[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-134457-en":3,"doc-seo-134457-105":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},134457,1099523882182,"Alex Sinclair","https://ap-avatar.wpscdn.com/davatar_6f874abed73319feea01a86fa6f0fab8",8,"Research & Report","Learning Linear Models Using Distributed Iterative Hessian Sketching","This work studies learning the Markov parameters of a linear time-invariant system from observed trajectories and characterizes the resulting optimization burden when sample complexity grows. Building on non-asymptotic system identification theory, it proposes a randomized, distributed Newton method using Hessian sketching to obtain ε-optimal estimates. The method achieves geometric convergence, is easily parallelized across compute nodes, and applies across a range of sketching matrices. Numerical experiments validate the theoretical guarantees.","Learning Linear Models Using Distributed Iterative Hessian Sketching  \nHan Wang HW 2786@COLUMBIA . EDU  \nColumbia University, New York, NY  \nJames Anderson JAMES . ANDERSON @COLUMBIA . EDU  \nColumbia University, New York, NY  \nEditors: R. Firoozi, N. Mehr, E. Yel, R. Antonova, J. Bohg, M. Schwager, M. Kochenderfer  \nAbstract  \nThis work considers the problem of learning the Markov parameters of a linear system from observed data. Recent non-asymptotic system identification results have characterized the sample complexity of this problem in the single and multi-rollout setting. In both instances, the number of samples required in order to obtain acceptable estimates can produce optimization problems with an intractably large number of decision variables for a second-order algorithm. We show that a randomized and distributed Newton algorithm based on Hessian-sketching can produce 􀀏-optimal solutions and converges geometrically. Moreover, the algorithm is trivially parallelizable. Our results hold for a variety of sketching matrices and we illustrate the theory with numerical examples.  \nKeywords: Distributed optimization; System identification; Sketching; Randomized algorithms  \n1. Introduction  \nObtaining a dynamic model of a system or process is fundamental to most of science and engineering. As the systems we study become increasingly complex, data-driven modeling has become the de facto framework for obtaining accurate models (Brunton and Kutz, 2019) . Fortunately, as systems become more interconnected and sensors become smaller and cheaper, there is no shortage of data to work with. Indeed, the volume of data available can overwhelm the (often limited) computational resources at our disposal, forcing us to consider data versus resource trade-offs (Chandrasekaran and Jordan, 2013) .  \nThere has recently been considerable interest in applying machine learning techniques to the problem of controlling a dynamical system. Two paradigms have emerged; model-based control, in which a model is first learnt from data and then a classical controller is synthesized from the model. In the model-free setting the control action is learnt directly from data without ever constructing an explicit model, see for example Fazel et al. (2018) . Our work is motivated by two observations; i) the asymptotic sample complexity of a model-based solution outperforms that of a model-free Least-Squares Temporal Difference Learning (Boyan, 1999) approach Tu and Recht (2019); ii) the recent body of work characterizing the sample complexity of learning linear system models from data, shows that for systems with a large number of inputs and outputs, the resulting optimization problems are intractable as they require a large number of rollouts or long trajectory horizon lengthsin order to produce accurate estimates Oymak and Ozay (2019); Zheng and Li (2020); Tsiamis and Pappas (2019); Dean et al. (2020) . The goal of this work is to construct and solve approximations of these optimization problems that are consistent with the sample complexity results, provide provably good solutions, and do so in an algorithmically tractable manner. Our approach is based on the  \n􀀍c 2022 H. Wang & J. Anderson.  \nSHORT TITLE  \nconcept of “sketching”(Drineas and Mahoney, 2016) . Broadly speaking, a “sketch” is an approximation of a large matrix by a smaller or more “simple” matrix. For a sketch to be useful, it must retain certain properties of the original matrix that allow it to be used for computation in place of the original. What is perhaps surprising, is that randomization is the enabling force used to construct sketches (Woodruff, 2014 ; Martinsson and Tropp, 2020 ; Mahoney, 2011) . Moreover, numerical linear algebra routines based on randomization (and sketching) can outperform their deterministic counterparts (Avron et al., 2010), and are more suited to distributed computing archtectures.  \n1.1. Problem Setting  \nNotation: Given a matrix A 2 Rm 􀀂 n ; we use kAkF to denot","cbCaiugwuuk4q9Sa","https://ap.wps.com/l/cbCaiugwuuk4q9Sa","pdf",6720632,1,14,"English","en",105,"# Introduction\n## Problem Setting","[{\"question\":\"What problem does the work address in learning linear system models?\",\"answer\":\"It addresses learning the Markov parameters of a linear time-invariant system from observed input-output data, where the required sample size can make second-order optimization intractable.\"},{\"question\":\"How does the proposed method achieve tractable learning?\",\"answer\":\"It uses a randomized, distributed Newton algorithm based on Hessian sketching to build approximations that remain solvable despite large optimization dimensions.\"},{\"question\":\"What convergence and parallelization properties are claimed?\",\"answer\":\"The algorithm is shown to converge geometrically to ε-optimal solutions and is trivially parallelizable for distributed computing.\"}]","Learning Linear Models Using Distributed Iterative Hessian Sketching | 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problem does the work address in learning linear system models?","Question",{"text":76,"@type":77},"It addresses learning the Markov parameters of a linear time-invariant system from observed input-output data, where the required sample size can make second-order optimization intractable.","Answer",{"name":79,"@type":74,"acceptedAnswer":80},"How does the proposed method achieve tractable learning?",{"text":81,"@type":77},"It uses a randomized, distributed Newton algorithm based on Hessian sketching to build approximations that remain solvable despite large optimization dimensions.",{"name":83,"@type":74,"acceptedAnswer":84},"What convergence and parallelization properties are claimed?",{"text":85,"@type":77},"The algorithm is shown to converge geometrically to ε-optimal solutions and is trivially parallelizable for distributed 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