{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,31]],"date-time":"2025-12-31T12:08:00Z","timestamp":1767182880029,"version":"3.37.3"},"reference-count":48,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2023,8,3]],"date-time":"2023-08-03T00:00:00Z","timestamp":1691020800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2023,8,3]],"date-time":"2023-08-03T00:00:00Z","timestamp":1691020800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11971090"],"award-info":[{"award-number":["11971090"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Comput Optim Appl"],"published-print":{"date-parts":[[2024,1]]},"DOI":"10.1007\/s10589-023-00512-0","type":"journal-article","created":{"date-parts":[[2023,8,3]],"date-time":"2023-08-03T19:02:18Z","timestamp":1691089338000},"page":"249-288","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Distributed stochastic compositional optimization problems over directed networks"],"prefix":"10.1007","volume":"87","author":[{"given":"Shengchao","family":"Zhao","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9955-4414","authenticated-orcid":false,"given":"Yongchao","family":"Liu","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2023,8,3]]},"reference":[{"key":"512_CR1","doi-asserted-by":"publisher","first-page":"519","DOI":"10.1137\/21M1406222","volume":"32","author":"K Balasubramanian","year":"2022","unstructured":"Balasubramanian, K., Ghadimi, S., Nguyen, A.: Stochastic multi-level composition optimization algorithms with levelindependent convergence rates. SIAM J. Optim. 32, 519\u2013544 (2022)","journal-title":"SIAM J. Optim."},{"key":"512_CR2","doi-asserted-by":"publisher","first-page":"7405","DOI":"10.1109\/TIT.2013.2275131","volume":"59","author":"P Bianchi","year":"2013","unstructured":"Bianchi, P., Fort, G., Hachem, W.: Performance of a distributed stochastic approximation algorithm. IEEE Trans. Inf. Theory 59, 7405\u20137418 (2013)","journal-title":"IEEE Trans. Inf. Theory"},{"key":"512_CR3","doi-asserted-by":"publisher","first-page":"4937","DOI":"10.1109\/TSP.2021.3092377","volume":"69","author":"T Chen","year":"2021","unstructured":"Chen, T., Sun, Y., Yin, W.: Solving stochastic compositional optimization is nearly as easy as solving stochastic optimization. IEEE Trans. Signal Process. 69, 4937\u20134948 (2021)","journal-title":"IEEE Trans. Signal Process."},{"key":"512_CR4","doi-asserted-by":"publisher","first-page":"463","DOI":"10.1214\/aoms\/1177728716","volume":"25","author":"KL Chung","year":"1954","unstructured":"Chung, K.L.: On a stochastic approximation method. Ann. Math. Stat. 25, 463\u2013483 (1954)","journal-title":"Ann. Math. Stat."},{"key":"512_CR5","unstructured":"Dai, B., He, N., Pan, Y., Boots, B., Song, L.: Learning from Conditional Distributions via Dual Embeddings. In: Proceedings of the 20th International Conference on Artificial Intelligence and Statistics, vol.\u00a054, PMLR, pp.\u00a01458\u20131467 (2017)"},{"key":"512_CR6","doi-asserted-by":"publisher","first-page":"737","DOI":"10.1007\/s10463-016-0559-8","volume":"69","author":"D Dentcheva","year":"2017","unstructured":"Dentcheva, D., Penev, S., Ruszczy\u0144ski, A.: Statistical estimation of composite risk functionals and risk optimization problems. Ann. Inst. Stat. Math. 69, 737\u2013760 (2017)","journal-title":"Ann. Inst. Stat. Math."},{"key":"512_CR7","volume-title":"Methods of Stochastic Programming","author":"YM Ermoliev","year":"1976","unstructured":"Ermoliev, Y.M.: Methods of Stochastic Programming. Nauka, Moscow (1976)"},{"key":"512_CR8","doi-asserted-by":"publisher","first-page":"2231","DOI":"10.1137\/120863277","volume":"23","author":"YM Ermoliev","year":"2013","unstructured":"Ermoliev, Y.M., Norkin, V.I.: Sample average approximation method for compound stochastic optimization problems. SIAM J. Optim. 23, 2231\u20132263 (2013)","journal-title":"SIAM J. Optim."},{"key":"512_CR9","doi-asserted-by":"publisher","first-page":"1327","DOI":"10.1214\/aoms\/1177698258","volume":"39","author":"V Fabian","year":"1968","unstructured":"Fabian, V.: On asymptotic normality in stochastic approximation. Ann. Math. Stat. 39, 1327\u20131332 (1968)","journal-title":"Ann. Math. Stat."},{"key":"512_CR10","unstructured":"Finn, C., Abbeel, P., Levine, S.: Model-agnostic meta-learning for fast adaptation of deep networks. In: Proceedings of the 34th International Conference on Machine Learning, vol.\u00a070, pp. 1126\u20131135 (2017)"},{"key":"512_CR11","unstructured":"Gao, H., Huang, H.: Fast training method for stochastic compositional optimization problems. In: Advances in Neural Information Processing Systems, vol.\u00a034, pp. 25334\u201325345 (2021)"},{"key":"512_CR12","unstructured":"Gao, H., Li, J., Huang, H.: On the convergence of local stochastic compositional gradient descent with momentum. In: Proceedings of the 39th International Conference on Machine Learning, vol.\u00a0162, pp. 7017\u20137035 (2022)"},{"key":"512_CR13","doi-asserted-by":"publisher","first-page":"960","DOI":"10.1137\/18M1230542","volume":"30","author":"S Ghadimi","year":"2020","unstructured":"Ghadimi, S., Ruszczynski, A., Wang, M.: A single timescale stochastic approximation method for nested stochastic optimization. SIAM J. Optim. 30, 960\u2013979 (2020)","journal-title":"SIAM J. Optim."},{"key":"512_CR14","unstructured":"Ghadimi, S., Wang, M.: Approximation methods for bilevel programming, arXiv preprint arXiv:1802.02246 (2018)"},{"key":"512_CR15","unstructured":"Guo, Z., Hu, Q., Zhang, L., Yang, T.: Randomized stochastic variance-reduced methods for multi-task stochastic bilevel optimization, arXiv preprint arXiv:2105.02266 (2021)"},{"key":"512_CR16","doi-asserted-by":"publisher","first-page":"147","DOI":"10.1137\/20M1387341","volume":"33","author":"M Hong","year":"2023","unstructured":"Hong, M., Wai, H.-T., Wang, Z., Yang, Z.: A two-timescale stochastic algorithm framework for bilevel optimization: complexity analysis and application to actor-critic. SIAM J. Optim. 33, 147\u2013180 (2023)","journal-title":"SIAM J. Optim."},{"key":"512_CR17","doi-asserted-by":"publisher","first-page":"2103","DOI":"10.1137\/19M1284865","volume":"30","author":"Y Hu","year":"2020","unstructured":"Hu, Y., Chen, X., He, N.: Sample complexity of sample average approximation for conditional stochastic optimization. SIAM J. Optim. 30, 2103\u20132133 (2020)","journal-title":"SIAM J. Optim."},{"key":"512_CR18","doi-asserted-by":"crossref","unstructured":"Huo, Z., Gu, B., Liu, J., Huang, H.: Accelerated method for stochastic composition optimization with nonsmooth regularization. In: Proceedings of the 32nd AAAI Conference on Artificial Intelligence, pp. 3287\u20133294 (2018)","DOI":"10.1609\/aaai.v32i1.11795"},{"key":"512_CR19","unstructured":"Ji, K., Yang, J., Liang, Y.: Bilevel optimization: Convergence analysis and enhanced design. In: Proceedings of the 38th International Conference on Machine Learning, vol.\u00a0139, pp. 4882\u20134892 (2021)"},{"key":"512_CR20","unstructured":"Jiang, W., Wang, B., Wang, Y., Zhang, L., Yang, T.: Optimal algorithms for stochastic multi-level compositional optimization. In: Proceedings of the 39th International Conference on Machine Learning, vol 162, pp. 10195\u201310216 (2022)"},{"key":"512_CR21","doi-asserted-by":"publisher","first-page":"2159","DOI":"10.1137\/16M1086133","volume":"56","author":"J Lei","year":"2018","unstructured":"Lei, J., Chen, H.F., Fang, H.T.: Asymptotic properties of primal-dual algorithm for distributed stochastic optimization over random networks with imperfect communications. SIAM J. Control. Optim. 56, 2159\u20132188 (2018)","journal-title":"SIAM J. Control. Optim."},{"key":"512_CR22","first-page":"5813","volume":"44","author":"L Liu","year":"2021","unstructured":"Liu, L., Liu, J., Tao, D.: Variance reduced methods for non-convex composition optimization. IEEE Trans Pattern Anal Mach Intell. 44, 5813\u20135825 (2021)","journal-title":"IEEE Trans Pattern Anal Mach Intell."},{"key":"512_CR23","doi-asserted-by":"crossref","unstructured":"Morral, G., Bianchi, P., Fort, G., Jakubowicz, J.: Distributed stochastic approximation: the price of non-double stochasticity. In: 2012 Conference Record of the Forty Sixth Asilomar Conference on Signals, Systems and Computers, pp. 1473\u20131477 (2012)","DOI":"10.1109\/ACSSC.2012.6489272"},{"key":"512_CR24","doi-asserted-by":"publisher","first-page":"92","DOI":"10.1109\/MSP.2020.2975210","volume":"37","author":"A Nedic","year":"2020","unstructured":"Nedic, A.: Distributed gradient methods for convex machine learning problems in networks: distributed optimization. IEEE Signal Process. Mag. 37, 92\u2013101 (2020)","journal-title":"IEEE Signal Process. Mag."},{"key":"512_CR25","volume-title":"Introduction to Optimization","author":"BT Polyak","year":"1987","unstructured":"Polyak, B.T.: Introduction to Optimization. Optimization Software, New York (1987)"},{"key":"512_CR26","doi-asserted-by":"publisher","first-page":"838","DOI":"10.1137\/0330046","volume":"30","author":"BT Polyak","year":"1992","unstructured":"Polyak, B.T., Juditsky, A.B.: Acceleration of stochastic approximation by averaging. SIAM J. Control. Optim. 30, 838\u2013855 (1992)","journal-title":"SIAM J. Control. Optim."},{"key":"512_CR27","doi-asserted-by":"crossref","unstructured":"Pu, S., Shi, W., Xu, J., Nedic, A.: A push-pull gradient method for distributed optimization in networks. In: IEEE Conference on Decision and Control, pp. 3385\u20133390 (2018)","DOI":"10.1109\/CDC.2018.8619047"},{"key":"512_CR28","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1109\/TAC.2020.2972824","volume":"66","author":"S Pu","year":"2021","unstructured":"Pu, S., Shi, W., Xu, J., Nedic, A.: Push-pull gradient methods for distributed optimization in networks. IEEE Trans. Autom. Control 66, 1\u201316 (2021)","journal-title":"IEEE Trans. Autom. Control"},{"key":"512_CR29","unstructured":"Qi, Q., Luo, Y., Xu, Z., Ji, S., Yang, T.: Stochastic optimization of areas under precision-recall curves with provable convergence. In: Advances in Neural Information Processing Systems, vol 34, pp. 1752\u20131765 (2021)"},{"key":"512_CR30","doi-asserted-by":"publisher","first-page":"1245","DOI":"10.1109\/TCNS.2017.2698261","volume":"5","author":"G Qu","year":"2018","unstructured":"Qu, G., Li, N.: Harnessing smoothness to accelerate distributed optimization. IEEE Trans. Control Netw. Syst. 5, 1245\u20131260 (2018)","journal-title":"IEEE Trans. Control Netw. Syst."},{"key":"512_CR31","unstructured":"Rakhlin, A., Shamir, O., Sridharan, K.: Making gradient descent optimal for strongly convex stochastic optimization. In: Proceedings of the 29th International Coference on Machine Learning, pp. 1571\u20131578 (2012)"},{"key":"512_CR32","doi-asserted-by":"publisher","DOI":"10.1016\/j.jocs.2020.101280","volume":"49","author":"J Ren","year":"2021","unstructured":"Ren, J., Haupt, J., Guo, Z.: Communication-efficient hierarchical distributed optimization for multi-agent policy evaluation. J. Comput. Sci. 49, 101280 (2021)","journal-title":"J. Comput. Sci."},{"key":"512_CR33","doi-asserted-by":"publisher","first-page":"2301","DOI":"10.1137\/20M1312952","volume":"59","author":"A Ruszczynski","year":"2021","unstructured":"Ruszczynski, A.: A stochastic subgradient method for nonsmooth nonconvex multilevel composition optimization. SIAM J. Control. Optim. 59, 2301\u20132320 (2021)","journal-title":"SIAM J. Control. Optim."},{"key":"512_CR34","first-page":"426","volume":"2","author":"AK Sahu","year":"2016","unstructured":"Sahu, A.K., Kar, S., Moura, J.M.F., Poor, H.V.: Distributed constrained recursive nonlinear least-squares estimation: algorithms and asymptotics. IEEE Trans. Signal Inf. Process. Netw. 2, 426\u2013441 (2016)","journal-title":"IEEE Trans. Signal Inf. Process. Netw."},{"key":"512_CR35","doi-asserted-by":"publisher","DOI":"10.1007\/0-387-32792-4","volume-title":"Non-Negative Matrices and Markov Chains","author":"E Seneta","year":"1981","unstructured":"Seneta, E.: Non-Negative Matrices and Markov Chains. Springer, New York (1981)"},{"key":"512_CR36","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.automatica.2021.110092","volume":"136","author":"X Sha","year":"2022","unstructured":"Sha, X., Zhang, J., You, K., Zhang, K., Basar, T.: Fully asynchronous policy evaluation in distributed reinforcement learning over networks. Automatica 136, 1\u201311 (2022)","journal-title":"Automatica"},{"key":"512_CR37","doi-asserted-by":"publisher","first-page":"1775","DOI":"10.1109\/TSP.2022.3160238","volume":"70","author":"Z Song","year":"2022","unstructured":"Song, Z., Shi, L., Pu, S., Yan, M.: Compressed gradient tracking for decentralized optimization over general directed networks. IEEE Trans. Signal Process. 70, 1775\u20131787 (2022)","journal-title":"IEEE Trans. Signal Process."},{"key":"512_CR38","unstructured":"Wang, B., Yuan, Z., Ying, Y., Yang, T.: Memory-based optimization methods for model-agnostic meta-learning, arXiv preprint arXiv:2106.04911 (2021)"},{"key":"512_CR39","doi-asserted-by":"publisher","first-page":"419","DOI":"10.1007\/s10107-016-1017-3","volume":"161","author":"M Wang","year":"2017","unstructured":"Wang, M., Fang, E.X., Liu, H.: Stochastic compositional gradient descent: algorithms for minimizing compositions of expected-value functions. Math. Program. 161, 419\u2013449 (2017)","journal-title":"Math. Program."},{"key":"512_CR40","first-page":"1","volume":"18","author":"M Wang","year":"2017","unstructured":"Wang, M., Liu, J., Fang, E.: Accelerating stochastic composition optimization. J. Mach. Learn. Res. 18, 1\u201323 (2017)","journal-title":"J. Mach. Learn. Res."},{"key":"512_CR41","doi-asserted-by":"publisher","first-page":"315","DOI":"10.1109\/LCSYS.2018.2834316","volume":"2","author":"R Xin","year":"2018","unstructured":"Xin, R., Khan, U.A.: A linear algorithm for optimization over directed graphs with geometric convergence. IEEE Control Syst. Lett. 2, 315\u2013320 (2018)","journal-title":"IEEE Control Syst. Lett."},{"key":"512_CR42","doi-asserted-by":"publisher","first-page":"1869","DOI":"10.1109\/JPROC.2020.3024266","volume":"108","author":"R Xin","year":"2020","unstructured":"Xin, R., Pu, S., Nedic, A., Khan, U.A.: A general framework for decentralized optimization with first-order methods. Proc. IEEE 108, 1869\u20131889 (2020)","journal-title":"Proc. IEEE"},{"key":"512_CR43","doi-asserted-by":"publisher","first-page":"616","DOI":"10.1137\/18M1164846","volume":"29","author":"S Yang","year":"2019","unstructured":"Yang, S., Wang, M., Fang, E.X.: Multilevel stochastic gradient methods for nested composition optimization. SIAM J. Optim. 29, 616\u2013659 (2019)","journal-title":"SIAM J. Optim."},{"key":"512_CR44","unstructured":"Yang, S., Zhang, X., Wang, M.: Decentralized gossip-based stochastic bilevel optimization over communication networks. In: Advances in Neural Information Processing Systems, vol. 35, pp. 238\u2013252 (2022)"},{"key":"512_CR45","doi-asserted-by":"publisher","first-page":"1131","DOI":"10.1137\/19M1285457","volume":"31","author":"J Zhang","year":"2021","unstructured":"Zhang, J., Xiao, L.: Multilevel composite stochastic optimization via nested variance reduction. SIAM J. Optim. 31, 1131\u20131157 (2021)","journal-title":"SIAM J. Optim."},{"key":"512_CR46","unstructured":"Zhang, J., Xiao, L.: A stochastic composite gradient method with incremental variance reduction. In: Advances in Neural Information Processing Systems, pp. 9078\u20139088 (2019)"},{"key":"512_CR47","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.sysconle.2022.105252","volume":"165","author":"S Zhao","year":"2022","unstructured":"Zhao, S., Chen, X.-M., Liu, Y.: Asymptotic properties of dual averaging algorithm for constrained distributed stochastic optimization. Syst. Control Lett. 165, 1\u201314 (2022)","journal-title":"Syst. Control Lett."},{"key":"512_CR48","unstructured":"Zhao, S., Liu, Y.: Asymptotic properties of $$\\cal{S}-\\cal{AB}$$ method with diminishing stepsize, arXiv preprint arXiv:2109.07981 (2021)"}],"container-title":["Computational Optimization and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10589-023-00512-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10589-023-00512-0\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10589-023-00512-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,1,4]],"date-time":"2024-01-04T12:05:21Z","timestamp":1704369921000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10589-023-00512-0"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,8,3]]},"references-count":48,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2024,1]]}},"alternative-id":["512"],"URL":"https:\/\/doi.org\/10.1007\/s10589-023-00512-0","relation":{},"ISSN":["0926-6003","1573-2894"],"issn-type":[{"type":"print","value":"0926-6003"},{"type":"electronic","value":"1573-2894"}],"subject":[],"published":{"date-parts":[[2023,8,3]]},"assertion":[{"value":"23 March 2022","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"20 July 2023","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"3 August 2023","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors have no competing interests to declare that are relevant to the content of this article.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}]}}