{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,13]],"date-time":"2026-01-13T04:59:30Z","timestamp":1768280370125,"version":"3.49.0"},"reference-count":90,"publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"2","license":[{"start":{"date-parts":[[2026,2,1]],"date-time":"2026-02-01T00:00:00Z","timestamp":1769904000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/ieeexplore.ieee.org\/Xplorehelp\/downloads\/license-information\/IEEE.html"},{"start":{"date-parts":[[2026,2,1]],"date-time":"2026-02-01T00:00:00Z","timestamp":1769904000000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-029"},{"start":{"date-parts":[[2026,2,1]],"date-time":"2026-02-01T00:00:00Z","timestamp":1769904000000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-037"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["U23A20382"],"award-info":[{"award-number":["U23A20382"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Collaborative Innovation Center of Novel Software Technology and Industrialization"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["IEEE Trans. Pattern Anal. Mach. Intell."],"published-print":{"date-parts":[[2026,2]]},"DOI":"10.1109\/tpami.2025.3621320","type":"journal-article","created":{"date-parts":[[2025,10,14]],"date-time":"2025-10-14T17:40:04Z","timestamp":1760463604000},"page":"1949-1960","source":"Crossref","is-referenced-by-count":0,"title":["Optimizing Unnormalized Statistical Models Through Compositional Optimization"],"prefix":"10.1109","volume":"48","author":[{"ORCID":"https:\/\/orcid.org\/0009-0007-4595-1060","authenticated-orcid":false,"given":"Wei","family":"Jiang","sequence":"first","affiliation":[{"name":"School of Computer Science and Engineering, Nanjing University of Science and Technology, Nanjing, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0009-0002-9142-2016","authenticated-orcid":false,"given":"Jiayu","family":"Qin","sequence":"additional","affiliation":[{"name":"Department of Computer Science and Engineering, University at Buffalo, the State University of New York, Buffalo, NY, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0009-0009-2581-969X","authenticated-orcid":false,"given":"Lingyu","family":"Wu","sequence":"additional","affiliation":[{"name":"National Key Laboratory for Novel Software Technology, and School of Artificial Intelligence, Nanjing University, Nanjing, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Changyou","family":"Chen","sequence":"additional","affiliation":[{"name":"Department of Computer Science and Engineering, University at Buffalo, the State University of New York, Buffalo, NY, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Tianbao","family":"Yang","sequence":"additional","affiliation":[{"name":"Department of Computer Science and Engineering, Texas A&#x0026;M University, College Station, TX, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5138-3182","authenticated-orcid":false,"given":"Lijun","family":"Zhang","sequence":"additional","affiliation":[{"name":"National Key Laboratory for Novel Software Technology, and School of Artificial Intelligence, Nanjing University, Nanjing, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"263","reference":[{"key":"ref1","article-title":"A tutorial on energy-based learning","volume-title":"Predicting Structured Data","author":"LeCun","year":"2006"},{"key":"ref2","first-page":"3603","article-title":"Implicit generation and modeling with energy based models","volume-title":"Proc. Adv. Neural Inf. Process. Syst.","author":"Du"},{"key":"ref3","first-page":"10957","article-title":"Training deep energy-based models with f-divergence minimization","volume-title":"Proc. 37th Int. Conf. Mach. Learn.","author":"Yu"},{"key":"ref4","article-title":"Your classifier is secretly an energy based model and you should treat it like one","volume-title":"Proc. Int. Conf. Learn. Representations","author":"Grathwohl"},{"key":"ref5","doi-asserted-by":"publisher","DOI":"10.1162\/089976602760128018"},{"key":"ref6","doi-asserted-by":"publisher","DOI":"10.1145\/1390156.1390290"},{"key":"ref7","doi-asserted-by":"publisher","DOI":"10.1609\/aaai.v34i04.5973"},{"key":"ref8","volume-title":"Monte Carlo Statistical Methods","author":"Christian","year":"2004"},{"key":"ref9","first-page":"3732","article-title":"Learning the stein discrepancy for training and evaluating energy-based models without sampling","volume-title":"Proc. 37th Int. Conf. Mach. Learn.","author":"Grathwohl"},{"key":"ref10","first-page":"7518","article-title":"Flow contrastive estimation of energy-based models","volume-title":"Proc. IEEE\/CVF Conf. Comput. Vis. Pattern Recognit.","author":"Ruiqi"},{"key":"ref11","first-page":"19808","article-title":"Bounds all around: Training energy-based models with bidirectional bounds","volume-title":"Proc. Adv. Neural Inf. Process. Syst.","author":"Geng"},{"key":"ref12","first-page":"5233","article-title":"Learning non-convergent non-persistent short-run MCMC toward energy-based model","volume-title":"Proc. Adv. Neural Inf. Process. Syst.","author":"Nijkamp"},{"issue":"11","key":"ref13","first-page":"307","article-title":"Noise-contrastive estimation of unnormalized statistical models, with applications to natural image statistics","volume":"13","author":"Gutmann","year":"2012","journal-title":"J. Mach. Learn. Res."},{"key":"ref14","doi-asserted-by":"publisher","DOI":"10.5555\/2969033.2969125"},{"key":"ref15","article-title":"Analyzing and improving the optimization landscape of noise-contrastive estimation","volume-title":"Proc. Int. Conf. Learn. Representations","author":"Liu"},{"key":"ref16","first-page":"15105","article-title":"Learning unnormalized statistical models via compositional optimization","volume-title":"Proc. 40th Int. Conf. Mach. Learn.","author":"Jiang"},{"key":"ref17","first-page":"297","article-title":"Noise-contrastive estimation: A new estimation principle for unnormalized statistical models","volume-title":"Proc. 13th Int. Conf. Artif. Intell. Statist.","author":"Gutmann"},{"key":"ref18","first-page":"2265","article-title":"Learning word embeddings efficiently with noise-contrastive estimation","volume-title":"Proc. Adv. Neural Inf. Process. Syst.","author":"Mnih"},{"key":"ref19","article-title":"Learning deep representations by mutual information estimation and maximization","volume-title":"Proc. Int. Conf. Learn. Representations","author":"Hjelm"},{"key":"ref20","first-page":"4182","article-title":"Data-efficient image recognition with contrastive predictive coding","volume-title":"Proc. 37th Int. Conf. Mach. Learn.","author":"Henaff"},{"key":"ref21","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-030-58621-8_45"},{"key":"ref22","article-title":"A mutual information maximization perspective of language representation learning","volume-title":"Proc. Int. Conf. Learn. Representations","author":"Kong"},{"key":"ref23","article-title":"Pitfalls of gaussians as a noise distribution in NCE","volume-title":"Proc. Int. Conf. Learn. Representations","author":"Lee"},{"key":"ref24","doi-asserted-by":"publisher","DOI":"10.18653\/v1\/P18-1094"},{"key":"ref25","first-page":"726","article-title":"Conditional noise-contrastive estimation of unnormalised models","volume-title":"Proc. 35th Int. Conf. Mach. Learn.","author":"Ceylan"},{"key":"ref26","first-page":"52444","article-title":"A unified view on learning unnormalized distributions via noise-contrastive estimation","volume-title":"Proc. 42nd Int. Conf. Mach. Learn.","author":"Ryu"},{"issue":"24","key":"ref27","first-page":"695","article-title":"Estimation of non-normalized statistical models by score matching","volume":"6","author":"Hyv\u00e4rinen","year":"2005","journal-title":"J. Mach. Learn. Res."},{"key":"ref28","first-page":"11895","article-title":"Generative modeling by estimating gradients of the data distribution","volume-title":"Proc. Adv. Neural Inf. Process. Syst.","author":"Song"},{"key":"ref29","first-page":"19175","article-title":"Efficient learning of generative models via finite-difference score matching","volume-title":"Proc. Adv. Neural Inf. Process. Syst.","author":"Pang"},{"key":"ref30","doi-asserted-by":"publisher","DOI":"10.1109\/CVPR.2018.00954"},{"key":"ref31","doi-asserted-by":"publisher","DOI":"10.1109\/CVPR.2019.00887"},{"key":"ref32","first-page":"2635","article-title":"A theory of generative convnet","volume-title":"Proc. 33rd Int. Conf. Int. Conf. Mach. Learn.","author":"Xie"},{"issue":"40","key":"ref33","first-page":"1","article-title":"Residual energy-based models for text","volume":"22","author":"Bakhtin","year":"2021","journal-title":"J. Mach. Learn. Res."},{"key":"ref34","first-page":"574","article-title":"Sliced score matching: A scalable approach to density and score estimation","volume-title":"Proc. 35th Conf. Uncertainty Artif. Intell.","author":"Song"},{"key":"ref35","first-page":"1352","article-title":"Reinforcement learning with deep energy-based policies","volume-title":"Proc. 34th Int. Conf. Mach. Learn.","author":"Haarnoja"},{"key":"ref36","first-page":"1126","article-title":"Regularized estimation of image statistics by score matching","volume-title":"Proc. Adv. Neural Inf. Process. Syst.","author":"Kingma"},{"key":"ref37","doi-asserted-by":"publisher","DOI":"10.1162\/NECO_a_00142"},{"key":"ref38","first-page":"419","article-title":"A fast and simple algorithm for training neural probabilistic language models","volume-title":"Proc. 29th Int. Conf. Int. Conf. Mach. Learn.","author":"Mnih"},{"key":"ref39","doi-asserted-by":"publisher","DOI":"10.1111\/j.2517-6161.1994.tb02000.x"},{"key":"ref40","doi-asserted-by":"publisher","DOI":"10.1201\/b10905-6"},{"key":"ref41","doi-asserted-by":"publisher","DOI":"10.1609\/aaai.v32i1.11834"},{"key":"ref42","doi-asserted-by":"publisher","DOI":"10.1109\/TPAMI.2018.2879081"},{"key":"ref43","doi-asserted-by":"publisher","DOI":"10.1007\/s10107-016-1017-3"},{"key":"ref44","first-page":"105:1","article-title":"Accelerating stochastic composition optimization","volume":"18","author":"Wang","year":"2017","journal-title":"J. Mach. Learn. Res."},{"key":"ref45","doi-asserted-by":"publisher","DOI":"10.1137\/18M1230542"},{"key":"ref46","first-page":"2613","article-title":"SARAH: A novel method for machine learning problems using stochastic recursive gradient","volume-title":"Proc. 34th Int. Conf. Mach. Learn.","author":"Nguyen"},{"key":"ref47","article-title":"Spider: Near-optimal non-convex optimization via stochastic path integrated differential estimator","author":"Fang","year":"2018"},{"key":"ref48","first-page":"15210","article-title":"Momentum-based variance reduction in non-convex SGD","volume-title":"Proc. Adv. Neural Inf. Process. Syst.","author":"Cutkosky"},{"key":"ref49","first-page":"9075","article-title":"A stochastic composite gradient method with incremental variance reduction","volume-title":"Proc. Adv. Neural Inf. Process. Syst.","author":"Zhang"},{"key":"ref50","doi-asserted-by":"publisher","DOI":"10.1109\/TSP.2021.3092377"},{"key":"ref51","article-title":"An online method for a class of distributionally robust optimization with non-convex objectives","author":"Qi","year":"2021"},{"key":"ref52","first-page":"10195","article-title":"Optimal algorithms for stochastic multi-level compositional optimization","volume-title":"Proc. 39th Int. Conf. Mach. Learn.","author":"Jiang"},{"key":"ref53","first-page":"21962","article-title":"Projection-free variance reduction methods for stochastic constrained multi-level compositional optimization","volume-title":"Proc. 41st Int. Conf. Mach. Learn.","author":"Jiang"},{"key":"ref54","doi-asserted-by":"publisher","DOI":"10.1109\/TPAMI.2025.3552197"},{"key":"ref55","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511802256"},{"key":"ref56","doi-asserted-by":"publisher","DOI":"10.1007\/978-0-387-21736-9"},{"key":"ref57","first-page":"1714","article-title":"Accelerating stochastic composition optimization","volume-title":"Proc. Adv. Neural Inf. Process. Syst.","author":"Wang"},{"key":"ref58","doi-asserted-by":"publisher","DOI":"10.1137\/19M1285457"},{"key":"ref59","first-page":"23292","article-title":"Finite-sum coupled compositional stochastic optimization: Theory and applications","volume-title":"Proc. 39th Int. Conf. Mach. Learn.","author":"Wang"},{"key":"ref60","doi-asserted-by":"publisher","DOI":"10.52202\/079017-0694"},{"key":"ref61","first-page":"313","article-title":"The power of adaptivity in SGD: Self-tuning step sizes with unbounded gradients and affine variance","volume-title":"Proc. 35th Annu. Conf. Learn. Theory","author":"Faw"},{"key":"ref62","first-page":"5396","article-title":"Generalized-smooth nonconvex optimization is as efficient as smooth nonconvex optimization","volume-title":"Proc. 40th Int. Conf. Mach. Learn.","author":"Chen"},{"key":"ref63","doi-asserted-by":"publisher","DOI":"10.52202\/079017-1068"},{"key":"ref64","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-46128-1_50"},{"key":"ref65","first-page":"242","article-title":"A convergence theory for deep learning via over-parameterization","volume-title":"Proc. 36th Int. Conf. Mach. Learn.","author":"Allen-Zhu"},{"key":"ref66","article-title":"Gradient descent provably optimizes over-parameterized neural networks","volume-title":"Proc. Int. Conf. Learn. Representations","author":"Du"},{"key":"ref67","first-page":"1752","article-title":"Stochastic optimization of areas under precision-recall curves with provable convergence","volume-title":"Proc. Adv. Neural Inf. Process. Syst.","author":"Qi"},{"key":"ref68","doi-asserted-by":"publisher","DOI":"10.1109\/TIT.2011.2182178"},{"key":"ref69","doi-asserted-by":"publisher","DOI":"10.1007\/s10107-022-01822-7"},{"key":"ref70","article-title":"Nice: Non-linear independent components estimation","volume-title":"Proc. Int. Conf. Learn. Representations","author":"Dinh"},{"key":"ref71","first-page":"681","article-title":"Bayesian learning via stochastic gradient Langevin dynamics","volume-title":"Proc. 28th Int. Conf. Mach. Learn.","author":"Welling"},{"key":"ref72","article-title":"Gradient-guided importance sampling for learning binary energy-based models","author":"Liu","year":"2022"},{"issue":"25","key":"ref73","first-page":"723","article-title":"A kernel two-sample test","volume":"13","author":"Gretton","year":"2012","journal-title":"J. Mach. Learn. Res."},{"key":"ref74","article-title":"Learning multiple layers of features from tiny images","author":"Krizhevsky","year":"2009"},{"key":"ref75","doi-asserted-by":"publisher","DOI":"10.5244\/C.30.87"},{"key":"ref76","doi-asserted-by":"publisher","DOI":"10.2118\/18761-MS"},{"key":"ref77","article-title":"LSUN: Construction of a large-scale image dataset using deep learning with humans in the loop","author":"Yu","year":"2015"},{"key":"ref78","first-page":"14680","article-title":"Likelihood ratios for out-of-distribution detection","volume-title":"Proc. Adv. Neural Inf. Process. Syst.","author":"Ren"},{"key":"ref79","first-page":"4117","article-title":"Hierarchical VAEs know what they don\u2019t know","volume-title":"Proc. 38th Int. Conf. Mach. Learn.","author":"Havtorn"},{"key":"ref80","doi-asserted-by":"publisher","DOI":"10.1109\/5.726791"},{"key":"ref81","first-page":"4905","article-title":"Telescoping density-ratio estimation","volume-title":"Proc. Adv. Neural Inf. Process. Syst.","author":"Rhodes"},{"key":"ref82","doi-asserted-by":"publisher","DOI":"10.1109\/CVPR.2016.90"},{"key":"ref83","first-page":"1735","article-title":"Autoregressive energy machines","volume-title":"Proc. 36th Int. Conf. Mach. Learn.","author":"Nash"},{"key":"ref84","doi-asserted-by":"publisher","DOI":"10.1109\/CVPR46437.2021.01516"},{"key":"ref85","article-title":"Learning energy-based generative models via coarse-to-fine expanding and sampling","volume-title":"Proc. Int. Conf. Learn. Representations","author":"Zhao"},{"key":"ref86","article-title":"On stochastic moving-average estimators for non-convex optimization","author":"Guo","year":"2021"},{"key":"ref87","first-page":"20571","article-title":"STORM : Fully adaptive SGD with recursive momentum for nonconvex optimization","volume-title":"Proc. Adv. Neural Inf. Process. Syst.","author":"Levy"},{"key":"ref88","first-page":"32499","article-title":"Multi-block-single-probe variance reduced estimator for coupled compositional optimization","volume-title":"Proc. Adv. Neural Inf. Process. Syst.","author":"Jiang"},{"key":"ref89","article-title":"Joint energy models","author":"Grathwohl","year":"2020"},{"key":"ref90","article-title":"PyTorch-FID: FID Score for PyTorch","author":"Seitzer","year":"2020"}],"container-title":["IEEE Transactions on Pattern Analysis and Machine Intelligence"],"original-title":[],"link":[{"URL":"http:\/\/xplorestaging.ieee.org\/ielx8\/34\/11345188\/11202712.pdf?arnumber=11202712","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,1,12]],"date-time":"2026-01-12T22:00:57Z","timestamp":1768255257000},"score":1,"resource":{"primary":{"URL":"https:\/\/ieeexplore.ieee.org\/document\/11202712\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,2]]},"references-count":90,"journal-issue":{"issue":"2"},"URL":"https:\/\/doi.org\/10.1109\/tpami.2025.3621320","relation":{},"ISSN":["0162-8828","2160-9292","1939-3539"],"issn-type":[{"value":"0162-8828","type":"print"},{"value":"2160-9292","type":"electronic"},{"value":"1939-3539","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,2]]}}}