[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83259-en":3,"doc-seo-83259-105":30,"detail-sidebar-cat-0-en-105":95},{"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},83259,13056703019662,"Evangeline","https://ap-avatar.wpscdn.com/avatar/be000253a8e92610077?_k=1778726343310543188",8,"Research & Report","Avoiding Unsafe Sets When Training With Langevin Dynamics","Training with noisy gradient descent can be modeled as overdamped Langevin dynamics on the loss landscape, motivating a safety question about bounding the probability that the parameter trajectory lies in a designated failure region. For a smooth, strongly convex loss in d dimensions with an energy-gap separated region, the equilibrium failure mass is exponentially small in d and yields time-uniform bounds after burn-in of order d. A shape-free bound relaxes to a static level, while an Ornstein–Uhlenbeck example shows transient swelling can still occur. A local relaxation rate based on the failure region’s spectral measure and a maximum-principle ceiling prevent such swelling uniformly in time.","arXiv :2607 .07538v 1 [ cs .LG] 8 Jul 2026  \nAVOIDING UNSAFE SETS WHEN TRAINING WITH LANGEVIN DYNAMICS  \nADAM OBERMAN  \nAbstract . Training a model with noisy gradient descent can be idealized as overdamped Langevin dynamics on the loss landscape, and a natural safety question is to bound the probability νt (AH ) = P (Qt ∈ AH ) that the trajectory lies in a designated failure region AH . We study this for a smooth, strongly convex loss in d dimensions and a failure region separated from the minimizer by an energy gap. Three bounds emerge. At the end of training, the equilibrium mass π (AH ) is exponentially small in d, with a complementary energy-barrier rate when the noise is small. Along the trajectory, a shape-free bound    \nνt (AH ) ≤ π(AH ) 􀀒 1 + qχ20/π (AH ) e −mt􀀓  \nshows that the in-set probability relaxes to (twice) the static value after a burn-in time of orderd, using only the global spectral gap m of the loss. A worked Ornstein-Uhlenbeck example shows this burn-in is necessary: an angular slice of the equilibrium shell can transiently swell by a factor exponential in d, even though its equilibrium mass is tiny. To rule such swelling out we introduce a local relaxation rate attached to the failure region, defined through the spectral measure of its centered indicator rather than a Dirichlet-form Rayleigh quotient. For geometrically isolated regions this rate exceeds the global one, shrinking the burn-in proportionally, and combined with a maximum-principle ceiling it caps the trajectory probability uniformly in time. The picture is that strong convexity sets how fast training relaxes, but the shape of the unsafe set decides whether the trajectory bulges through it on the way home.  \n1. Introduction  \nA model trained by noisy gradient descent can be idealized as a diffusion on its loss landscape, and a basic safety question is whether the training trajectory ever enters a designated bad region of parameter space. Write Qt ∈ Rd for the parameters at training time t and AH ⊆ Rd for a failure region: a set of parameters whose induced behavior we would like the trained model to avoid. Even when training ends safely, the trajectory can pass through A H on its way to the optimum, so the object of interest is the in-set probability  \nνt(AH) = P (Qt ∈ AH)  \nat every training time t, not only at convergence.  \nSeveral safety concerns share this shape. In code generation, A H is the set of parameters that emita hidden backdoor or a known-insecure pattern, and one wants the chance that training ever lands there to be negligible. In alignment, AH is a region of misaligned or deceptive behavior that a model might drift through before settling into a benign optimum. The motivating instance for this work is the Scientist AI (SAI) Predictor safety case of [BRG+26], which separates an honest non-agentic Predictor from a scaffold that gates its outputs through a guardrail, and bounds the probability that a consequence-invariant training process produces a dangerous Predictor (one whose guarded deployment causes a designated harm event above a normative threshold) uniformly in t by  \nνt(AH) ≤ Cbad Rshell.  \nHere Rshell is the conditional fraction of dangerous Predictors inside a narrow loss band under the initialization, argued exponentially small on the grounds that danger requires many coordinated  \nerrors, and Cbad is the within-band enrichment factor of training, assumed bounded as a stated Date: July 9, 2026 .  \n2 ADAM OBERMAN  \nrequirement on the process. That argument treats the dynamics generating ν t abstractly, as a distribution over training trajectories indexed by t; the present paper supplies those dynamics and bounds νt(AH) directly.  \nWe model a training run as the overdamped Langevin dynamics  \ndQt = −∇J(Qt)dt + σ dWt ,  \nwhere J is the training loss on Rd , Wt is standard Brownian motion, and σ > 0 is a noise level set by the optimization (informed, for example, by batch size and learning rate) .  \nThis paper proves an","cbCailJ1vHMHIwEp","https://ap.wps.com/l/cbCailJ1vHMHIwEp","pdf",587076,3,1,16,"English","en",105,"# Introduction\n## The Langevin idealization\n## Transient swelling","[{\"question\":\"What safety quantity does the paper aim to bound during training?\",\"answer\":\"It bounds the in-set probability ν_t(A_H)=P(Q_t∈A_H) for every training time t, where A_H is a designated failure region of parameters.\"},{\"question\":\"Under what assumptions are the probability bounds derived?\",\"answer\":\"The results assume a smooth, strongly convex loss in d dimensions and a failure region separated from the minimizer by an energy gap.\"},{\"question\":\"Why is a burn-in period needed in the shape-free bound?\",\"answer\":\"Because along the trajectory the in-set probability relaxes toward the static equilibrium value only after a burn-in time of order d, even though the equilibrium mass is small.\"},{\"question\":\"How does the paper prevent transient swelling of the failure-region probability?\",\"answer\":\"It introduces a local relaxation rate tied to the failure region’s spectral measure and combines it with a maximum-principle ceiling to cap the trajectory probability uniformly over time.\"}]",1784186326,40,{"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":90,"head_meta":92,"extra_data":94,"updated_unix":28},"avoiding-unsafe-sets-when-training-with-langevin-dynamics","",{"@graph":36,"@context":89},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,50],{"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":20},"https://docshare.wps.com/document/research-report/",{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/avoiding-unsafe-sets-when-training-with-langevin-dynamics/83259/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-24","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81,85],{"name":72,"@type":73,"acceptedAnswer":74},"What safety quantity does the paper aim to bound during training?","Question",{"text":75,"@type":76},"It bounds the in-set probability ν_t(A_H)=P(Q_t∈A_H) for every training time t, where A_H is a designated failure region of parameters.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"Under what assumptions are the probability bounds derived?",{"text":80,"@type":76},"The results assume a smooth, strongly convex loss in d dimensions and a failure region separated from the minimizer by an energy gap.",{"name":82,"@type":73,"acceptedAnswer":83},"Why is a burn-in period needed in the shape-free bound?",{"text":84,"@type":76},"Because along the trajectory the in-set probability relaxes toward the static equilibrium value only after a burn-in time of order d, even though the equilibrium mass is small.",{"name":86,"@type":73,"acceptedAnswer":87},"How does the paper prevent transient swelling of the failure-region probability?",{"text":88,"@type":76},"It introduces a local relaxation rate tied to the failure region’s spectral measure and combines it with a maximum-principle ceiling to cap the trajectory probability uniformly over time.","https://schema.org",{"og:url":51,"og:type":91,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":93,"canonical":51},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":96},[97,101,105,109,114,119,123,126,131,134,138],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Story & 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