{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,13]],"date-time":"2026-01-13T06:25:37Z","timestamp":1768285537165,"version":"3.49.0"},"reference-count":56,"publisher":"Association for Computing Machinery (ACM)","issue":"1","license":[{"start":{"date-parts":[[2022,2,24]],"date-time":"2022-02-24T00:00:00Z","timestamp":1645660800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.acm.org\/publications\/policies\/copyright_policy#Background"}],"funder":[{"name":"NSF","award":["2112533, 1740776"],"award-info":[{"award-number":["2112533, 1740776"]}]},{"DOI":"10.13039\/100018014","name":"Raytheon Technologies","doi-asserted-by":"crossref","id":[{"id":"10.13039\/100018014","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["dl.acm.org"],"crossmark-restriction":true},"short-container-title":["Proc. ACM Meas. Anal. Comput. Syst."],"published-print":{"date-parts":[[2022,2,24]]},"abstract":"<jats:p>Stochastic approximation (SA) and stochastic gradient descent (SGD) algorithms are work-horses for modern machine learning algorithms. Their constant stepsize variants are preferred in practice due to fast convergence behavior. However, constant stepsize SA algorithms do not converge to the optimal solution, but instead have a stationary distribution, which in general cannot be analytically characterized. In this work, we study the asymptotic behavior of the appropriately scaled stationary distribution, in the limit when the constant stepsize goes to zero. Specifically, we consider the following three settings: (1) SGD algorithm with a smooth and strongly convex objective, (2) linear SA algorithm involving a Hurwitz matrix, and (3) nonlinear SA algorithm involving a contractive operator. When the iterate is scaled by 1\/\u03b1, where \u03b1 is the constant stepsize, we show that the limiting scaled stationary distribution is a solution of an implicit equation. Under a uniqueness assumption (which can be removed in certain settings) on this equation, we further characterize the limiting distribution as a Gaussian distribution whose covariance matrix is the unique solution of a suitable Lyapunov equation. For SA algorithms beyond these cases, our numerical experiments suggest that unlike central limit theorem type results: (1) the scaling factor need not be 1\/\u03b1, and (2) the limiting distribution need not be Gaussian. Based on the numerical study, we come up with a heuristic formula to determine the right scaling factor, and make insightful connection to the Euler-Maruyama discretization scheme for approximating stochastic differential equations.<\/jats:p>","DOI":"10.1145\/3508039","type":"journal-article","created":{"date-parts":[[2022,2,28]],"date-time":"2022-02-28T23:44:29Z","timestamp":1646091869000},"page":"1-24","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":6,"title":["Stationary Behavior of Constant Stepsize SGD Type Algorithms"],"prefix":"10.1145","volume":"6","author":[{"given":"Zaiwei","family":"Chen","sequence":"first","affiliation":[{"name":"Georgia Institute of Technology, Atlanta, GA, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Shancong","family":"Mou","sequence":"additional","affiliation":[{"name":"Georgia Institute of Technology, Atlanta, GA, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Siva Theja","family":"Maguluri","sequence":"additional","affiliation":[{"name":"Georgia Institute of Technology, Atlanta, GA, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"320","published-online":{"date-parts":[[2022,2,28]]},"reference":[{"key":"e_1_2_1_1_1","volume-title":"Sur les op\u00e9rations dans les ensembles abstraits et leur application aux \u00e9quations int\u00e9grales. Fund. math","author":"Banach Stefan","year":"1922","unstructured":"Stefan Banach . 1922. Sur les op\u00e9rations dans les ensembles abstraits et leur application aux \u00e9quations int\u00e9grales. Fund. math , Vol. 3 , 1 ( 1922 ), 133--181. Stefan Banach. 1922. Sur les op\u00e9rations dans les ensembles abstraits et leur application aux \u00e9quations int\u00e9grales. Fund. math , Vol. 3, 1 (1922), 133--181."},{"key":"e_1_2_1_2_1","volume-title":"First-order methods in optimization","author":"Beck Amir","unstructured":"Amir Beck . 2017. First-order methods in optimization . Vol. 25 . SIAM. Amir Beck. 2017. First-order methods in optimization . Vol. 25. SIAM."},{"key":"e_1_2_1_3_1","volume-title":"Adaptive algorithms and stochastic approximations","author":"Benveniste Albert","unstructured":"Albert Benveniste , Michel M\u00e9tivier , and Pierre Priouret . 2012. Adaptive algorithms and stochastic approximations . Vol. 22 . Springer Science & Business Media . Albert Benveniste, Michel M\u00e9tivier, and Pierre Priouret. 2012. Adaptive algorithms and stochastic approximations. Vol. 22. Springer Science & Business Media."},{"key":"e_1_2_1_4_1","volume-title":"Neuro-dynamic programming","author":"Bertsekas Dimitri P","unstructured":"Dimitri P Bertsekas and John N Tsitsiklis . 1996. Neuro-dynamic programming . Athena Scientific . Dimitri P Bertsekas and John N Tsitsiklis. 1996. Neuro-dynamic programming .Athena Scientific."},{"key":"e_1_2_1_5_1","volume-title":"Conference On Learning Theory . 1691--1692","author":"Bhandari Jalaj","year":"2018","unstructured":"Jalaj Bhandari , Daniel Russo , and Raghav Singal . 2018 . A Finite Time Analysis of Temporal Difference Learning With Linear Function Approximation . In Conference On Learning Theory . 1691--1692 . Jalaj Bhandari, Daniel Russo, and Raghav Singal. 2018. A Finite Time Analysis of Temporal Difference Learning With Linear Function Approximation. In Conference On Learning Theory . 1691--1692."},{"key":"e_1_2_1_6_1","volume-title":"Convergence of constant step stochastic gradient descent for non-smooth non-convex functions. Preprint arXiv:2005.08513","author":"Bianchi Pascal","year":"2020","unstructured":"Pascal Bianchi , Walid Hachem , and Sholom Schechtman . 2020. Convergence of constant step stochastic gradient descent for non-smooth non-convex functions. Preprint arXiv:2005.08513 ( 2020 ). Pascal Bianchi, Walid Hachem, and Sholom Schechtman. 2020. Convergence of constant step stochastic gradient descent for non-smooth non-convex functions. Preprint arXiv:2005.08513 (2020)."},{"key":"e_1_2_1_7_1","volume-title":"Stochastic approximation: a dynamical systems viewpoint","author":"Borkar Vivek S","unstructured":"Vivek S Borkar . 2009. Stochastic approximation: a dynamical systems viewpoint . Vol. 48 . Springer . Vivek S Borkar. 2009. Stochastic approximation: a dynamical systems viewpoint. Vol. 48. Springer."},{"key":"e_1_2_1_8_1","doi-asserted-by":"publisher","DOI":"10.1137\/16M1080173"},{"key":"e_1_2_1_9_1","doi-asserted-by":"publisher","DOI":"10.1109\/INDIANCC.2019.8715554"},{"key":"e_1_2_1_10_1","series-title":"SIAM review","volume-title":"Iterated random functions","author":"Diaconis Persi","year":"1999","unstructured":"Persi Diaconis and David Freedman . 1999. Iterated random functions . SIAM review , Vol. 41 , 1 ( 1999 ), 45--76. Persi Diaconis and David Freedman. 1999. Iterated random functions. SIAM review , Vol. 41, 1 (1999), 45--76."},{"key":"e_1_2_1_11_1","doi-asserted-by":"publisher","DOI":"10.1214\/19-AOS1850"},{"key":"e_1_2_1_12_1","volume-title":"On Riemannian Stochastic Approximation Schemes with Fixed Step-Size. In International Conference on Artificial Intelligence and Statistics. PMLR, 1018--1026","author":"Durmus Alain","year":"2021","unstructured":"Alain Durmus , Pablo Jim\u00e9nez , \u00c9ric Moulines , and SAID Salem . 2021 . On Riemannian Stochastic Approximation Schemes with Fixed Step-Size. In International Conference on Artificial Intelligence and Statistics. PMLR, 1018--1026 . Alain Durmus, Pablo Jim\u00e9nez, \u00c9ric Moulines, and SAID Salem. 2021. On Riemannian Stochastic Approximation Schemes with Fixed Step-Size. In International Conference on Artificial Intelligence and Statistics. PMLR, 1018--1026."},{"key":"e_1_2_1_13_1","volume-title":"Probability: theory and examples","author":"Durrett Rick","unstructured":"Rick Durrett . 2019. Probability: theory and examples . Vol. 49 . Cambridge university press . Rick Durrett. 2019. Probability: theory and examples . Vol. 49. Cambridge university press."},{"key":"e_1_2_1_14_1","doi-asserted-by":"publisher","DOI":"10.1007\/s11134-012-9305-y"},{"key":"e_1_2_1_15_1","volume-title":"On asymptotic normality in stochastic approximation. The Annals of Mathematical Statistics","author":"Fabian Vaclav","year":"1968","unstructured":"Vaclav Fabian . 1968. On asymptotic normality in stochastic approximation. The Annals of Mathematical Statistics ( 1968 ), 1327--1332. Vaclav Fabian. 1968. On asymptotic normality in stochastic approximation. The Annals of Mathematical Statistics (1968), 1327--1332."},{"key":"e_1_2_1_16_1","article-title":"Convergence rates for the stochastic gradient descent method for non-convex objective functions","volume":"21","author":"Fehrman Benjamin","year":"2020","unstructured":"Benjamin Fehrman , Benjamin Gess , and Arnulf Jentzen . 2020 . Convergence rates for the stochastic gradient descent method for non-convex objective functions . Journal of Machine Learning Research , Vol. 21 (2020). Benjamin Fehrman, Benjamin Gess, and Arnulf Jentzen. 2020. Convergence rates for the stochastic gradient descent method for non-convex objective functions. Journal of Machine Learning Research , Vol. 21 (2020).","journal-title":"Journal of Machine Learning Research"},{"key":"e_1_2_1_17_1","doi-asserted-by":"publisher","DOI":"10.4310\/CMS.2018.v16.n3.a8"},{"key":"e_1_2_1_18_1","doi-asserted-by":"publisher","DOI":"10.1080\/00029890.1973.11993339"},{"key":"e_1_2_1_19_1","doi-asserted-by":"crossref","unstructured":"D. Gamarnik and A. Zeevi. 2006. Validity of Heavy Traffic Steady-State Approximations in Generalized Jackson Networks. The Annals of Applied Probability (2006) 56--90.  D. Gamarnik and A. Zeevi. 2006. Validity of Heavy Traffic Steady-State Approximations in Generalized Jackson Networks. The Annals of Applied Probability (2006) 56--90.","DOI":"10.1214\/105051605000000638"},{"key":"e_1_2_1_20_1","volume-title":"Deep learning","author":"Goodfellow Ian","unstructured":"Ian Goodfellow , Yoshua Bengio , Aaron Courville , and Yoshua Bengio . 2016. Deep learning . Vol. 1 . MIT press Cambridge . Ian Goodfellow, Yoshua Bengio, Aaron Courville, and Yoshua Bengio. 2016. Deep learning. Vol. 1. MIT press Cambridge."},{"key":"e_1_2_1_21_1","volume-title":"International Conference on Artificial Intelligence and Statistics. PMLR, 1315--1323","author":"Gower Robert","year":"2021","unstructured":"Robert Gower , Othmane Sebbouh , and Nicolas Loizou . 2021 . SGD for structured nonconvex functions: Learning rates, minibatching and interpolation . In International Conference on Artificial Intelligence and Statistics. PMLR, 1315--1323 . Robert Gower, Othmane Sebbouh, and Nicolas Loizou. 2021. SGD for structured nonconvex functions: Learning rates, minibatching and interpolation. In International Conference on Artificial Intelligence and Statistics. PMLR, 1315--1323."},{"key":"e_1_2_1_22_1","volume-title":"Nonlinear dynamical systems and control: a Lyapunov-based approach","author":"Haddad Wassim M","unstructured":"Wassim M Haddad and VijaySekhar Chellaboina . 2011. Nonlinear dynamical systems and control: a Lyapunov-based approach . Princeton University Press . Wassim M Haddad and VijaySekhar Chellaboina. 2011. Nonlinear dynamical systems and control: a Lyapunov-based approach. Princeton University Press."},{"key":"e_1_2_1_23_1","volume-title":"Stochastic Differential Systems, Stochastic Control Theory and Applications","author":"Harrison J.M.","unstructured":"J.M. Harrison . 1988. Brownian Models of Queueing Networks with Heterogeneous Customer Populations . In Stochastic Differential Systems, Stochastic Control Theory and Applications . Springer , 147--186. J.M. Harrison. 1988. Brownian Models of Queueing Networks with Heterogeneous Customer Populations. In Stochastic Differential Systems, Stochastic Control Theory and Applications. Springer, 147--186."},{"key":"e_1_2_1_24_1","unstructured":"J. M. Harrison. 1998. Heavy traffic analysis of a system with parallel servers: Asymptotic optimality of discrete review policies. Ann. App. Probab. (1998) 822--848.  J. M. Harrison. 1998. Heavy traffic analysis of a system with parallel servers: Asymptotic optimality of discrete review policies. Ann. App. Probab. (1998) 822--848."},{"key":"e_1_2_1_25_1","doi-asserted-by":"crossref","unstructured":"J. M. Harrison and M. J. L\u00f3pez. 1999. Heavy traffic resource pooling in parallel-server systems. Queueing Systems (1999) 339--368.  J. M. Harrison and M. J. L\u00f3pez. 1999. Heavy traffic resource pooling in parallel-server systems. Queueing Systems (1999) 339--368.","DOI":"10.1023\/A:1019188531950"},{"key":"e_1_2_1_26_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.automatica.2018.10.017"},{"key":"e_1_2_1_27_1","doi-asserted-by":"publisher","DOI":"10.4310\/AMSA.2019.v4.n1.a1"},{"key":"e_1_2_1_28_1","doi-asserted-by":"publisher","DOI":"10.1287\/stsy.2019.0056"},{"key":"e_1_2_1_29_1","volume-title":"Sushil Mahavir Varma, and Siva Theja Maguluri","author":"Hurtado-Lange Daniela","year":"2020","unstructured":"Daniela Hurtado-Lange , Sushil Mahavir Varma, and Siva Theja Maguluri . 2020 . Logarithmic Heavy Traffic Error Bounds in Generalized Switch and Load Balancing Systems. Preprint arXiv:2003.07821 (2020). Daniela Hurtado-Lange, Sushil Mahavir Varma, and Siva Theja Maguluri. 2020. Logarithmic Heavy Traffic Error Bounds in Generalized Switch and Load Balancing Systems. Preprint arXiv:2003.07821 (2020)."},{"key":"e_1_2_1_30_1","doi-asserted-by":"publisher","DOI":"10.1214\/aoms\/1177706705"},{"key":"e_1_2_1_31_1","volume-title":"Nonlinear systems","author":"Khalil Hassan K","unstructured":"Hassan K Khalil and Jessy W Grizzle . 2002. Nonlinear systems . Vol. 3 . Prentice hall Upper Saddle River, NJ. Hassan K Khalil and Jessy W Grizzle. 2002. Nonlinear systems . Vol. 3. Prentice hall Upper Saddle River, NJ."},{"key":"e_1_2_1_32_1","volume-title":"First-order and Stochastic Optimization Methods for Machine Learning","author":"Lan Guanghui","unstructured":"Guanghui Lan . 2020. First-order and Stochastic Optimization Methods for Machine Learning . Springer Nature . Guanghui Lan. 2020. First-order and Stochastic Optimization Methods for Machine Learning. Springer Nature."},{"key":"e_1_2_1_33_1","doi-asserted-by":"publisher","DOI":"10.1007\/s11222-021-10016-8"},{"key":"e_1_2_1_34_1","volume-title":"International Conference on Machine Learning. PMLR, 2101--2110","author":"Li Qianxiao","year":"2017","unstructured":"Qianxiao Li , Cheng Tai , and E Weinan . 2017 . Stochastic modified equations and adaptive stochastic gradient algorithms . In International Conference on Machine Learning. PMLR, 2101--2110 . Qianxiao Li, Cheng Tai, and E Weinan. 2017. Stochastic modified equations and adaptive stochastic gradient algorithms. In International Conference on Machine Learning. PMLR, 2101--2110."},{"key":"e_1_2_1_35_1","volume-title":"The 22nd International Conference on Artificial Intelligence and Statistics. PMLR, 983--992","author":"Li Xiaoyu","year":"2019","unstructured":"Xiaoyu Li and Francesco Orabona . 2019 . On the convergence of stochastic gradient descent with adaptive stepsizes . In The 22nd International Conference on Artificial Intelligence and Statistics. PMLR, 983--992 . Xiaoyu Li and Francesco Orabona. 2019. On the convergence of stochastic gradient descent with adaptive stepsizes. In The 22nd International Conference on Artificial Intelligence and Statistics. PMLR, 983--992."},{"key":"e_1_2_1_36_1","doi-asserted-by":"publisher","DOI":"10.1007\/s11134-017-9562-x"},{"key":"e_1_2_1_37_1","doi-asserted-by":"publisher","DOI":"10.1287\/15-SSY193"},{"key":"e_1_2_1_38_1","volume-title":"On the almost sure convergence of stochastic gradient descent in non-convex problems. Preprint arXiv:2006.11144","author":"Mertikopoulos Panayotis","year":"2020","unstructured":"Panayotis Mertikopoulos , Nadav Hallak , Ali Kavis , and Volkan Cevher . 2020. On the almost sure convergence of stochastic gradient descent in non-convex problems. Preprint arXiv:2006.11144 ( 2020 ). Panayotis Mertikopoulos, Nadav Hallak, Ali Kavis, and Volkan Cevher. 2020. On the almost sure convergence of stochastic gradient descent in non-convex problems. Preprint arXiv:2006.11144 (2020)."},{"key":"e_1_2_1_39_1","volume-title":"Heavy Traffic Queue Length Behaviour in a Switch under Markovian Arrivals. Preprint arXiv:2006.06150","author":"Mou Shancong","year":"2020","unstructured":"Shancong Mou and Siva Theja Maguluri . 2020. Heavy Traffic Queue Length Behaviour in a Switch under Markovian Arrivals. Preprint arXiv:2006.06150 ( 2020 ). Shancong Mou and Siva Theja Maguluri. 2020. Heavy Traffic Queue Length Behaviour in a Switch under Markovian Arrivals. Preprint arXiv:2006.06150 (2020)."},{"key":"e_1_2_1_40_1","first-page":"269","article-title":"Easy proof of the Jacobian for the n-dimensional polar coordinates","volume":"14","author":"Muleshkov Angel","year":"2016","unstructured":"Angel Muleshkov and Tan Nguyen . 2016 . Easy proof of the Jacobian for the n-dimensional polar coordinates . Pi Mu Epsilon Journal , Vol. 14 (2016), 269 -- 273 . Angel Muleshkov and Tan Nguyen. 2016. Easy proof of the Jacobian for the n-dimensional polar coordinates . Pi Mu Epsilon Journal , Vol. 14 (2016), 269--273.","journal-title":"Pi Mu Epsilon Journal"},{"key":"e_1_2_1_41_1","volume-title":"A stochastic approximation method. The Annals of Mathematical Statistics","author":"Robbins Herbert","year":"1951","unstructured":"Herbert Robbins and Sutton Monro . 1951. A stochastic approximation method. The Annals of Mathematical Statistics ( 1951 ), 400--407. Herbert Robbins and Sutton Monro. 1951. A stochastic approximation method. The Annals of Mathematical Statistics (1951), 400--407."},{"key":"e_1_2_1_42_1","volume-title":"Handbook of computational finance","author":"Sauer Timothy","unstructured":"Timothy Sauer . 2012. Numerical solution of stochastic differential equations in finance . In Handbook of computational finance . Springer , 529--550. Timothy Sauer. 2012. Numerical solution of stochastic differential equations in finance. In Handbook of computational finance . Springer, 529--550."},{"key":"e_1_2_1_43_1","volume-title":"International conference on machine learning . PMLR, 71--79","author":"Shamir Ohad","year":"2013","unstructured":"Ohad Shamir and Tong Zhang . 2013 . Stochastic gradient descent for non-smooth optimization: Convergence results and optimal averaging schemes . In International conference on machine learning . PMLR, 71--79 . Ohad Shamir and Tong Zhang. 2013. Stochastic gradient descent for non-smooth optimization: Convergence results and optimal averaging schemes. In International conference on machine learning . PMLR, 71--79."},{"key":"e_1_2_1_44_1","doi-asserted-by":"publisher","DOI":"10.1287\/stsy.2019.0050"},{"key":"e_1_2_1_45_1","volume-title":"Finite-Time Error Bounds For Linear Stochastic Approximation and TD Learning. In Conference on Learning Theory. 2803--2830","author":"Srikant R","year":"2019","unstructured":"R Srikant and Lei Ying . 2019 . Finite-Time Error Bounds For Linear Stochastic Approximation and TD Learning. In Conference on Learning Theory. 2803--2830 . R Srikant and Lei Ying. 2019. Finite-Time Error Bounds For Linear Stochastic Approximation and TD Learning. In Conference on Learning Theory. 2803--2830."},{"key":"e_1_2_1_46_1","doi-asserted-by":"publisher","DOI":"10.1214\/aoap\/1075828046"},{"key":"e_1_2_1_47_1","volume-title":"Learning to predict by the methods of temporal differences. Machine learning","author":"Sutton Richard S","year":"1988","unstructured":"Richard S Sutton . 1988. Learning to predict by the methods of temporal differences. Machine learning , Vol. 3 , 1 ( 1988 ), 9--44. Richard S Sutton. 1988. Learning to predict by the methods of temporal differences. Machine learning , Vol. 3, 1 (1988), 9--44."},{"key":"e_1_2_1_48_1","volume-title":"Reinforcement learning: An introduction","author":"Sutton Richard S","unstructured":"Richard S Sutton and Andrew G Barto . 2018. Reinforcement learning: An introduction . MIT press . Richard S Sutton and Andrew G Barto. 2018. Reinforcement learning: An introduction .MIT press."},{"key":"e_1_2_1_49_1","unstructured":"John N Tsitsiklis and Benjamin Van Roy. 1997. Analysis of temporal-difference learning with function approximation. In Advances in neural information processing systems. 1075--1081.  John N Tsitsiklis and Benjamin Van Roy. 1997. Analysis of temporal-difference learning with function approximation. In Advances in neural information processing systems. 1075--1081."},{"key":"e_1_2_1_50_1","volume-title":"Asymptotic statistics","author":"Van der Vaart Aad W","unstructured":"Aad W Van der Vaart . 2000. Asymptotic statistics . Vol. 3 . Cambridge university press . Aad W Van der Vaart. 2000. Asymptotic statistics . Vol. 3. Cambridge university press."},{"key":"e_1_2_1_51_1","volume-title":"Machine learning","author":"Watkins Christopher JCH","year":"1992","unstructured":"Christopher JCH Watkins and Peter Dayan . 1992. Q-learning. Machine learning , Vol. 8 , 3--4 ( 1992 ), 279--292. Christopher JCH Watkins and Peter Dayan. 1992. Q-learning. Machine learning , Vol. 8, 3--4 (1992), 279--292."},{"key":"e_1_2_1_52_1","volume-title":"Improved Bounds for Discretization of Langevin Diffusions: Near-Optimal Rates without Convexity. https:\/\/arxiv.org\/pdf\/1907.11331","author":"Wenlong Mou","year":"2019","unstructured":"Mou Wenlong , Flammarion Nicolas , Wainwright Martin J., and Bartlett Peter L. 2019. Improved Bounds for Discretization of Langevin Diffusions: Near-Optimal Rates without Convexity. https:\/\/arxiv.org\/pdf\/1907.11331 ( 2019 ). Mou Wenlong, Flammarion Nicolas, Wainwright Martin J., and Bartlett Peter L. 2019. Improved Bounds for Discretization of Langevin Diffusions: Near-Optimal Rates without Convexity. https:\/\/arxiv.org\/pdf\/1907.11331 (2019)."},{"key":"e_1_2_1_53_1","volume-title":"Diffusion approximations for open multiclass queueing networks: Sufficient conditions involving state space collapse. Queueing Systems Theory and Applications","author":"Williams R. J.","year":"1998","unstructured":"R. J. Williams . 1998. Diffusion approximations for open multiclass queueing networks: Sufficient conditions involving state space collapse. Queueing Systems Theory and Applications ( 1998 ), 27 -- 88. R. J. Williams. 1998. Diffusion approximations for open multiclass queueing networks: Sufficient conditions involving state space collapse. Queueing Systems Theory and Applications (1998), 27 -- 88."},{"key":"e_1_2_1_54_1","volume-title":"International Conference on Artificial Intelligence and Statistics. PMLR, 1475--1485","author":"Xie Yuege","year":"2020","unstructured":"Yuege Xie , Xiaoxia Wu , and Rachel Ward . 2020 . Linear convergence of adaptive stochastic gradient descent . In International Conference on Artificial Intelligence and Statistics. PMLR, 1475--1485 . Yuege Xie, Xiaoxia Wu, and Rachel Ward. 2020. Linear convergence of adaptive stochastic gradient descent. In International Conference on Artificial Intelligence and Statistics. PMLR, 1475--1485."},{"key":"e_1_2_1_55_1","doi-asserted-by":"publisher","DOI":"10.3233\/ASY-201622"},{"key":"e_1_2_1_56_1","volume-title":"An Analysis of Constant Step Size SGD in the Non-convex Regime: Asymptotic Normality and Bias . Preprint arXiv:2006.07904","author":"Yu Lu","year":"2020","unstructured":"Lu Yu , Krishnakumar Balasubramanian , Stanislav Volgushev , and Murat A Erdogdu . 2020. An Analysis of Constant Step Size SGD in the Non-convex Regime: Asymptotic Normality and Bias . Preprint arXiv:2006.07904 ( 2020 ). Lu Yu, Krishnakumar Balasubramanian, Stanislav Volgushev, and Murat A Erdogdu. 2020. An Analysis of Constant Step Size SGD in the Non-convex Regime: Asymptotic Normality and Bias . Preprint arXiv:2006.07904 (2020)."}],"container-title":["Proceedings of the ACM on Measurement and Analysis of Computing Systems"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/3508039","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/dl.acm.org\/doi\/pdf\/10.1145\/3508039","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/dl.acm.org\/doi\/pdf\/10.1145\/3508039","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,6,17]],"date-time":"2025-06-17T20:12:29Z","timestamp":1750191149000},"score":1,"resource":{"primary":{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/3508039"}},"subtitle":["An Asymptotic Characterization"],"short-title":[],"issued":{"date-parts":[[2022,2,24]]},"references-count":56,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2022,2,24]]}},"alternative-id":["10.1145\/3508039"],"URL":"https:\/\/doi.org\/10.1145\/3508039","relation":{},"ISSN":["2476-1249"],"issn-type":[{"value":"2476-1249","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,2,24]]},"assertion":[{"value":"2022-02-28","order":2,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}]}}