[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-118204-en":3,"doc-seo-118204-105":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":4,"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":27,"seo_description":14,"update_tm":28,"read_time":29},118204,8796095461610,"Oliver","https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c",8,"Research & Report","On the Theory of Lipschitz Continuous Machine Learning - Doctoral Thesis","Machine learning theory provides mathematical foundations and performance limits for data-driven modelling. Through rigorous analysis of algorithm properties, theoretical machine learning supports the development of dependable methods for real-world use. Lipschitz regularity is a key instrument for establishing robustness, worst-case error bounds, and generalisation for many learning frameworks. This thesis studies Lipschitz continuous machine learning theory, emphasizing dynamical system identification. It derives sample complexity results for estimating Lipschitz constants under minimal assumptions, then analyzes Lipschitz interpolation via asymptotic consistency and convergence-rate bounds. Finally, it applies relaxed Lipschitz-type regularity with neural identification for time-series mean reversion, giving probabilistic guarantees in a financial pairs-trading application.","On the Theory of Lipschitz Continuous Machine Learning  \nJulien Walden Huang  \nSt John's College University of Oxford  \nA thesis submitted for the degree of Doctor of Philosophy Trinity 2023  \nThis thesis is dedicated to my family: my parents, Daphne and Mathieu.  \nAcknowledgements  \nFirstly, I would like to express my gratitude to my first supervisor, Professor JanPeter Calliess, with whom I worked closely throughout my DPhil journey. Our meetings and brain-storming sessions were always both incredibly intellectually intense and fun. I enjoyed them tremendously. Your mentorship was instrumental in shaping my research and growth over the past four years.  \nSecondly, I would like to thank my second supervisor, Professor Stephen Roberts, foryour crucial contributions in providing a broader perspective to my work and guiding my research to its best possible outcome. Your guidance and support throughout the DPhil were invaluable.  \nI also extend my gratitude to my viva committee members, Professor Kostas Margellos and Professor Daniel Limon for examining my thesis. Your constructive feedback and insightful comments have significantly enhanced the overall quality of this thesis.  \nI am incredibly grateful to the Oxford-Man Institute of Quantitative Finance for their funding and support during the DPhil.  \nI was fortunate to have been able to work alongside amazing friends at the OMI. In particular, I would like to thank my cubicle neighbour, Daniel, for his friendship as we travelled our DPhil journeys together. Outside of the OMI, many thanks tomy friends in Oxford, especially the members of the Ping Pong lunch group and Romain, for all the laughs and great moments.  \nTo Isabelle, thank you for your unwavering love, support and kindness during this DPhil. Your presence and belief in me made every step of this academic adventure more meaningful.  \nLast but certainly not least, I want to thank my family. To my sister Daphne and brother Mathieu, for always being present and for the fun times we shared during the COVID-19 pandemic. To my Grandfather, for your love and for encouraging me to start the DPhil. And finally, to my parents for their unwavering support and love throughout my life, which have been the foundation of my success. Dad, you were the main source of inspiration for this whole academic journey and Mom, your constant encouragement and belief in me made this adventure possible. Thank you from the bottom of my heart for everything.  \nAbstract  \nThe field of machine learning theory plays an essential role in establishing the mathematical foundations and performance boundaries of data-driven modelling techniques. By providing a rigorous analysis of the underlying properties of an algorithm, theoretical machine learning guides the development of reliable methods that can be utilised in realworld applications. In this context, Lipschitz regularity has been a particularly useful tool, aiding in establishing robustness, worst-case error bounds, and generalisation capabilities for a wide range of machine learning frameworks. Building on this foundation, this thesis explores the theoretical properties of the general class of Lipschitz continuous machine learning frameworks with a specific focus on dynamical system identification.  \nThe first part of this thesis investigates a fundamental problem of this class of machine learning frameworks which is the estimation of the Lipschitz constant of the target function from data. We derive optimal sample complexity rates for this problem in both the noiseless and the noisy settings under minimal parametric assumptions on the target function. A novel Lipschitz constant estimation technique shown to be computationally efficient and sample optimal is also proposed.  \nThe second part of the thesis focuses on a popular non-parametric system identification method utilised in control: Lipschitz interpolation. It derives a series of theoretical resultson the asymptotic properties of the framework under a","cbCaiolELHXFhKWi","https://ap.wps.com/l/cbCaiolELHXFhKWi","pdf",4407355,1,218,"English","en",105,"# Table of Contents\n## 1 Introduction\n## 2 Background","[{\"question\":\"What is the main research focus of this thesis?\",\"answer\":\"The thesis investigates theoretical properties of Lipschitz continuous machine learning frameworks, with a particular emphasis on dynamical system identification.\"},{\"question\":\"How does the thesis address the estimation of a Lipschitz constant?\",\"answer\":\"It studies estimating the Lipschitz constant of the target function from data and derives optimal sample complexity rates in noiseless and noisy settings, under minimal parametric assumptions.\"},{\"question\":\"What roles do Lipschitz interpolation and neural network-based identification play?\",\"answer\":\"Lipschitz interpolation is analyzed for asymptotic properties, including consistency and uniform non-parametric convergence rates. Separately, relaxed Lipschitz-type regularity is combined with neural network-based identification for time-series mean reversion, leading to bounds and a financial pairs-trading application.\"}]","On the Theory of Lipschitz Continuous Machine Learning - Doctoral Thesis | PDF",1785682169,549,{"code":4,"msg":31,"data":32},"ok",{"site_id":24,"language":23,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"on-the-theory-of-lipschitz-continuous-machine-learning-doctoral-thesis","",{"@graph":36,"@context":85},[37,54,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":20},"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":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":53},"https://docshare.wps.com/document/on-the-theory-of-lipschitz-continuous-machine-learning-doctoral-thesis/118204/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":23,"description":14,"dateModified":62,"datePublished":62,"encodingFormat":61,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-08-02",true,{"@type":65,"interactionType":66,"userInteractionCount":4},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What is the main research focus of this thesis?","Question",{"text":75,"@type":76},"The thesis investigates theoretical properties of Lipschitz continuous machine learning frameworks, with a particular emphasis on dynamical system identification.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the thesis address the estimation of a Lipschitz constant?",{"text":80,"@type":76},"It studies estimating the Lipschitz constant of the target function from data and derives optimal sample complexity rates in noiseless and noisy settings, under minimal parametric assumptions.",{"name":82,"@type":73,"acceptedAnswer":83},"What roles do Lipschitz interpolation and neural network-based identification play?",{"text":84,"@type":76},"Lipschitz interpolation is analyzed for asymptotic properties, including consistency and uniform non-parametric convergence rates. Separately, relaxed Lipschitz-type regularity is combined with neural network-based identification for time-series mean reversion, leading to bounds and a financial pairs-trading application.","https://schema.org",{"og:url":52,"og:type":87,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":89,"canonical":52},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":20,"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":53,"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"]