[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83028-en":3,"doc-seo-83028-105":30,"detail-sidebar-cat-0-en-105":92},{"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":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},83028,7971461740909,"Levi","https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d",8,"Research & Report","Stability Annealing Selects the Implicit Bias of Smoothed Sign Descent: A Rate-Indexed Barrier Path on Separable Data","Adaptive gradient methods can favor max-margin separators different from gradient descent, but a fixed positive numerical stability constant eventually changes the update geometry again. This paper studies the rate-controlled middle case for full-batch linear classification on separable data. For memoryless stability-annealed smoothed-sign descent with weighted exponential loss, it proves normalized iterates converge to a Burg-type convex barrier minimizer over a margin slice. The dynamics are rewritten as entropic mirror ascent on a concave dual, with a KL recursion controlling the dual gap, yielding explicit normalized-iterate envelopes.","Stability Annealing Selects the Implicit Bias of Smoothed Sign Descent: A Rate-Indexed Barrier Path on Separable Data  \nXiangwu Wang1 , Chengwei Cao2 , Yicheng Song3 , Ran Bi1 , Peilin Yu1  \n1The University of Hong Kong  \n2University of California, San Diego  \n3Beijing University of Aeronautics and Astronautics  \narXiv :2607 .060 13v 1 [ cs .LG] 7 Jul 2026  \nAbstract  \nAdaptive gradient methods can favor max-margin separators that differ from gradient descent, yet a fixed positive numerical stability constant eventually changes the update geometry again. This paper studies the rate-controlled middle case for full-batch linear classification on separable data. For memoryless stability-annealed smoothed-sign descent with weighted exponential loss, we prove that the normalized iterates converge to the minimizer of a convex Burg-type barrier over a margin slice. The proof rewrites the dynamics exactly as entropic mirror ascent on a concave dual objective, controls the dual gap by a KL recursion, and yields an explicit S1/2 normalized-iterate envelope. The static barrier geometry is fully characterized, including KKT conditions and both endpoint limits. Experiments validate the exact dual identities to floating-point error, illustrate the predicted path and rate diagram, and show an empirical fixed-ϵ crossover scaling in cumulative time. We further report robustness and boundary diagnostics for logistic tails, fixed-ϵ crossover, and adaptive-method variants, delineating the scope of the proved smoothed-sign theory.  \nIntroduction  \nImplicit bias results for separable classification usually compare endpoints: gradient descent converges in direction to an ℓ2-margin separator, while sign-like, normalized, and adaptive methods are linked to non-Euclidean or ℓ∞ -type geometry (Soudry et al. 2018; Nacson, Srebro, and Soudry 2019; Lyu and Li 2020; Kingma and Ba 2015; Wilson et al. 2017; Zhang, Zou, and Cao 2024; Fan, Schmidt, and Thrampoulidis 2025). The numerical stability constantin Adam-like updates complicates that endpoint story. When the stability term is absent or negligible, the update resembles a sign direction. When it is fixed and positive, sufficiently small gradients eventually see a more gradient-descent-like denominator. The motivating contradiction is that both endpoints are plausible, but neither specifies what separator is selected when the stability term is deliberately annealed at a prescribed exponential rate.  \nWe study this question first for the memoryless smoothedsign proxy  \nwt+1 = wt + ηt ~~ ~~|~~ ~~∇∇(Lwt()w|~~ ~~)~~ ~~ϵt , ϵt = ϵ0 exp(−κSt),  \n(1)  \nu2  \n1.00 0.75 0.50 0.25  \n0.00 −0.25 −0.50 −0.75 −1.00  \nRate-indexed barrier path  \n−1.0 −0.5 0.0 0.5 1.0  \nu1  \n0.9  \n0.5  \n0.1  \nκ/ γ∞  \nFigure 1: Two-dimensional barrier geometry. Contours show the Burg-type barrier, the shaded region is one feasible margin slice, markers are uκ colored by κ/γ∞ , the arrow shows the small-κEuclidean direction, and the star is the ℓ∞ -margin endpoint. The selected dataset is illustrative only and is excluded from aggregate statistics.  \nwhere St = Pts10 ηs , and absolute values and divisions are coordinatewise. Equation (1) removes Adam’s exponential moving averages while retaining the coordinatewise competition between gradient magnitude and stability. The central object is not a generic transition curve. It is the rate-indexed constrained barrier path shown in Figure 1 .  \nThis paper makes four contributions. First, it gives an exact rate-indexed implicit-bias theorem for stability-annealed smoothed-sign descent under weighted exponential loss. Second, it identifies the limiting separator as a Burg-barrier minimizer over a margin slice and characterizes the endpoint geometry. Third, it proves convergence through an exact entropic mirror-ascent representation and a KL recursion. Fourth, it reports theorem-compatible numerical diagnostics, including floating-point dual residuals and a longhorizon showcase, together with boundar","cbCaip91PSRtygI9","https://ap.wps.com/l/cbCaip91PSRtygI9","pdf",1288079,5,1,17,"English","en",105,"# Abstract\n# Introduction\n# Related Work and Problem Setup\n# Rate-indexed Barrier Path\n# Contributions and Theorem Scope","[{\"question\":\"What does stability annealing determine in smoothed-sign descent for separable data?\",\"answer\":\"It selects the implicit bias by inducing a rate-indexed constrained barrier path, where the limiting separator is characterized as a Burg-barrier minimizer over a margin slice.\"},{\"question\":\"How are the normalized iterates shown to converge?\",\"answer\":\"The paper rewrites the dynamics exactly as entropic mirror ascent on a concave dual objective, then controls the dual gap using a KL recursion to establish convergence to the barrier minimizer.\"},{\"question\":\"What is validated through experiments in the study?\",\"answer\":\"Experiments validate the dual identities to floating-point error, illustrate the predicted path and rate diagram, and demonstrate an empirical fixed-ε crossover scaling in cumulative time, along with boundary diagnostics for logistic tails and adaptive-method variants.\"}]",1784184739,43,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":87,"head_meta":89,"extra_data":91,"updated_unix":28},"stability-annealing-selects-the-implicit-bias-of-smoothed-sign-descent-a-rate-indexed-barrier-path-on-separable-data","",{"@graph":36,"@context":86},[37,54,69],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":53},"https://docshare.wps.com/document/stability-annealing-selects-the-implicit-bias-of-smoothed-sign-descent-a-rate-indexed-barrier-path-on-separable-data/83028/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":24,"description":14,"dateModified":62,"datePublished":63,"encodingFormat":61,"isAccessibleForFree":64,"interactionStatistic":65},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-07-24","2026-07-16",true,{"@type":66,"interactionType":67,"userInteractionCount":20},"InteractionCounter",{"@type":68},"ViewAction",{"@type":70,"mainEntity":71},"FAQPage",[72,78,82],{"name":73,"@type":74,"acceptedAnswer":75},"What does stability annealing determine in smoothed-sign descent for separable data?","Question",{"text":76,"@type":77},"It selects the implicit bias by inducing a rate-indexed constrained barrier path, where the limiting separator is characterized as a Burg-barrier minimizer over a margin slice.","Answer",{"name":79,"@type":74,"acceptedAnswer":80},"How are the normalized iterates shown to converge?",{"text":81,"@type":77},"The paper rewrites the dynamics exactly as entropic mirror ascent on a concave dual objective, then controls the dual gap using a KL recursion to establish convergence to the barrier minimizer.",{"name":83,"@type":74,"acceptedAnswer":84},"What is validated through experiments in the study?",{"text":85,"@type":77},"Experiments validate the dual identities to floating-point error, illustrate the predicted path and rate diagram, and demonstrate an empirical fixed-ε crossover scaling in cumulative time, along with boundary diagnostics for logistic tails and adaptive-method 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