[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85005-en":3,"doc-seo-85005-105":30,"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":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},85005,13056703019404,"Miles","https://ap-avatar.wpscdn.com/davatar_29158cc5080c5b710cf443261637dec0",8,"Research & Report","Gradient-free Riemannian Langevin Sampler","Efficient sampling of multimodal probability distributions is addressed where standard Markov Chain Monte Carlo can mix poorly and become trapped in modes. The Gradient-free Riemannian Langevin Sampler (GRiLS) improves exploration by introducing a Riemannian metric that reshapes local geometry to facilitate cross-mode transitions while avoiding gradient evaluations of the target density. The method builds a gradient-free MCMC procedure for complex, derivative-free targets by estimating the target mean and covariance via an interacting particle ensemble. Benchmarks show improved mixing versus existing gradient-based and gradient-free approaches.","arXiv :2607 .075 19v 1 [ cs .LG] 8 Jul 2026  \nGradient-free Riemannian Langevin Sampler  \nRicardo Baptista∗, Olivier Zahm†  \nJuly 9, 2026  \nAbstract  \nWe address the problem of efficiently sampling multimodal probability distributions, where standard Markov Chain Monte Carlo methods often suffer from poor mixing and mode trapping. To mitigate these issues, we propose Gradient-free Riemannian Langevin Sampler (GRiLS), a novel proposal that improves exploration without requiring gradient evaluations of the target density. Our approach introduces a Riemannian metric which reshapes the local geometry in order to facilitate transitions across modes. The resulting gradient-free MCMC algorithm is particularly suitable for complex, computationally expensive targets where derivatives are unavailable or impractical. The GRiLS proposal requires knowing the mean and covariance of the target density, which we estimate using an ensemble of interacting particles. Empirical results on multimodal benchmarks demonstrate that GRiLS achieves improved mixing compared to existing gradient-based and gradient-free MCMC approaches.  \nKeywords: Gradient-free MCMC, Riemannian Langevin dynamics, Lamperti transform, Interacting particle system.  \n1 Introduction  \nThe accurate and efficient sampling of high-dimensional probability distributions is a central task in computational statistics. For sampling distributions on Rd whose density µ is known up to a normalization constant, Markov Chain Monte Carlo (MCMC) remains a predominant method [25 , 54] . Given a proposal density q (·|·), MCMC algorithms build a Markov chain {x1 , x2 , ... } by drawing a proposal candidate x† ∼ q (·|xk ) and directly accepting this as the next state xk+1 = x† (unadjusted algorithms) or rejecting it with a certain probability and setting xk+1 = xk (Metropolis adjusted algorithms) . MCMC offers strong theoretical guarantees, especially when the target density µ is logconcave [10 , 17 , 18] . However, one major difficulty in practical applications is when the target is not log-concave, e.g., under multimodality. In such scenarios, the Markov chain may be trapped in a mode, leading to poor mixing, high autocorrelation, and large bias. The computational efficiency of various samplers, often quantified by the time required to traverse energy barriers between modes, deteriorates exponentially as the modes become more separated [34] .  \n∗ Department of Statistical Sciences, University of Toronto, Canada, [r.baptista@utoronto.ca](r.baptista@utoronto.ca)[ ](r.baptista@utoronto.ca)†UGA, Inria, CNRS, Grenoble INP*, LJK, 38000 Grenoble, France, [olivier.zahm@inria.fr](olivier.zahm@inria.fr)  \nTo overcome these mixing challenges, practitioners have explored a variety of techniques, including tempered MCMC (also known as Parallel Tempering [49]), transport map accelerated MCMC [15 , 52], adaptive biasing force methods [12 , 62], dimension reduction techniques [45 , 67] and preconditioned Langevin dynamics [16 , 28 , 42 , 66] to name just a few. We focus here on the latter approach, which defines the proposal density via a time discretization of a Langevin dynamic that has been preconditioned in order to improve the convergence of the continuous-time dynamic toward equilibrium. This preconditioning is achieved by equipping Rd with a suitable Riemannian metric that locally reduces the geodesic distances between modes, thereby facilitating the transitions of particles across different regions of high probability. Given an arbitrary field of symmetric definite matrices W : x →7 W (x) ∈ Rd ×d , we endow Rd with the Riemannian metric ⟨u, v⟩x = u⊤ W (x)−1v. The Riemannian Langevin dynamic is given by  \ndXt =􀀀divW(Xt ) + W(Xt)∇lnµ(Xt)􀀁dt +p2W(Xt)dBt , (1)  \nwhere divW(x) = (Pdj=1 ∂jWi,j (x))1≤i≤d is the divergence of W and (Bt)t≥0 is the standard Brownian motion in Rd , see e.g. [11 , 36] . Popular choices for W include the constant metric [30 , 33 , 64], the inverse negative Hessian of the log-densi","cbCaii9tZ019PwAA","https://ap.wps.com/l/cbCaii9tZ019PwAA","pdf",4659153,3,1,32,"English","en",105,"# Abstract\n# Introduction\n## Sampling and MCMC challenges\n## Riemannian Langevin dynamics and preconditioning\n## GRiLS construction and proposal mechanism","[{\"question\":\"What problem does GRiLS target in multimodal sampling?\",\"answer\":\"GRiLS targets the poor mixing and mode trapping that standard MCMC can suffer when the target distribution is not log-concave, leading to high autocorrelation and bias. It aims to traverse energy barriers between separated modes more efficiently.\"},{\"question\":\"How does GRiLS avoid gradient evaluations of the target density?\",\"answer\":\"GRiLS selects a specific Riemannian metric using the target mean and covariance, which yields a gradient-free Langevin dynamic. This eliminates the ∇lnµ term in the Riemannian Langevin equation.\"},{\"question\":\"What information does GRiLS require to run the sampler?\",\"answer\":\"GRiLS requires the mean and covariance of the target density. These quantities are estimated using an ensemble of interacting particles, which supports the construction of the metric and the resulting proposal.\"}]",1784200183,81,{"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":86,"head_meta":88,"extra_data":90,"updated_unix":28},"gradient-free-riemannian-langevin-sampler","",{"@graph":36,"@context":85},[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/gradient-free-riemannian-langevin-sampler/85005/",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","2026-07-23","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 problem does GRiLS target in multimodal sampling?","Question",{"text":75,"@type":76},"GRiLS targets the poor mixing and mode trapping that standard MCMC can suffer when the target distribution is not log-concave, leading to high autocorrelation and bias. It aims to traverse energy barriers between separated modes more efficiently.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does GRiLS avoid gradient evaluations of the target density?",{"text":80,"@type":76},"GRiLS selects a specific Riemannian metric using the target mean and covariance, which yields a gradient-free Langevin dynamic. This eliminates the ∇lnµ term in the Riemannian Langevin equation.",{"name":82,"@type":73,"acceptedAnswer":83},"What information does GRiLS require to run the sampler?",{"text":84,"@type":76},"GRiLS requires the mean and covariance of the target density. These quantities are estimated using an ensemble of interacting particles, which supports the construction of the metric and the resulting proposal.","https://schema.org",{"og:url":51,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":51},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":46,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":46,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]