[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84350-en":3,"doc-seo-84350-105":29,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},84350,1099514068365,"Aurelia","https://ap-avatar.wpscdn.com/avatar/10000253d8d9f28188e?_k=1776742907772140068",8,"Research & Report","Learning AC0 under Locally Sampleable Graphical Models","Learning constant-depth circuits has deep impact on computational learning theory. Linial, Mansour, and Nisan established a quasipolynomial-time learner for AC0 under the uniform distribution using low-degree techniques. Extending learning to broader correlated distributions, prior work achieved quasipolynomial learners for bounded-degree Gibbs graphical models requiring strong spatial mixing and polynomial growth. This paper provides quasipolynomial-time learning for AC0 under graphical models with efficient local samplers, removing the polynomial-growth constraint via new low-degree approximation based on truncated Glauber dynamics. Applications cover two-spin systems near sampling thresholds.","arXiv :2607 .08303v 1 [ cs .LG] 9 Jul 2026  \nLearning AC0 under Locally Sampleable Graphical Models Weiming Feng∗ Xiongxin Yang† Yixiao Yu‡ Yiyao Zhang‡  \nAbstract  \nThe problem of learning constant-depth circuits holds profound implications for computational learning theory. In a seminal result, by introducing the low-degree algorithm, Linial, Mansour, and Nisan (J. ACM 1993) presented a quasipolynomial-time learner for AC0 under the uniform distribution. However, obtaining comparable learning guarantees for broader classes of correlated distributions has remained a longstanding challenge. Recently, Chandrasekaran, Gaitonde, Moitra, and Vasilyan (arXiv 2026) extended these guarantees to Gibbs distributionson bounded-degree graphical models with both strong spatial mixing and polynomial growth. In this paper, we give a quasipolynomial-time learner for AC0 under graphical models that admit efficient local samplers, circumventing the polynomial-growth requirement in prior work. The key ingredient is a new low-degree approximation for Gibbs distributions, established by simulating and suitably truncating the classical Glauber dynamics. As applications, this framework yields learners for two-spin systems, including the hard-core model and Ising model, on arbitrary bounded-degree graphs, in regimes approaching their respective sampling thresholds.  \nContents  \n1 Introduction 2  \n2 Technical Overview 5  \n3 Preliminaries 9  \n4 Low-Degree Approximation under General Product Distributions 12  \n5 Low-Degree Approximation under Gibbs Distributions 17  \n6 Applications 32  \nReferences 43  \n∗School of Computing and Data Science, The University of Hong Kong. Email: wfeng@hku.hk  \n†Department of Computer Science, University of California, Santa Barbara. Email: [xiongxinyang@ucsb.edu](xiongxinyang@ucsb.edu)  \n‡State Key Laboratory for Novel Software Technology, New Cornerstone Science Laboratory, Nanjing University.  \nEmails: [yixiaoyu@smail.nju.edu.cn](yixiaoyu@smail.nju.edu.cn) , [zhangyiyao@smail.nju.edu.cn](zhangyiyao@smail.nju.edu.cn)  \n1 Introduction  \nLearning small-depth Boolean circuits is a classical and fundamental problem in learning theory. In a seminal result, Linial, Mansour and Nisan [LMN93] showed that every polynomial-size AC0 circuit has a quasipolynomial-degree Fourier approximation under the uniform distribution, leading to a quasipolynomial-time learning algorithm. The same technique also applies broadly to product distributions since independence gives a Fourier basis, and low-degree approximation turns learning into low-dimensional regression [FJS91; BOW10] .  \nMany natural distributions, however, are far from product. The graphical models is a succient way to represent complex joint distributions, which are widely used in physics, probability, and computer science. Given a underlying graph G = (V, E), a Gibbs distribution µ is a probability distribution on the configuration space {±1}V, where each vertex v ∈ V is associated with a Boolean random variable and adjacent variables are coupled through the edges in E. We consider the following problem of learning AC0 function f : {±1}V → {±1} under graphical models.  \nLearning AC0 function f under Gibbs distribution µ Input: N samples (x1, f (x1 )), . . . ,(xN, f (xN)), where xi ∼ µ .  \nOutput: A hypothesis h : supp (µ) → {±1} such that Px∼µ [f (x)  h (x)] ≤ ε .  \nA recent work of Chandrasekaran, Gaitonde, Moitra and Vasilyan [CGMV26] made an important step toward this problem: they gave a learning algorithm for AC0 under bounded-degree Gibbs distributions satisfying strong spatial mixing and polynomial growth. Strong spatial mixing is a natural property of Gibbs distributions, which ensures that the correlation between random variables decays with their graph distance. Polynomial growth requires that for every vertex v ∈ V, the number of vertices within distance ℓ of v is at most poly(ℓ) . Combining these two properties, they showed that every AC0 function f can be approxim","cbCaif5ULdo88eow","https://ap.wps.com/l/cbCaif5ULdo88eow","pdf",665262,1,45,"English","en",105,"# Introduction\n## Technical Overview\n## Preliminaries\n## Low-Degree Approximation under General Product Distributions\n## Low-Degree Approximation under Gibbs Distributions\n## Applications","[{\"question\":\"What learning problem is addressed for AC0 circuits under graphical models?\",\"answer\":\"The goal is to learn an AC0 function f from labeled samples drawn from a Gibbs distribution on {±1}^V, outputting a hypothesis h with small misclassification error under that same distribution.\"},{\"question\":\"How does this work improve over prior results for AC0 learning under Gibbs distributions?\",\"answer\":\"It gives a quasipolynomial-time learner for Gibbs models that admit efficient local samplers, avoiding the polynomial-growth requirement used in earlier work while retaining strong learning guarantees.\"},{\"question\":\"What is the main technical ingredient enabling the new learning guarantee?\",\"answer\":\"A new low-degree approximation for Gibbs distributions is obtained by simulating the classical Glauber dynamics and applying suitable truncation to control the approximation error.\"}]",1784194992,113,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":27},"learning-ac0-under-locally-sampleable-graphical-models","",{"@graph":35,"@context":85},[36,53,68],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/learning-ac0-under-locally-sampleable-graphical-models/84350/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-17","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What learning problem is addressed for AC0 circuits under graphical models?","Question",{"text":75,"@type":76},"The goal is to learn an AC0 function f from labeled samples drawn from a Gibbs distribution on {±1}^V, outputting a hypothesis h with small misclassification error under that same distribution.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does this work improve over prior results for AC0 learning under Gibbs distributions?",{"text":80,"@type":76},"It gives a quasipolynomial-time learner for Gibbs models that admit efficient local samplers, avoiding the polynomial-growth requirement used in earlier work while retaining strong learning guarantees.",{"name":82,"@type":73,"acceptedAnswer":83},"What is the main technical ingredient enabling the new learning guarantee?",{"text":84,"@type":76},"A new low-degree approximation for Gibbs distributions is obtained by simulating the classical Glauber dynamics and applying suitable truncation to control the approximation 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