[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83977-en":3,"doc-seo-83977-105":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},83977,7971461740886,"Theodore","https://ap-avatar.wpscdn.com/davatar_3d24733baf745e90a7e4bdd5f77d97b2",8,"Research & Report","On the Linear Convergence of Bregman Proximal Gradient Methods with Applications to Kullback-Leibler Regression","Bregman Proximal Gradient methods (BPGM) use a mirror map to capture the geometry of convex objectives. The paper introduces Restricted Relative Strong Convexity and proves linear convergence rates for BPGM under this condition. The theory is applied to regularized and unregularized Kullback–Leibler regression, covering cases with unique and non-unique minimizers. It shows Burg’s entropy may provide only limited linear convergence, while a smoothed Burg-entropy restores the needed geometry, supported by numerical experiments.","arXiv :2607 .05539v1 [math .OC] 6 Jul 2026  \nON THE LINEAR CONVERGENCE OF BREGMAN PROXIMAL GRADIENT METHODS WITH APPLICATIONS TO  \nKULLBACK–LEIBLER REGRESSION  \nJONATHAN CHIRINOS-RODR´IGUEZ1 , ∗ , CHRISTIAN DANIELE2 , C´EDRIC F´EVOTTE1 ,  \nAND EMMANUEL SOUBIES 1  \n1 University of Toulouse, Toulouse INP, CNRS, IRIT, France  \nEmail address: {jonathan-eduardo .chirinos-rodriguez, emmanuel .soubies,  \n[cedric.fevotte](cedric.fevotte}@irit.fr)[}](cedric.fevotte}@irit.fr)[@irit.fr](cedric.fevotte}@irit.fr)  \n2 MaLGa Center, DIBRIS, University of Genoa, Italy  \nLaboratoire J. A. Dieudonn´e, Universit´e Cˆote d’Azur, France Email address: [christian.daniele@edu.unige.it](christian.daniele@edu.unige.it)  \nAbstract. Bregman Proximal Gradient methods (BPGM) exploit the underlying geometry of the objective function through a carefully chosen mirror map. In this work, we introduce a novel notion of strong convexity, termed Restricted Relative Strong Convexity, and establish linear convergence rates for BPGM under this condition. We then exploit the proposed theoretical framework to provide an in-depth analysis of the convergence of BPGM for (regularized) Kullback–Leibler regression problems, covering scenarios with both unique and non-unique minimizers, as well as regularized and unregularized formulations. Specifically, we demonstrate that using the popular Burg’s entropy as a distancegenerating function may only yield linear convergence for certain KL regression problems.  \nIn contrast, we show that employing a smoothed version of the Burg’s entropy induces the suitable geometry required to guarantee linear convergence. We conclude with numerical experiments that nicely align with our theoretical findings.  \n1. Introduction  \nMany problems in fields such as machine learning, computational imaging, signal processing, and variational inverse problems can be formulated as the minimization of a convex composite functional. Specifically, these are problems of the form  \n(1) arg min J(x) := F(x) + R(x),  \nx∈C  \nwhere F : RN → ( −∞ , ∞ ] is proper, convex and differentiable, and R: RN → ( −∞ , ∞ ] is proper, convex and lower semicontinuous, but not necessarily differentiable. Moreover, C ⊆ int(dom J) denotes a constraint set, which we assume to be closed and convex. Finally, we assume that the solution set, denoted by S, is nonempty, and we denote ¯J := infx∈CJ(x) > −∞ the optimal value for Problem (1) .  \n∗ Corresponding author.  \n2 J. CHIRINOS-RODRIGUEZ, C. DANIELE, C. F´EVOTTE, AND E. SOUBIES  \nProximal Gradient Method (PGM) . In order to find solutions of (1), a method of choice is the so-called PGM, also known as forward-backward splitting algorithm [21, 40, 51] . Given x0 ∈ C, its iterates write:  \n(PGM) xk+1 = arg min  \nx∈C  \nF (xk ) + D∇F(xk ), x − xkE + 1τ ∥x − xk ∥22 + R(x),  \nwhere τ > 0 denotes a constant step-size. The convergence of (PGM) is classically ensured by assuming Lipschitz smoothness of the gradient (or simply L-smoothness) of F. Under this assumption, and for suitable choices of the step-size τ > 0, sublinear convergence rates—of the objective function values and sometimes of the iterates—, i.e., of order O(1/k) are wellestablished [50], with further accelerated versions providing rates of order O(1/k2 ) studied in [10, 49] .  \nIt is also well-known that if the differentiable term F is, in addition, strongly convex, then one can establish a linear rate of convergence, i.e. , of the form O (ρk ) for some ρ ∈ (0 , 1), see e.g. [17, 50] . We recall that F is strongly convex if there exists µ > 0 such that, for all x, x′ ∈ int(dom F)  \n(2) ⟨∇F(x) − ∇F(x′), x − x′⟩ ≥ µ∥x − x′∥22.  \nWhile L-smoothness is satisfied by a broad class of functions (possibly with a large, i.e. , loose, constant L), strong convexity is much more restrictive [39] . To illustrate this, consider the least-squares regression problem, where the smooth term equals F = (1/2)∥A ·−y∥22 for some linear map A ∈ RM×N . In this case, it is easy to see that F is","cbCaifQV9VyWgmnZ","https://ap.wps.com/l/cbCaifQV9VyWgmnZ","pdf",37393187,1,34,"English","en",105,"# Introduction\n## Proximal Gradient Method (PGM) and convergence overview\n## Strong convexity versus practical failure cases\n## Weaker conditions enabling linear convergence (Kurdyka–Łojasiewicz, RSC)\n## Restricted Strong Convexity (RSC) definition and implications","[{\"question\":\"What key condition is introduced to guarantee linear convergence of Bregman Proximal Gradient methods?\",\"answer\":\"The work introduces Restricted Relative Strong Convexity, a notion that requires strong convexity only between a point and its projection onto the solution set, rather than global strong convexity.\"},{\"question\":\"How is the theory applied in the paper?\",\"answer\":\"The paper develops an in-depth convergence analysis for (regularized) Kullback–Leibler regression, treating both unique and non-unique minimizers and both regularized and unregularized formulations.\"},{\"question\":\"Why does Burg’s entropy not always ensure linear convergence for KL regression?\",\"answer\":\"Using Burg’s entropy as the distance-generating function may only yield linear convergence for certain KL regression scenarios, while a smoothed version induces the geometry needed for the linear rate.\"}]",1784191808,86,{"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},"on-the-linear-convergence-of-bregman-proximal-gradient-methods-with-applications-to-kullback-leibler-regression","",{"@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/on-the-linear-convergence-of-bregman-proximal-gradient-methods-with-applications-to-kullback-leibler-regression/83977/",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 key condition is introduced to guarantee linear convergence of Bregman Proximal Gradient methods?","Question",{"text":75,"@type":76},"The work introduces Restricted Relative Strong Convexity, a notion that requires strong convexity only between a point and its projection onto the solution set, rather than global strong convexity.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How is the theory applied in the paper?",{"text":80,"@type":76},"The paper develops an in-depth convergence analysis for (regularized) Kullback–Leibler regression, treating both unique and non-unique minimizers and both regularized and unregularized formulations.",{"name":82,"@type":73,"acceptedAnswer":83},"Why does Burg’s entropy not always ensure linear convergence for KL regression?",{"text":84,"@type":76},"Using Burg’s entropy as the distance-generating function may only yield linear convergence for certain KL regression scenarios, while a smoothed version induces the geometry needed for the linear 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