[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-118836-en":3,"doc-seo-118836-105":30,"detail-sidebar-cat-0-en-105":92},{"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":27,"seo_description":14,"update_tm":28,"read_time":29},118836,34359740700684,"Finn","https://ap-avatar.wpscdn.com/avatar/1f400023980c374ae676?_k=1777273430885731487",8,"Research & Report","A Gentle Introduction to Gradient-Based Optimization and Variational Inequalities for Machine Learning","Rapid advances in machine learning have been powered by gradient-based optimization, yet further progress increasingly depends on shifting attention from pure pattern recognition to decision-making and multi-agent settings. In these domains, the central mathematical objects change from optimization minima to equilibria and game-theoretic structures. This set of notes introduces gradient-based algorithms beyond standard gradient descent, starting from saddle points and monotone games and extending to general variational inequalities, emphasizing intuition alongside convergence proofs for several methods.","arXiv :2309 .04877v 1 [ cs .LG] 9 Sep 2023  \nA Gentle Introduction to Gradient-Based Optimization and Variational Inequalities for Machine Learning  \nNeha S. Wadia 1, * Yatin Dandi2, and Michael I. Jordan3  \n1 Center for Computational Mathematics, Flatiron Institute  \n2 SPOC Laboratory and IdePHICS Laboratory, Ecole Polytechnique F´ed´erale de Lausanne (EPFL)  \n3 Department of EECS and Department of Statistics, University of California, Berkeley  \nSeptember 12, 2023  \n* [neha.wadia@berkeley.edu](neha.wadia@berkeley.edu)  \nAbstract  \nThe rapid progress in machine learning in recent years has been based on a highly productive connection to gradient-based optimization. Further progress hinges in part on a shift in focus from pattern recognition to decision-making and multi-agent problems. In these broader settings, new mathematical challenges emerge that involve equilibria and game theory instead of optima. Gradientbased methods remain essential—given the high dimensionality and large scale of machine-learning problems—but simple gradient descent is no longer the point of departure for algorithm design. We provide a gentle introduction to a broader framework for gradient-based algorithms in machine learning, beginning with saddle points and monotone games, and proceeding to general variationalinequalities. While we provide convergence proofs for several of the algorithms that we present, our main focus is that of providing motivation and intuition.  \nContents  \n1 Introduction 4  \n1.1 The Challenges of Decision-Making Processes ...................... 4  \n1.2 Multi-Way Markets ..................................... 6  \n1.3 Challenges at the Intersection of Machine Learning and Economics .......... 7  \n1.4 Two Illustrative Examples ................................. 7  \n1.4.1 Strategic Classification ............................... 7  \n1.4.2 Distribution-Free Uncertainty Quantification for Decision-Making ....... 8  \n1.5 Overview of the Lectures .................................. 9  \n1.6 The Subgradient Method and a First Convergence Proof ................ 10  \n2 Computing Optima in Discrete and Continuous Time 13  \n2.1 Convergence Guarantees for Gradient Descent on Convex Functions ......... 13  \n2.2 Gradient Descent on Nonconvex Functions: Escaping Saddle Points Efficiently ... 15  \n2.3 Variational, Hamiltonian, and Symplectic Perspectives on Acceleration ........ 16  \n2.4 Variational Inequalities: From Minima to Nash Equilibria and Fixed Points ..... 21  \n2.4.1 Two-Player Zero-Sum Games ........................... 21  \n2.4.2 Variational Inequalities .............................. 21  \n2.4.3 Nash Equilibria ................................... 22  \n2.4.4 Computing Nash Equilibria ............................ 23  \n3 Computing Equilibria 24  \n3.1 Monotone Operators .................................... 24  \n3.2 Fixed-Point Finding Algorithms .............................. 25  \n3.2.1 A Naive Algorithm ................................. 25  \n3.2.2 The Proximal Point Method ............................ 27  \n3.2.3 The Extragradient Algorithm ........................... 29  \n3.2.4 High-Resolution Continuous-Time Limits .................... 30  \nAcknowledgements  \nThese notes are based on three lectures delivered by Michael Jordan at the summer school “Statistical Physics and Machine Learning” held in Les Houches, France, in July 2022 . Neha Wadia and Yatin Dandi were students at the school. The authors are grateful to Florent Krzakala and Lenka Zdeborov´a for organizing the school.  \nThe authors thank Sai Praneeth Karimireddy for useful discussions clarifying the material on monotone operators, and Sidak Pal Singh for collaboration in the early stages of this manuscript.  \nMJ was supported in part by the Mathematical Data Science program of the Office of Naval Research under grant number N00014-18-1-2764 and by the Vannevar Bush Faculty Fellowship program under grant number N00014-21-1-2941 . NW’s attendance at the summer school was supported","cbCaiiErPkhtZAdg","https://ap.wps.com/l/cbCaiiErPkhtZAdg","pdf",669297,1,36,"English","en",105,"# Introduction\n## The Challenges of Decision-Making Processes\n## Multi-Way Markets\n## Challenges at the Intersection of Machine Learning and Economics\n## Two Illustrative Examples\n## Overview of the Lectures\n## The Subgradient Method and a First Convergence Proof\n# Computing Optima in Discrete and Continuous Time\n## Convergence Guarantees for Gradient Descent on Convex Functions\n## Gradient Descent on Nonconvex Functions: Escaping Saddle Points Efficiently\n## Variational, Hamiltonian, and Symplectic Perspectives on Acceleration\n## Variational Inequalities: From Minima to Nash Equilibria and Fixed Points","[{\"question\":\"What motivates moving beyond pattern recognition in machine learning?\",\"answer\":\"The notes argue that broader settings such as decision-making and multi-agent problems require formulating behavior in terms of tradeoffs and equilibria rather than only optima.\"},{\"question\":\"How do variational inequalities relate to Nash equilibria and fixed points?\",\"answer\":\"Variational inequalities provide a framework connecting minima to Nash equilibria and fixed points, including specializations to two-player zero-sum games and Nash equilibrium computation.\"},{\"question\":\"What algorithms are highlighted for computing equilibria?\",\"answer\":\"The notes emphasize monotone operators and fixed-point finding algorithms, including the proximal point method and the extragradient algorithm, along with continuous-time limits.\"}]","A Gentle Introduction to Gradient-Based Optimization and Variational Inequalities for Machine Learning | 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motivates moving beyond pattern recognition in machine learning?","Question",{"text":76,"@type":77},"The notes argue that broader settings such as decision-making and multi-agent problems require formulating behavior in terms of tradeoffs and equilibria rather than only optima.","Answer",{"name":79,"@type":74,"acceptedAnswer":80},"How do variational inequalities relate to Nash equilibria and fixed points?",{"text":81,"@type":77},"Variational inequalities provide a framework connecting minima to Nash equilibria and fixed points, including specializations to two-player zero-sum games and Nash equilibrium computation.",{"name":83,"@type":74,"acceptedAnswer":84},"What algorithms are highlighted for computing equilibria?",{"text":85,"@type":77},"The notes emphasize monotone operators and fixed-point finding algorithms, including the proximal point method and the extragradient algorithm, along with continuous-time 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