{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T00:55:26Z","timestamp":1740099326638,"version":"3.37.3"},"publisher-location":"Cham","reference-count":21,"publisher":"Springer International Publishing","isbn-type":[{"type":"print","value":"9783030168407"},{"type":"electronic","value":"9783030168414"}],"license":[{"start":{"date-parts":[[2019,4,3]],"date-time":"2019-04-03T00:00:00Z","timestamp":1554249600000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2020]]},"DOI":"10.1007\/978-3-030-16841-4_7","type":"book-chapter","created":{"date-parts":[[2019,4,2]],"date-time":"2019-04-02T12:12:08Z","timestamp":1554207128000},"page":"58-77","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Perturbed Proximal Descent to Escape Saddle Points for Non-convex and Non-smooth Objective Functions"],"prefix":"10.1007","author":[{"given":"Zhishen","family":"Huang","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1932-8159","authenticated-orcid":false,"given":"Stephen","family":"Becker","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2019,4,3]]},"reference":[{"key":"7_CR1","first-page":"1","volume":"137","author":"H Attouch","year":"2011","unstructured":"Attouch, H., Bolte, J., Svaiter, B.F.: Convergence of descent methods for semi-algebraic and tame problems: proximal algorithms, forward-backward splitting, and regularized Gauss-Seidel methods. Math. Program. 137, 1\u201339 (2011)","journal-title":"Math. Program."},{"key":"7_CR2","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-48311-5","volume-title":"Convex Analysis and Monotone Operator Theory in Hilbert Spaces","author":"HH Bauschke","year":"2017","unstructured":"Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces, 2nd edn. Springer, New York (2017)","edition":"2"},{"key":"7_CR3","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611974997","volume-title":"First-Order Methods in Optimization: MOS-SIAM Series on Optimization","author":"A Beck","year":"2017","unstructured":"Beck, A.: First-Order Methods in Optimization: MOS-SIAM Series on Optimization. Society for Industrial and Applied Mathematics, Philadelphia (2017)"},{"issue":"1\u20132","key":"7_CR4","doi-asserted-by":"publisher","first-page":"459","DOI":"10.1007\/s10107-013-0701-9","volume":"146","author":"J Bolte","year":"2014","unstructured":"Bolte, J., Sabach, S., Teboulle, M.: Proximal alternating linearized minimization for nonconvex and nonsmooth problems. Math. Prog. 146(1\u20132), 459\u2013494 (2014)","journal-title":"Math. Prog."},{"key":"7_CR5","unstructured":"Bot, R.I., Csetnek, E.R., Nguyen, D.-K.: A proximal minimization algorithm for structured nonconvex and nonsmooth problems. arXiv preprint \n                  arXiv:1805.11056v1\n                  \n                 [math.OC] (2018)"},{"issue":"2","key":"7_CR6","doi-asserted-by":"publisher","first-page":"1751","DOI":"10.1137\/17M1114296","volume":"28","author":"Y Carmon","year":"2018","unstructured":"Carmon, Y., Duchi, J., Hinder, O., Sidford, A.: Accelerated methods for nonconvex optimization. SIAM J. Optim. 28(2), 1751\u20131772 (2018)","journal-title":"SIAM J. Optim."},{"issue":"4","key":"7_CR7","doi-asserted-by":"publisher","first-page":"1168","DOI":"10.1137\/050626090","volume":"4","author":"PL Combettes","year":"2005","unstructured":"Combettes, P.L., Wajs, V.R.: Signal recovery by proximal forward-backward splitting. SIAM Multiscale Model. Simul. 4(4), 1168\u20131200 (2005)","journal-title":"SIAM Multiscale Model. Simul."},{"issue":"1","key":"7_CR8","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s10107-016-1026-2","volume":"162","author":"FE Curtis","year":"2017","unstructured":"Curtis, F.E., Robinson, D.P., Samadi, M.: A trust region algorithm with a worst-case iteration complexity of \n                  \n                    \n                  \n                  $$\\cal{O}(\\epsilon ^{\\frac{3}{2}})$$\n                  \n                    \n                      \n                        O\n                        (\n                        \n                          \u03f5\n                          \n                            3\n                            2\n                          \n                        \n                        )\n                      \n                    \n                  \n                 for nonconvex optimization. Math. Program. 162(1), 1\u201332 (2017)","journal-title":"Math. Program."},{"key":"7_CR9","unstructured":"Dauphin, Y.N.,\u00a0Pascanu, R.,\u00a0Gulcehre, C.,\u00a0Cho, K.,\u00a0Ganguli, S.,\u00a0Bengio, Y.: Identifying and attacking the saddle point problem in high-dimensional non-convex optimization. In: Advances in Neural Information Processing Systems, pp. 2933\u20132941 (2014)"},{"key":"7_CR10","unstructured":"Du, S.S., Jin, C., Lee, J.D., Jordan, M.I., Singh, A., Poczos, B.: Gradient descent can take exponential time to escape saddle points. In: Advances in Neural Information Processing Systems, pp. 1067\u20131077 (2017)"},{"issue":"2","key":"7_CR11","doi-asserted-by":"publisher","first-page":"219","DOI":"10.1162\/neco.1995.7.2.219","volume":"7","author":"F Girosi","year":"1995","unstructured":"Girosi, F., Jones, M., Poggio, T.: Regularization theory and neural networks architectures. Neural Comput. 7(2), 219\u2013269 (1995)","journal-title":"Neural Comput."},{"key":"7_CR12","first-page":"1145","volume":"29","author":"SJ Reddi","year":"2016","unstructured":"Reddi, S.J., Sra, S., Poczos, B., Smola, A.J.: Proximal stochastic methods for nonsmooth nonconvex finite-sum optimization. Adv. Neural Inf. Process. Syst. 29, 1145\u20131153 (2016)","journal-title":"Adv. Neural Inf. Process. Syst."},{"key":"7_CR13","unstructured":"Jin, C.,\u00a0Ge, R.,\u00a0Netrapalli, P., Kakade, S.M., Jordan, M.I.: How to escape saddle points efficiently. In: ICML (2017)"},{"key":"7_CR14","unstructured":"Lee, J.D.,\u00a0Simchowitz, M., Jordan, M.I.,\u00a0Recht, B.: Gradient descent only converges to minimizers. In: Conference on Learning Theory, pp. 1246\u20131257 (2016)"},{"key":"7_CR15","doi-asserted-by":"crossref","unstructured":"Liu, Y.,\u00a0Yin, W.: An envelope for Davis-Yin splitting and strict saddle point avoidance. arXiv preprint \n                  arXiv:1804.08739\n                  \n                 (2018)","DOI":"10.1007\/s10957-019-01477-z"},{"key":"7_CR16","unstructured":"Nesterov, Y.: A method for unconstrained convex minimization problem with the rate of convergence \n                  \n                    \n                  \n                  $$\\cal{O}$$\n                  \n                    \n                      O\n                    \n                  \n                (1\/\n                  \n                    \n                  \n                  $$k^{2}$$\n                  \n                    \n                      \n                        k\n                        2\n                      \n                    \n                  \n                ). In: Doklady AN SSSR (translated as Soviet Math. Docl.), vol. 269, pp. 543\u2013547 (1983)"},{"key":"7_CR17","doi-asserted-by":"publisher","first-page":"177","DOI":"10.1007\/s10107-006-0706-8","volume":"108","author":"Y Nesterov","year":"2006","unstructured":"Nesterov, Y., Polyak, B.T.: Cubic regularization of Newton method and its global performance. Math. Program. 108, 177\u2013205 (2006)","journal-title":"Math. Program."},{"key":"7_CR18","unstructured":"Shor, N.Z.: An application of the method of gradient descent to the solution of the network transportation problem. Materialy Naucnovo Seminara po Teoret i Priklad. Voprosam Kibernet. i Issted. Operacii, Nucnyi Sov. po Kibernet, Akad. Nauk Ukrain. SSSR, vyp 1, 9\u201317 (1962)"},{"issue":"3","key":"7_CR19","doi-asserted-by":"publisher","first-page":"443","DOI":"10.1007\/s10589-017-9912-y","volume":"67","author":"L Stella","year":"2017","unstructured":"Stella, L., Themelis, A., Patrinos, P.: Forward-backward Quasi-Newton methods for nonsmooth optimization problems. Comput. Optim. Appl. 67(3), 443\u2013487 (2017)","journal-title":"Comput. Optim. Appl."},{"key":"7_CR20","unstructured":"Xu, Y.,\u00a0Jin, R.,\u00a0Yang, T.: First-order stochastic algorithms for escaping from saddle points in almost linear time. arXiv preprint (2018). \n                  arXiv:1711.01944v3\n                  \n                 [math.OC]"},{"key":"7_CR21","unstructured":"Zhu, Z.,\u00a0Li, Y.: Neon2: finding local minima via first-order oracles. arXiv preprint (2018). \n                  arXiv:1711.06673\n                  \n                 [cs.LG]"}],"container-title":["Proceedings of the International Neural Networks Society","Recent Advances in Big Data and Deep Learning"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/978-3-030-16841-4_7","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,16]],"date-time":"2019-05-16T05:00:16Z","timestamp":1557982816000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/978-3-030-16841-4_7"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,4,3]]},"ISBN":["9783030168407","9783030168414"],"references-count":21,"URL":"https:\/\/doi.org\/10.1007\/978-3-030-16841-4_7","relation":{},"ISSN":["2661-8141","2661-815X"],"issn-type":[{"type":"print","value":"2661-8141"},{"type":"electronic","value":"2661-815X"}],"subject":[],"published":{"date-parts":[[2019,4,3]]},"assertion":[{"value":"3 April 2019","order":1,"name":"first_online","label":"First Online","group":{"name":"ChapterHistory","label":"Chapter History"}},{"value":"INNSBDDL","order":1,"name":"conference_acronym","label":"Conference Acronym","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"INNS Big Data and Deep Learning conference","order":2,"name":"conference_name","label":"Conference Name","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"Sestri Levante","order":3,"name":"conference_city","label":"Conference City","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"Italy","order":4,"name":"conference_country","label":"Conference Country","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"2019","order":5,"name":"conference_year","label":"Conference Year","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"18 April 2019","order":7,"name":"conference_start_date","label":"Conference Start Date","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"19 April 2019","order":8,"name":"conference_end_date","label":"Conference End Date","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"innsbddl2019","order":10,"name":"conference_id","label":"Conference ID","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"https:\/\/innsbddl2019.org\/","order":11,"name":"conference_url","label":"Conference URL","group":{"name":"ConferenceInfo","label":"Conference Information"}}]}}