{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,6]],"date-time":"2025-11-06T20:35:49Z","timestamp":1762461349136,"version":"build-2065373602"},"reference-count":59,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2025,9,22]],"date-time":"2025-09-22T00:00:00Z","timestamp":1758499200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2025,9,22]],"date-time":"2025-09-22T00:00:00Z","timestamp":1758499200000},"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":["12101262","12201546","12001072","12101097"],"award-info":[{"award-number":["12101262","12201546","12001072","12101097"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12171063"],"award-info":[{"award-number":["12171063"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100021171","name":"Basic and Applied Basic Research Foundation of Guangdong Province","doi-asserted-by":"publisher","award":["2022A1515010263"],"award-info":[{"award-number":["2022A1515010263"]}],"id":[{"id":"10.13039\/501100021171","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Sci Comput"],"published-print":{"date-parts":[[2025,11]]},"DOI":"10.1007\/s10915-025-03066-x","type":"journal-article","created":{"date-parts":[[2025,9,22]],"date-time":"2025-09-22T06:13:20Z","timestamp":1758521600000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Zeroth-order Proximal Clipped Gradient Method with Shifts for Distributed Stochastic Composite Optimization Problems with Infinite Variance"],"prefix":"10.1007","volume":"105","author":[{"given":"Zhen-Ping","family":"Yang","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Pin-Bo","family":"Chen","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yong","family":"Zhao","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4330-1519","authenticated-orcid":false,"given":"Lin","family":"Chen","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2025,9,22]]},"reference":[{"key":"3066_CR1","doi-asserted-by":"crossref","first-page":"2119","DOI":"10.1109\/TSP.2022.3162958","volume":"70","author":"Z Akhtar","year":"2022","unstructured":"Akhtar, Z., Rajawat, K.: Zeroth and first order stochastic frank-wolfe algorithms for constrained optimization. IEEE Trans. Signal Process. 70, 2119\u20132135 (2022)","journal-title":"IEEE Trans. Signal Process."},{"key":"3066_CR2","unstructured":"Armacki, A., Sharma, P., Joshi, G., Bajovic, D., Jakovetic, D., Kar, S.: High-probability convergence bounds for nonlinear stochastic gradient descent under heavy-tailed noise. arXiv:2310.18784 (2024)"},{"issue":"4","key":"3066_CR3","doi-asserted-by":"crossref","first-page":"563","DOI":"10.1109\/JSTSP.2014.2317284","volume":"8","author":"B Baingana","year":"2014","unstructured":"Baingana, B., Mateos, G., Giannakis, G.B.: Proximal-gradient algorithms for tracking cascades over social networks. IEEE J. Selected. Topics. Signal. Process. 8(4), 563\u2013575 (2014)","journal-title":"IEEE J. Selected. Topics. Signal. Process."},{"issue":"1","key":"3066_CR4","doi-asserted-by":"crossref","first-page":"35","DOI":"10.1007\/s10208-021-09499-8","volume":"22","author":"K Balasubramanian","year":"2022","unstructured":"Balasubramanian, K., Ghadimi, S.: Zeroth-order nonconvex stochastic optimization: handling constraints, high dimensionality, and saddle points. Found. Comput. Math. 22(1), 35\u201376 (2022)","journal-title":"Found. Comput. Math."},{"key":"3066_CR5","first-page":"3459","volume":"31","author":"K Balasubramanian","year":"2018","unstructured":"Balasubramanian, K., Ghadimi, S.: Zeroth-order (Non)-convex stochastic optimization via conditional gradient and gradient updates. Proc. 32nd Int. Conference Neural Inf. Process. Syst. 31, 3459\u20133468 (2018)","journal-title":"Proc. 32nd Int. Conference Neural Inf. Process. Syst."},{"key":"3066_CR6","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4419-9467-7","volume-title":"Convex Analysis and Monotone Operator Theory in Hilbert Spaces","author":"HH Bauschke","year":"2011","unstructured":"Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer, New York (2011)"},{"issue":"297","key":"3066_CR7","doi-asserted-by":"crossref","first-page":"33","DOI":"10.1080\/01621459.1962.10482149","volume":"57","author":"G Bennett","year":"1962","unstructured":"Bennett, G.: Probability inequalities for the sum of independent random variables. J. Am. Stat. Assoc. 57(297), 33\u201345 (1962)","journal-title":"J. Am. Stat. Assoc."},{"key":"3066_CR8","unstructured":"Chezhegov, S., Klyukin, Y., Semenov, A., Beznosikov, A., Gasnikov, A., Horv\u00e1th, S., Tak\u00e1\u010d, M., Gorbunov, E.: Gradient clipping improves AdaGrad when the noise is heavy-tailed. arXiv:2406.04443 (2024)"},{"issue":"8","key":"3066_CR9","doi-asserted-by":"crossref","first-page":"4289","DOI":"10.1109\/TSP.2012.2198470","volume":"60","author":"J Chen","year":"2012","unstructured":"Chen, J., Sayed, A.H.: Diffusion adaptation strategies for distributed optimization and learning over networks. IEEE Trans. Signal Process. 60(8), 4289\u20134305 (2012)","journal-title":"IEEE Trans. Signal Process."},{"key":"3066_CR10","doi-asserted-by":"crossref","unstructured":"Chen, P.Y., Zhang, H., Sharma, Y., Yi, J., Hsieh, C.J.: Zoo: Zeroth order optimization based black-box attacks to deep neural networks without training substitute models. In Proceedings of the 10th ACM Workshop on Artificial Intelligence and Security, 15-26 (2017)","DOI":"10.1145\/3128572.3140448"},{"issue":"6","key":"3066_CR11","doi-asserted-by":"crossref","first-page":"2854","DOI":"10.1109\/TAC.2016.2626578","volume":"62","author":"K Cohen","year":"2017","unstructured":"Cohen, K., Nedi\u0107, A., Srikant, R.: Distributed learning algorithms for spectrum sharing in spatial random access wireless networks. IEEE Trans. Autom. Control. 62(6), 2854\u20132869 (2017)","journal-title":"IEEE Trans. Autom. Control."},{"issue":"11","key":"3066_CR12","doi-asserted-by":"crossref","first-page":"5974","DOI":"10.1109\/TAC.2017.2705559","volume":"62","author":"K Cohen","year":"2017","unstructured":"Cohen, K., Nedi\u0107, A., Srikant, R.: On projected stochastic gradient descent algorithm with weighted averaging for least squares regression. IEEE Trans. Autom. Control. 62(11), 5974\u20135981 (2017)","journal-title":"IEEE Trans. Autom. Control."},{"issue":"1","key":"3066_CR13","doi-asserted-by":"crossref","first-page":"207","DOI":"10.1137\/18M1178244","volume":"29","author":"D Davis","year":"2019","unstructured":"Davis, D., Drusvyatskiy, D.: Stochastic model-based minimization of weakly convex functions. SIAM J. Optim. 29(1), 207\u2013239 (2019)","journal-title":"SIAM J. Optim."},{"issue":"3","key":"3066_CR14","first-page":"471","volume":"8","author":"JC Duchi","year":"2019","unstructured":"Duchi, J.C., Ruan, F.: Solving (most) of a set of quadratic equalities: composite optimization for robust phase retrieval. Inf. Inference: A J. IMA 8(3), 471\u2013529 (2019)","journal-title":"Inf. Inference: A J. IMA"},{"issue":"1","key":"3066_CR15","doi-asserted-by":"crossref","first-page":"109","DOI":"10.1016\/S0304-4149(00)00086-7","volume":"93","author":"KV Dzhaparidze","year":"2001","unstructured":"Dzhaparidze, K.V., Zanten, J.: On bernstein-type inequalities for martingales. Stoch. Process. Appl. 93(1), 109\u2013117 (2001)","journal-title":"Stoch. Process. Appl."},{"issue":"3","key":"3066_CR16","doi-asserted-by":"crossref","first-page":"473","DOI":"10.1016\/j.acha.2013.08.003","volume":"36","author":"YC Eldar","year":"2014","unstructured":"Eldar, Y.C., Mendelson, S.: Phase retrieval: stability and recovery guarantees. Appl. Comput. Harmon. Anal. 36(3), 473\u2013494 (2014)","journal-title":"Appl. Comput. Harmon. Anal."},{"key":"3066_CR17","unstructured":"Elesedy, B., Hutter, M: U-clip: On-average unbiased stochastic gradient clipping. arXiv:2302.02971 (2023)"},{"issue":"1","key":"3066_CR18","doi-asserted-by":"crossref","first-page":"100","DOI":"10.1214\/aop\/1176996452","volume":"3","author":"DA Freedman","year":"1975","unstructured":"Freedman, D.A.: On tail probabilities for martingales. Ann. Probab. 3(1), 100\u2013118 (1975)","journal-title":"Ann. Probab."},{"issue":"4","key":"3066_CR19","doi-asserted-by":"crossref","first-page":"2341","DOI":"10.1137\/120880811","volume":"23","author":"S Ghadimi","year":"2013","unstructured":"Ghadimi, S., Lan, G.: Stochastic first-and zeroth-order methods for nonconvex stochastic programming. SIAM J. Optim. 23(4), 2341\u20132368 (2013)","journal-title":"SIAM J. Optim."},{"key":"3066_CR20","first-page":"15042","volume":"33","author":"E Gorbunov","year":"2020","unstructured":"Gorbunov, E., Danilova, M., Gasnikov, A.: Stochastic optimization with heavy-tailed noise via accelerated gradient clipping. Proc. 35th Int. Conference Neural Int. Process. Syst. 33, 15042\u201315053 (2020)","journal-title":"Proc. 35th Int. Conference Neural Int. Process. Syst."},{"key":"3066_CR21","doi-asserted-by":"crossref","first-page":"2679","DOI":"10.1007\/s10957-024-02533-z","volume":"203","author":"E Gorbunov","year":"2024","unstructured":"Gorbunov, E., Danilova, M., Shibaev, I., Dvurechensky, P., Gasnikov, A.: High-probability complexity bounds for non-smooth stochastic convex optimization with heavy-tailed noise. J. Optim. Theory Appl. 203, 2679\u20132738 (2024)","journal-title":"J. Optim. Theory Appl."},{"issue":"2","key":"3066_CR22","doi-asserted-by":"crossref","first-page":"1210","DOI":"10.1137\/19M1259225","volume":"32","author":"E Gorbunov","year":"2022","unstructured":"Gorbunov, E., Dvurechensky, P., Gasnikov, A.: An accelerated method for derivative-free smooth stochastic convex optimization. SIAM J. Optim. 32(2), 1210\u20131238 (2022)","journal-title":"SIAM J. Optim."},{"key":"3066_CR23","first-page":"7241","volume":"162","author":"A Gasnikov","year":"2022","unstructured":"Gasnikov, A., Novitskii, A., Novitskii, V., Abdukhakimov, F., Kamzolov, D., Beznosikov, A., Takac, M., Dvurechensky, P., Gu, B.: The power of first-order smooth optimization for black-box non-smooth problems. Proc. 39th Int. Conference. Mach. Learning, PMLR 162, 7241\u20137265 (2022)","journal-title":"Proc. 39th Int. Conference. Mach. Learning, PMLR"},{"key":"3066_CR24","first-page":"15951","volume":"235","author":"E Gorbunov","year":"2024","unstructured":"Gorbunov, E., Sadiev, A., Danilova, M., Horv\u00e1th, S., Gidel, G., Dvurechensky, P., Gasnikov, A., Richt\u00e1rik, P.: High-probability convergence for composite and distributed stochastic minimization and variational inequalities with heavy-tailed noise. Proc. 41st Int. Conference. Mach. Learning, PMLR 235, 15951\u201316070 (2024)","journal-title":"Proc. 41st Int. Conference. Mach. Learning, PMLR"},{"issue":"01","key":"3066_CR25","doi-asserted-by":"crossref","first-page":"1503","DOI":"10.1609\/aaai.v33i01.33011503","volume":"33","author":"F Huang","year":"2019","unstructured":"Huang, F., Gu, B., Huo, Z., Chen, S., Huang, H.: Faster gradient-free proximal stochastic methods for nonconvex nonsmooth optimization. Proc. AAAI Conference Artificial Intelligence 33(01), 1503\u20131510 (2019)","journal-title":"Proc. AAAI Conference Artificial Intelligence"},{"issue":"2","key":"3066_CR26","doi-asserted-by":"crossref","first-page":"394","DOI":"10.1137\/21M145896X","volume":"33","author":"D Jakoveti\u0107","year":"2023","unstructured":"Jakoveti\u0107, D., Bajovi\u0107, D., Sahu, A.K., Kar, S., Milo\u0161sevi\u0107, N., Stamenkovi\u0107, D.: Nonlinear gradient mappings and stochastic optimization: a general framework with applications to heavy-tail noise. SIAM J. Optim. 33(2), 394\u2013423 (2023)","journal-title":"SIAM J. Optim."},{"key":"3066_CR27","first-page":"3100","volume":"97","author":"K Ji","year":"2019","unstructured":"Ji, K., Wang, Z., Zhou, Y., Liang, Y.: Improved zeroth-order variance reduced algorithms and analysis for nonconvex optimization. Proc. 36th Int. Conference Mach. Learning, PMLR 97, 3100\u20133109 (2019)","journal-title":"Proc. 36th Int. Conference Mach. Learning, PMLR"},{"issue":"2","key":"3066_CR28","doi-asserted-by":"crossref","first-page":"386","DOI":"10.1137\/20M1315403","volume":"32","author":"DS Kalogerias","year":"2022","unstructured":"Kalogerias, D.S., Powell, W.B.: Zeroth-order stochastic compositional algorithms for risk-aware learning. SIAM J. Optim. 32(2), 386\u2013416 (2022)","journal-title":"SIAM J. Optim."},{"issue":"2","key":"3066_CR29","doi-asserted-by":"crossref","first-page":"499","DOI":"10.1007\/s10957-023-02297-y","volume":"199","author":"A Khaled","year":"2023","unstructured":"Khaled, A., Sebbouh, O., Loizou, N., Gower, R.M., Richt\u00e1rik, P.: Unified analysis of stochastic gradient methods for composite convex and smooth optimization. J. Optim. Theory Appl. 199(2), 499\u2013540 (2023)","journal-title":"J. Optim. Theory Appl."},{"key":"3066_CR30","first-page":"17343","volume":"202","author":"A Koloskova","year":"2023","unstructured":"Koloskova, A., Hendrikx, H., Stich, S.U.: Revisiting gradient clipping: stochastic bias and tight convergence guarantees. Proc. 40th Int. Conference Mach. Learning, PMLR, 202, 17343\u201317363 (2023)","journal-title":"Proc. 40th Int. Conference Mach. Learning, PMLR,"},{"key":"3066_CR31","unstructured":"Kornilov, N., Dorn, Y., Lobanov, A., Kutuzov, N., Shibaev, I., Gorbunov, E., Gasnikov, A., Nazin, A.: Median clipping for zeroth-order non-smooth convex optimization and multi arm bandit problem with heavy-tailed symmetric noise. arXiv:2402.02461 (2024)"},{"issue":"1","key":"3066_CR32","doi-asserted-by":"crossref","first-page":"37","DOI":"10.1007\/s10287-023-00470-2","volume":"20","author":"N Kornilov","year":"2023","unstructured":"Kornilov, N., Gasnikov, A., Dvurechensky, P., Dvinskikh, D.: Gradient-free methods for non-smooth convex stochastic optimization with heavy-tailed noise on convex compact. CMS 20(1), 37 (2023)","journal-title":"CMS"},{"key":"3066_CR33","first-page":"64083","volume":"36","author":"N Kornilov","year":"2023","unstructured":"Kornilov, N., Shamir, O., Lobanov, A., Dvinskikh, D., Gasnikov, A., Shibaev, I., Gorbunov, E., Horv\u00e1th, S.: Accelerated zeroth-order method for non-smooth stochastic convex optimization problem with infinite variance. Proc. 36th Int. Conference Neural Inf. Proc. Syst. 36, 64083\u201364102 (2023)","journal-title":"Proc. 36th Int. Conference Neural Inf. Proc. Syst."},{"issue":"3","key":"3066_CR34","doi-asserted-by":"crossref","first-page":"731","DOI":"10.1007\/s10589-021-00313-3","volume":"80","author":"V Kungurtsev","year":"2021","unstructured":"Kungurtsev, V., Rinaldi, F.: A zeroth order method for stochastic weakly convex optimization. Comput. Optim. Appl. 80(3), 731\u2013753 (2021)","journal-title":"Comput. Optim. Appl."},{"key":"3066_CR35","unstructured":"Lan, G., Li, T., Xu, Y.: Projected gradient methods for nonconvex and stochastic optimization: new complexities and auto-conditioned stepsizes. arXiv:2412.14291 (2024)"},{"key":"3066_CR36","unstructured":"Li, S., Liu, Y.: High probability guarantees for nonconvex stochastic gradient descent with heavy tails. In Proceedings of the 39th International Conference on Machine Learning, PMLR, 162, 12931-12963 (2022)"},{"key":"3066_CR37","unstructured":"Lin, T., Zheng, Z., Jordan, M.I.: Gradient-free methods for deterministic and stochastic nonsmooth nonconvex optimization. In Proceedings of the 36th International Conference on Neural Information Processing Systems, 35, 26160-26175 (2022)"},{"key":"3066_CR38","unstructured":"Liu, S., Kailkhura, B., Chen, P. Y., Ting, P., Chang, S., Amini, L.: Zeroth-order stochastic variance reduction for nonconvex optimization. In Proceedings of the 32nd International Conference on Neural Information Processing Systems, 31, 3731-3741 (2018)"},{"key":"3066_CR39","unstructured":"Liu, Z., Zhou, Z.: Stochastic nonsmooth convex optimization with heavy-tailed noises: High-probability bound, in-expectation rate and initial distance adaptation. arXiv:2303.12277 (2023)"},{"key":"3066_CR40","unstructured":"Mai, V. V., Johansson, M.: Stability and convergence of stochastic gradient clipping: Beyond lipschitz continuity and smoothness. In Proceedings of the 38th International Conference on Machine Learning, PMLR, 139, 7325-7335 (2021)"},{"key":"3066_CR41","unstructured":"Nazari, P., Tarzanagh, D.A., Michailidis, G.: Adaptive first-and zeroth-order methods for weakly convex stochastic optimization problems. arXiv:2005.09261 (2020)"},{"issue":"11","key":"3066_CR42","doi-asserted-by":"crossref","first-page":"5538","DOI":"10.1109\/TAC.2017.2690401","volume":"62","author":"A Nedi\u0107","year":"2017","unstructured":"Nedi\u0107, A., Olshevsky, A., Uribe, C.A.: Fast convergence rates for distributed non-bayesian learning. IEEE Trans. Autom. Control 62(11), 5538\u20135553 (2017)","journal-title":"IEEE Trans. Autom. Control"},{"key":"3066_CR43","doi-asserted-by":"crossref","first-page":"527","DOI":"10.1007\/s10208-015-9296-2","volume":"17","author":"Y Nesterov","year":"2017","unstructured":"Nesterov, Y., Spokoiny, V.: Random gradient-free minimization of convex functions. Found. Comput. Math. 17, 527\u2013566 (2017)","journal-title":"Found. Comput. Math."},{"key":"3066_CR44","unstructured":"Nguyen, T.D., Nguyen, T.H., Ene, A., Nguyen, H.L.: High probability convergence of clipped-sgd under heavy-tailed noise. arXiv:2302.05437 (2023)"},{"key":"3066_CR45","first-page":"24191","volume":"35","author":"TD Nguyen","year":"2023","unstructured":"Nguyen, T.D., Nguyen, T.H., Ene, A., Le Nguyen, H.: Improved convergence in high probability of clipped gradient methods with heavy tailed noise. Proc. 37th Int. Conference Neural Inf. Proc. Syst. 35, 24191\u201324222 (2023)","journal-title":"Proc. 37th Int. Conference Neural Inf. Proc. Syst."},{"issue":"5","key":"3066_CR46","doi-asserted-by":"crossref","first-page":"2679","DOI":"10.1137\/22M1494270","volume":"45","author":"S Pougkakiotis","year":"2023","unstructured":"Pougkakiotis, S., Kalogerias, D.: A zeroth-order proximal stochastic gradient method for weakly convex stochastic optimization. SIAM J. Sci. Comput. 45(5), 2679\u20132702 (2023)","journal-title":"SIAM J. Sci. Comput."},{"key":"3066_CR47","first-page":"856","volume":"238","author":"N Puchkin","year":"2024","unstructured":"Puchkin, N., Gorbunov, E., Kutuzov, N., Gasnikov, A.: Breaking the heavy-tailed noise barrier in stochastic optimization problems. Proc. 27th Int. Conference Artificial Intelligence and Stat. 238, 856\u2013864 (2024)","journal-title":"Proc. 27th Int. Conference Artificial Intelligence and Stat."},{"issue":"12","key":"3066_CR48","doi-asserted-by":"crossref","first-page":"4007","DOI":"10.1109\/TAC.2016.2529958","volume":"61","author":"S Pu","year":"2016","unstructured":"Pu, S., Garcia, A., Lin, Z.: Noise reduction by swarming in social foraging. IEEE Trans. Autom. Control 61(12), 4007\u20134013 (2016)","journal-title":"IEEE Trans. Autom. Control"},{"issue":"1","key":"3066_CR49","doi-asserted-by":"crossref","first-page":"409","DOI":"10.1007\/s10107-020-01487-0","volume":"187","author":"S Pu","year":"2021","unstructured":"Pu, S., Nedi\u0107, A.: Distributed stochastic gradient tracking methods. Math. Program. 187(1), 409\u2013457 (2021)","journal-title":"Math. Program."},{"key":"3066_CR50","unstructured":"Reisizadeh, A., Li, H., Das, S., Jadbabaie, A.: Variance-reduced clipping for non-convex optimization. arXiv:2303.00883 (2023)"},{"key":"3066_CR51","doi-asserted-by":"crossref","DOI":"10.1002\/9781118631980","volume-title":"Simulation and the Monte Carlo method","author":"R Rubinstein","year":"2016","unstructured":"Rubinstein, R., Kroese, D.: Simulation and the Monte Carlo method, vol. 10. John Wiley & Sons, Hoboken, New Jersey (2016)"},{"key":"3066_CR52","first-page":"29563","volume":"202","author":"A Sadiev","year":"2023","unstructured":"Sadiev, A., Danilova, M., Gorbunov, E., Horv\u00e1th, S., Gidel, G., Dvurechensky, P., Gasnikov, A., Richt\u00e1rik, P.: High-probability bounds for stochastic optimization and variational inequalities: the case of unbounded variance. Proc. 40th Int. Conference Mach. Learning, PMLR 202, 29563\u201329648 (2023)","journal-title":"Proc. 40th Int. Conference Mach. Learning, PMLR"},{"issue":"1","key":"3066_CR53","first-page":"1703","volume":"18","author":"O Shamir","year":"2017","unstructured":"Shamir, O.: An optimal algorithm for bandit and zero-order convex optimization with two-point feedback. J. Mach. Learning Research 18(1), 1703\u20131713 (2017)","journal-title":"J. Mach. Learning Research"},{"key":"3066_CR54","first-page":"2951","volume":"25","author":"J Snoek","year":"2012","unstructured":"Snoek, J., Larochelle, H., Adams, R.: Practical Bayesian optimization of machine learning algorithms. Proc. 25th Int. Conference Neural Inf. Process. Syst. 25, 2951\u20132959 (2012)","journal-title":"Proc. 25th Int. Conference Neural Inf. Process. Syst."},{"issue":"3","key":"3066_CR55","doi-asserted-by":"crossref","first-page":"332","DOI":"10.1109\/9.119632","volume":"37","author":"JC Spall","year":"1992","unstructured":"Spall, J.C.: Multivariate stochastic approximation using a simultaneous perturbation gradient approximation. IEEE Trans. Autom. Control 37(3), 332\u2013341 (1992)","journal-title":"IEEE Trans. Autom. Control"},{"key":"3066_CR56","doi-asserted-by":"crossref","unstructured":"Taskar, B., Chatalbashev, V., Koller, D., Guestrin, C.: Learning structured prediction models: A large margin approach. In: Proceedings of the 22nd International Conference on Machine Learning, 896-903 (2005)","DOI":"10.1145\/1102351.1102464"},{"key":"3066_CR57","volume":"142","author":"X Yi","year":"2022","unstructured":"Yi, X., Zhang, S., Yang, T., Johansson, K.H.: Zeroth-order algorithms for stochastic distributed nonconvex optimization. Automatica 142, 110353 (2022)","journal-title":"Automatica"},{"key":"3066_CR58","unstructured":"Zhang, J., He, T., Sra, S., Jadbabaie, A.: Why gradient clipping accelerates training: A theoretical justification for adaptivity. In: Proceedings of the 10th International Conference on Learning Representations, https:\/\/openreview.net\/forum?id=BJgnXpVYwS (2020)"},{"key":"3066_CR59","first-page":"15511","volume":"33","author":"B Zhang","year":"2020","unstructured":"Zhang, B., Jin, J., Fang, C., Wang, L.: Improved analysis of clipping algorithms for non-convex optimization. Proc. 34th Int. Conference Neural Inf. Process. Syst. 33, 15511\u201315521 (2020)","journal-title":"Proc. 34th Int. Conference Neural Inf. Process. Syst."}],"container-title":["Journal of Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-025-03066-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10915-025-03066-x\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-025-03066-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,11,6]],"date-time":"2025-11-06T20:32:16Z","timestamp":1762461136000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10915-025-03066-x"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,9,22]]},"references-count":59,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2025,11]]}},"alternative-id":["3066"],"URL":"https:\/\/doi.org\/10.1007\/s10915-025-03066-x","relation":{},"ISSN":["0885-7474","1573-7691"],"issn-type":[{"type":"print","value":"0885-7474"},{"type":"electronic","value":"1573-7691"}],"subject":[],"published":{"date-parts":[[2025,9,22]]},"assertion":[{"value":"25 August 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"9 September 2025","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"12 September 2025","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"22 September 2025","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare no competing interests.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflicts of interest"}},{"value":"Not Applicable","order":3,"name":"Ethics","group":{"name":"EthicsHeading","label":"Ethical Approval and Consent to participate"}},{"value":"Not Applicable","order":4,"name":"Ethics","group":{"name":"EthicsHeading","label":"Human and Animal Ethics"}},{"value":"All authors consented the manuscript for publication","order":5,"name":"Ethics","group":{"name":"EthicsHeading","label":"Consent for publication"}}],"article-number":"36"}}