, 536\u2013581."}],"container-title":["Journal of Systems Science and Complexity"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11424-026-5503-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11424-026-5503-2","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11424-026-5503-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,28]],"date-time":"2026-04-28T00:03:36Z","timestamp":1777334616000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11424-026-5503-2"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,4]]},"references-count":38,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2026,4]]}},"alternative-id":["5503"],"URL":"https:\/\/doi.org\/10.1007\/s11424-026-5503-2","relation":{},"ISSN":["1009-6124","1559-7067"],"issn-type":[{"value":"1009-6124","type":"print"},{"value":"1559-7067","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,4]]},"assertion":[{"value":"1 November 2025","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"25 December 2025","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"28 April 2026","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"The authors declare no conflict of interest.","order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of Interest"}}]}}