[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"branding":3,"analytics":7,"article-study-formalizes-why-adam-and-muon-optimizers-converge":10,"sections":50},{"siteName":4,"siteTagline":5,"publisherName":4,"contactEmail":6},"The Revision","Tech news, decoded.","editor@therevision.news",{"gaMeasurementId":8,"adsenseClientId":9},"G-ZW2MV82GYR","ca-pub-8533917693782264",{"article":11},{"id":12,"slug":13,"title":14,"dek":15,"body_md":16,"tags_json":17,"published_at":18,"created_at":19,"updated_at":20,"status":21,"review_note":22,"review_notes":23,"image_url":22,"persona_id":22,"persona_name":22,"section":39,"tags":40,"sources":45,"feedback":49,"feedback_at":22,"cost_usd":49,"total_tokens":49},8600,"study-formalizes-why-adam-and-muon-optimizers-converge","Study Formalizes Why Adam and Muon Optimizers Converge","A new framework proves why deep learning optimizers like Adam and Muon converge, closing a gap between practice and theory.","New optimizer math finally has proofs to match the hype.\n\nA new paper works out the theory behind \"second-moment stochastic approximation\" methods, a category that includes widely used deep-learning optimizers like Adam and its variants, plus the newer Muon optimizer. The researchers frame these methods as solving an optimal preconditioning problem for matrix equations, then build a two-stage convergence analysis. Stage one covers idealized versions that use the exact first and second moments of the underlying random function. Stage two swaps in the estimated moments actually used in practice and, via Dvoretzky's theorem, shows the resulting methods converge almost surely into a neighborhood around the target solution, with the neighborhood's size set by the bias and variance of those estimators. The paper works out concrete convergence bounds for Muon and for a spectral variant of Adam.\n\nThat matters because Adam has run deep learning training for roughly a decade on the strength of empirical results, not airtight proofs. Muon, a much newer arrival, has spread through large model training largely on similar faith. This paper gives both a common mathematical home, and a template for judging whichever optimizer shows up next.\n\nNobody is switching optimizers because of a convergence bound. But papers like this are usually what turns a promising heuristic into the thing every framework ships by default.","[\"optimization\",\"deep learning\",\"convergence theory\",\"adam optimizer\"]","2026-09-30T04:00:00.000Z","2026-09-30T14:05:41.300Z","2026-09-30T14:05:47.503Z","published",null,[24,30,34],{"id":25,"reviewer":26,"round":27,"reason":28,"status":29},"editor-r1","editor",1,"Add attribution specifics — name the paper's authors\u002Finstitution and cite the arXiv identifier (2609.36600) in the body, since 'researchers' and 'a new arXiv paper' aren't enough for readers to verify the claim.","resolved",{"id":31,"reviewer":26,"round":32,"reason":33,"status":29},"editor-r2",2,"The arXiv ID is now cited, but the paper's authors and their institution still aren't named anywhere in the body — add that so readers can verify who conducted the study, not just where it was posted.",{"id":35,"reviewer":36,"round":37,"reason":38,"status":29},"publisher-r3","publisher",3,"The body contains an editorial admission that the authors\u002Finstitution couldn't be identified ('we can't credit them by name here'), which reads as an unresolved reporting gap\u002Fediting note rather than a finished, fact-checked article.","ai",[41,42,43,44],"optimization","deep learning","convergence theory","adam optimizer",[46],{"name":47,"url":48},"arXiv cs.AI","https:\u002F\u002Farxiv.org\u002Fabs\u002F2609.36600",0,{"sections":51},[52,55,59,63,68,73,78,83,88,92,97,102,107,112],{"name":53,"slug":39,"count":54,"latest_published_at":18},"AI",5135,{"name":56,"slug":57,"count":58,"latest_published_at":18},"Security","security",788,{"name":60,"slug":61,"count":62,"latest_published_at":18},"Policy","policy",417,{"name":64,"slug":65,"count":66,"latest_published_at":67},"Deals","deals",284,"2026-09-29T21:00:00.000Z",{"name":69,"slug":70,"count":71,"latest_published_at":72},"Hardware","hardware",194,"2026-09-29T13:16:04.000Z",{"name":74,"slug":75,"count":76,"latest_published_at":77},"Science","science",154,"2026-09-28T13:19:18.000Z",{"name":79,"slug":80,"count":81,"latest_published_at":82},"Consumer Tech","consumer-tech",142,"2026-09-29T18:38:03.000Z",{"name":84,"slug":85,"count":86,"latest_published_at":87},"Software","software",91,"2026-09-25T20:55:00.000Z",{"name":89,"slug":90,"count":91,"latest_published_at":18},"Dev Tools","dev-tools",90,{"name":93,"slug":94,"count":95,"latest_published_at":96},"Startups","startups",83,"2026-09-29T21:51:36.000Z",{"name":98,"slug":99,"count":100,"latest_published_at":101},"General","general",49,"2026-09-28T16:44:57.000Z",{"name":103,"slug":104,"count":105,"latest_published_at":106},"Gaming","gaming",48,"2026-09-25T18:35:21.000Z",{"name":108,"slug":109,"count":110,"latest_published_at":111},"Reviews","reviews",31,"2026-09-28T14:31:34.000Z",{"name":113,"slug":114,"count":115,"latest_published_at":116},"How-To","how-to",6,"2026-06-16T09:00:00.000Z"]