{"id":4017,"date":"2025-05-22T07:03:29","date_gmt":"2025-05-22T07:03:29","guid":{"rendered":"https:\/\/mailitics.com\/index.php\/2025\/05\/22\/2505-15013\/"},"modified":"2025-05-22T07:03:29","modified_gmt":"2025-05-22T07:03:29","slug":"2505-15013","status":"publish","type":"post","link":"https:\/\/mailitics.com\/index.php\/2025\/05\/22\/2505-15013\/","title":{"rendered":"Convergence of Adam in Deep ReLU Networks via Directional Complexity and Kakeya Bounds"},"content":{"rendered":"

Convergence of Adam in Deep ReLU Networks via Directional Complexity and Kakeya Bounds
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arXiv:2505.15013v1 Announce Type: new
\nAbstract: First-order adaptive optimization methods like Adam are the default choices for training modern deep neural networks. Despite their empirical success, the theoretical understanding of these methods in non-smooth settings, particularly in Deep ReLU networks, remains limited. ReLU activations create exponentially many region boundaries where standard smoothness assumptions break down. textbf{We derive the first (tilde{O}!bigl(sqrt{d_{mathrm{eff}}\/n}bigr)) generalization bound for Adam in Deep ReLU networks and the first global-optimal convergence for Adam in the non smooth, non convex relu landscape without a global PL or convexity assumption.} Our analysis is based on stratified Morse theory and novel results in Kakeya sets. We develop a multi-layer refinement framework that progressively tightens bounds on region crossings. We prove that the number of region crossings collapses from exponential to near-linear in the effective dimension. Using a Kakeya based method, we give a tighter generalization bound than PAC-Bayes approaches and showcase convergence using a mild uniform low barrier assumption.<\/div>\n

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\n Anupama Sridhar, Alexander Johansen
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Convergence of Adam in Deep ReLU Networks via Directional Complexity and Kakeya Bounds arXiv:2505.15013v1 Announce Type: new Abstract: First-order adaptive optimization methods like Adam are the default choices for training modern deep neural networks. Despite their empirical success, the theoretical understanding of these methods in non-smooth settings, particularly in Deep ReLU networks, remains limited. ReLU […]<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[62,113,112],"tags":[2740,4,1046],"class_list":["post-4017","post","type-post","status-publish","format-standard","hentry","category-aimldsaimlds","category-cs-lg","category-stat-ml","tag-adam","tag-deep","tag-relu"],"_links":{"self":[{"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/posts\/4017"}],"collection":[{"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/comments?post=4017"}],"version-history":[{"count":0,"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/posts\/4017\/revisions"}],"wp:attachment":[{"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/media?parent=4017"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/categories?post=4017"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/tags?post=4017"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}