大数据分析中的算法 (2019年春季)

• 本课程考核包括平时作业和程序，期中大项目，期末大项目，请谨慎选课

• 课程代码：00136720 (本科生），00100863 （本研合）

• 课程内容： 侧重数据分析中的数值代数和最优化算法

• 教师信息： 文再文，wenzw at pku dot edu dot cn

• 助教信息： 刘浩洋: liuhaoyang at pku dot edu dot cn ， 柳昊明: haomingliu at pku dot edu dot cn

• 课程微信群（下面二维码2月24日失效）

• 上课地点：理教103

• 上课时间：

  • 1~16周 每周周二10~11节, 18:40pm-20:30pm；单周周一1~2节，08:00am-9:50am

  • 老师答疑时间: 单独和老师联系或者每次课后

  • 助教答疑时间: 根据反馈待定

  • 北京大学校历

• 先修课程要求:

  • 年级要求: 大三，大四，研究生

  • 不是必须先修但有帮助: 数值代数，最优化（凸优化），概率论

  • 会Matlab或者Python程序编写

• 其它信息：

  • 大数据分析中的算法 (2018年春季)

  • 大数据分析中的算法 (2017年春季)

  • 大数据分析中的算法 (2016年春季)

  • 大数据分析中的算法 (2015年春季)

  • 凸优化 (2018年秋季)

  • 凸优化 (2017年秋季)

  • 凸优化 (2016年秋季)

  • 凸优化 (2015年秋季)

  • 凸优化 (2014年秋季)

• 参考材料：

  • “Mining of Massive Datasets”, Jure Leskovec, Anand Rajaraman, Jeff Ullman, Cambridge University Press

  • “Convex optimization”, Stephen Boyd and Lieven Vandenberghe

  • “Numerical Optimization”, Jorge Nocedal and Stephen Wright, Springer

  • “Optimization Theory and Methods”, Wenyu Sun, Ya-Xiang Yuan

  • “Matrix Computations”, Gene H. Golub and Charles F. Van Loan, The Johns Hopkins University Press

  • The Matrix Cookbook, Kaare Brandt Petersen, Michael Syskind Pedersen
    

  • Deep Learning, Ian Goodfellow and Yoshua Bengio and Aaron Courville

  • Theories of Deep Learning (STATS 385), Stanford University, Fall 2017

• 数据分析相关的网站：

  • Mining Massive Data Sets, Stanford University

  • Jure Leskovec @ Stanford - Stanford Computer Science

  • Mining Massive Data Sets: Hadoop Labs, Stanford University

  • Big Data Analytics, Columbia University

  • Theoretical Foundations of Big Data Analysis, The Simons Institute for the Theory of Computing, UC Berkeley

  • Introduction to Data Science, University of Washington

  • Core Methods in Data Science, University of Washington

  • Machine Learning and Artificial Intelligence, Sanjeev Arora and Elad Hazan, Princeton

  • Reinforcement Learning, David Silver

  • Deep Reinforcement Learning, Sergey Levine, John Schulman, Chelsea Finn

  • courses from Berkeley Artificial Intelligence Research (BAIR) Lab

  • Large-scale Machine Learning and Optimization, Dimitris Papailiopoulos

• 课程信息 （大致安排）

  • Acknowledgement: quite a few materials are taken from slides or lecture notes online.
    Please email me if the usage of any part is not appropriate and they will be deleted immediately.

  • 第1周，2月18日，课程简介: lecture notes
    

  • 第1周，2月19日，线性规划，二次锥规划，半定规划简介: lecture notes
    read: “Convex optimization”, Stephen Boyd and Lieven Vandenberghe, chapters 2 and 3
    思考题： 有哪些问题和应用可以化成线性规划，二次锥规划，半定规划？ 模型语言： CVX, YALMIP
    LP, SOCP, SDP典型算法软件: SDPT3, MOSEK, CPLEX, GUROBI
    Prof. Boyd lecture notes on Disciplined convex programming and CVX
    “Convex optimization”, Stephen Boyd and Lieven Vandenberghe, chapters 4 and 5
    

  • 第2周，2月26日，凸优化对偶理论: lecture notes
    

  • 第3周，3月4日，线性规划单纯形算法，内点算法: lecture notes
    “Numerical Optimization”, Jorge Nocedal and Stephen Wright, Springer, chapters 13 and 14

  • 第3周，3月5日，压缩感知和稀疏优化基本理论: lecture notes
    references: An Introduction to Compressive Sensing
    E. Candes and T. Tao. Decoding by linear programming. IEEE Transactions on Information Theory, 51:4203–4215, 2005
    Compressive Sensing Resources
    

  • 第4周，3月12日，压缩感知和稀疏优化基本理论， 稀疏优化的算法

  • 第5周，3月18日，稀疏优化的算法 lecture notes
    “Proximal Algorithms”, N. Parikh and S. Boyd, Foundations and Trends in Optimization, 1(3):123-231, 2014.
    Amir Beck, Marc Teboulle, A Fast Iterative Shrinkage-Thresholding Algorithm for Linear Inverse Problems
    “Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers”, S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, Foundations and Trends in Machine Learning, 3(1):1–122, 2011
    Junfeng Yang, Yin Zhang, Alternating direction algorithms for l1-problems in Compressed Sensing, SIAM Journal on Scientific Computing

  • 第5周，3月19日， 推荐系统与低秩矩阵恢复的算法

  • 第6周，3月26日， 推荐系统与低秩矩阵恢复的算法 lecture notes
    SVD: read section 2.5 in “Matrix Computations”, Gene H. Golub and Charles F. Van Loan, The Johns Hopkins University Press
    chapter 9 in “Mining Massive Data Sets”, Stanford University
    “Guaranteed Minimum-Rank Solutions of Linear Matrix Equations via Nuclear Norm Minimization”, Benjamin Recht, Maryam Fazel, Pablo A. Parrilo

  • 第7周，4月1日， optimal transport, lecture notes
    slides and computational resources on optimal transport

  • 第7周，4月2日， optimal transport

  • 第8周，4月9日， 随机优化算法 lecture notes
    Joel A Tropp, An introduction to matrix concentration inequalities
    Optimization Methods for Large-Scale Machine Learning, Léon Bottou, Frank E. Curtis, Jorge Nocedal
    Introductory Lectures on Stochastic Optimization, by John Duchi
    Deep Learning, Ian Goodfellow and Yoshua Bengio and Aaron Courville
    

  • 第9周，4月15日， 随机优化算法，随机特征值分解算法

  • 第9周，4月16日， 随机特征值分解算法 lecture notes
    Petros Drineas, Ravi Kannan, and Michael W. Mahoney, Fast Monte Carlo Algorithms for Matrices I: Approximating Matrix Multiplication, SIAM J. Comput., 36(1), 132–157
    Petros Drineas, Ravi Kannan, and Michael W. Mahoney, Fast Monte Carlo Algorithms for Matrices II: Computing a Low-Rank Approximation to a Matrix, SIAM J. Comput., 36(1), 158–183
    N. Halko, P. G. Martinsson, and J. A. Tropp, Finding Structure with Randomness: Probabilistic Algorithms for Constructing Approximate Matrix Decompositions,SIAM Rev., 53(2), 217–288.
    

  • 第10周，4月23日，相位恢复以及低温电子显微镜 lecture notes
    E. J. Candes, Y. Eldar, T. Strohmer and V. Voroninski. Phase retrieval via matrix completion. SIAM J. on Imaging Sciences 6(1), 199–225.
    E. J. Candès, X. Li and M. Soltanolkotabi. Phase retrieval via Wirtinger flow: theory and algorithms. IEEE Transactions on Information Theory 61(4), 1985–2007.
    A. Singer, Y. Shkolnisky, Three-Dimensional Structure Determination from Common Lines in Cryo-EM by Eigenvectors and Semidefinite Programming, SIAM Journal on Imaging Sciences, 4 (2), pp. 543-572 (2011).
    L. Wang, A. Singer, Z. Wen, ‘‘Orientation Determination from Cryo-EM images using Least Unsquared Deviations”, SIAM Journal on Imaging Sciences, 6(4), pp. 2450–2483 (2013).

  • 第11周，4月29日，4月30日，放假取消
    

  • 第12周，5月7日，高维数据降维 lecture notes
    网络流问题选讲 lecture notes
    A Global Geometric Framework for Nonlinear Dimensionality Reduction, Joshua B. Tenenbaum, Vin de Silva, John C. Langford
    Unsupervised Learning of Image Manifolds by Semidefinite Programming, KILIAN Q. WEINBERGER AND LAWRENCE K. SAUL
    A Duality View of Spectral Methods for Dimensionality Reduction, Lin Xiao, Jun Sun, Stephen Boyd
    Dimensionality Reduction: A Comparative Review, Laurens van der Maaten, Eric Postma, Jaap van den Herik
    support vector machine lecture notes
    LIBLINEAR: A Library for Large Linear Classification
    LIBSVM: A library for support vector machines
    Working set selection using the second order information for training SVM R.-E. Fan, P.-H. Chen, and C.-J. Lin.
    An Introduction to Support Vector Machines and Other Kernel-Based Learning Methods, CRISTIANINI, Nello, and John SHAWE-TAYLOR, Cambridge, UK: Cambridge University Press
    图和网络流问题: 最短路径问题 lecture notes
    “Network Flows: Theory, Algorithms, and Applications”, Ahuja, Magnanti, and Orlin
    图和网络流问题: 最大流问题，组合优化与线性规划 lecture notes
    “Network Flows: Theory, Algorithms, and Applications”, Ahuja, Magnanti, and Orlin
    链接分析： page rank lecture notes
    chapter 5, link analysis, Mining of Massive Datasets

  • 第13周，5月13日，高维数据降维，Submodular 优化与数据挖掘

  • 第13周，5月14日，Submodular 优化与数据挖掘 lecture notes
    Learning with Submodular Functions: A Convex Optimization Perspective, Francis Bach
    Learning with Submodular Functions, Francis Bach
    EE596B - Submodular Functions, Optimization, and Applications to Machine Learning, Prof. Jeff A. Bilmes

  • 第14周，5月21日，dynamic programming 1 lecture notes
    Richard S. Sutton and Andrew G. Barto, Reinforcement Learning: An Introduction
    Prof. Dimitri Bertsekas's lecture slides
    Dimitri P. Bertsekas, Abstract Dynamic Programming

  • 第15周，5月27日，dynamic programming 2

  • 第15周，5月28日，dynamic programming and Reinforcement Learning lecture notes
    

  • 第16周，6月4日，Reinforcement Learning lecture notes
    

• 作业和课程项目，重要日期和提交内容：

  • 3月4日上课前, 作业一: Convex Optimization 习题： 4.11, 4.12, 4.25, 4.27, Convex Optimization, Additional exercises，习题： 3.11, 3.12

  • 3月5日前，课程项目二提交分组和选题信息， 具体课题在中期报告提交前还可以修改,

  • 3月18日上课前, 作业二: Convex Optimization 习题： 5.11, 5.17, 5.19, “Numerical Optimization”, Jorge Nocedal and Stephen Wright, Springer，习题： 13.9, 14.6,

  • 4月1日上课前，作业三文件

  • 4月16日晚12点前, 课程项目一
    demo: sparse_l1_example.m

  • 4月23日晚12点, (可以推迟到5月7日晚12点提交) homework-sto.pdf

  • 5月7日晚12点, 提交项目二中期报告， 请点击此链接上传
    中期报告不超过3页纸（单面）。需要描述你们组的选题，已经完成的工作，你们将要做哪些事情
    提交的文件名为 “proj2rep1-name1-name2.pdf”. 比如如果成员为Pengyu Qian 和 Junzi Zhang，则文件名为 proj2rep1-PengyuQian-JunziZhang.pdf

  • 5月13日晚12点前, homework-svd.pdf

  • 课程项目一没有达到要求的同学补测：5月14日晚12点前, project1w.pdf

  • 5月21日晚12点前, homework-phase.pdf
    

  • 5月28日上课前， homework7-2018.pdf

  • 6月14日晚12点前，项目二书面报告 (包括latex源文件，程序等等打包， 请点击此链接上传
    提交的文件请全部打包，文件名为 “proj2-name1-name2.zip”

• 课程项目

  • 课程项目二: 请见下面课程项目二细节

• 成绩评定
  

  • 迟交一天(24小时）打折10%， 不接受晚交4天的作业和项目（任何时候理由都不接受）

  • 大作业，包括习题和程序： 40%

  • 课程项目一： 30%

  • 课程项目二： 30%

  • 作业要求：i) 计算题要求写出必要的推算步骤，证明题要写出关键推理和论证。数值试验题应该同时提交书面报告和程序，其中书面报告有详细的推导和数值结果及分析。 ii) 可以同学间讨论或者找助教答疑，但不允许在讨论中直接抄袭，应该过后自己独立完成。 iii) 严禁从其他学生，从互联网，从往年的答案，其它课程等等任何途径直接抄袭。iv) 如果有讨论或从其它任何途径取得帮助，请列出来源。

课程项目二细节

• 分组: 自由组合一般两人一组，可以一个人。如果是两人组队，期末项目报告里请明确说明各自分工

• 选题

  • 大致分为文献综述，模型项目，算法项目，理论问题

  • 题目请从 “课程项目文献” 里选择。 题目也可以自由选择。如果该项目在其它课程里或其他老师组里同时进行，请明确说明。 如果该课题是以往工作的延续，请明确说明。

  • 文献综述：挑选一篇感兴趣的文献，写文献综述

  • 模型项目: 挑选你感兴趣的应用问题，探索用最优化算法或数值代数算法解决。

  • 算法项目: 挑选一个或者一类问题，提出新的算法或者已有算法的变形

  • 理论问题：分析一些模型或算法的理论性质

• 重要日期和提交内容：

  • 3月5日前，提交分组和选题信息， 具体课题在中期报告提交前还可以修改,

  • 5月7日晚12点（不接受晚交报告）, 中期报告请点击此链接上传
    中期报告不超过3页纸（单面）。需要描述你们组的选题，已经完成的工作，你们将要做哪些事情
    提交的文件名为 “proj2rep1-name1-name2.pdf”. 比如如果成员为Pengyu Qian 和 Junzi Zhang，则文件名为 proj2rep2-PengyuQian-JunziZhang.pdf

  • 6月14日晚12点（不接受晚交报告），书面报告 (包括latex源文件，程序等等打包, 请点击此链接上传
    提交的文件请全部打包，文件名为 “proj2-name1-name2.zip”. 比如如果成员为Pengyu Qian 和 Junzi Zhang，则文件名为 proj2-PengyuQian-JunziZhang.zip

挑战项目 (持续更新)

• 实现线性规划单纯形法，能成功求解netlib测试集里所有问题:
  LP test problems from netlib
  成功求解：判断是否有可行解，是否无界，最优性条件(primal and dual infeasibility, duality gap)的违反度是否达到1e-6以下。参考 SDPT3 user guide 里sec. 3.3（第11-12页).
  故事：You have to figure out who your customer is going to be – An interview with Bob Bixby

• 实现线性规划内点法，能成功求解netlib测试集里所有问题（标准如上题）。

• 设计能充分利用GPU并行的线性规划单纯形法或内点法，测试性能是否有显著提升。可以挑一个大规模线性规划例子测试。
  

• 选择一个问题：压缩感知，低秩矩阵恢复，鲁棒主成分分析，相位恢复，社区检测，读懂凸优化模型跟原始问题解的等价性证明

• 选择一个问题：低秩矩阵恢复，鲁棒主成分分析，相位恢复，社区检测，考虑其非凸模型，读懂其局部极小点性质相关问题论文

• 证明ADAM 算法的收敛性性质

• 选择一个线性代数问题，设计随机算法

• 选择一个机器学习模型或算法，建立半定规划模型，分析理论性质

课程项目文献

• Programming related

  • Implement some optimization algorithms based on the Julia Language

  • Solve one practical application based on Tensorflow or Pytorch

  • project on medical diagnosis using deep neural network

• Optimization theory and algorithms

  • papers from NIPS 2018

  • A list of papers maintained by Ju Sun, Provable Nonconvex Methods/Algorithms

  • Marco Mondelli, Andrea Montanari, On the Connection Between Learning Two-Layers Neural Networks and Tensor Decomposition

  • Shiyu Liang, Ruoyu Sun, Yixuan Li, R. Srikant, Understanding the Loss Surface of Neural Networks for Binary Classification

  • Haoyu Fu, Yuejie Chi, Yingbin Liang, Local Geometry of One-Hidden-Layer Neural Networks for Logistic Regression

  • Chulhee Yun, Suvrit Sra, Ali Jadbabaie, A Critical View of Global Optimality in Deep Learning

  • Itay Safran, Ohad Shamir, Spurious Local Minima are Common in Two-Layer ReLU Neural Networks

  • Thomas Laurent, James von Brecht, The Multilinear Structure of ReLU Networks

  • Digvijay Boob, Guanghui Lan, Theoretical properties of the global optimizer of two layer neural network

  • Yi Zhou, Yingbin Liang, Critical Points of Neural Networks: Analytical Forms and Landscape Properties

  • M. Wang. Primal-Dual Pi Learning: Sample Complexity and Sublinear Run Time for Ergodic Markov Decision Problems

  • Y. Chen and M. Wang. Lower Bound On the Computational Complexity of Discounted Markov Decision Problems

  • M. Wang. Randomized Linear Programming Solves the Discounted Markov Decision Problem In Nearly-Linear Running Time

  • Ian Post, Yinyu Ye, The simplex method is strongly polynomial for deterministic Markov decision processes

  • Thomas Dueholm Hansen, Peter Bro Miltersen, Uri Zwick, Strategy iteration is strongly polynomial for 2-player turn-based stochastic games with a constant discount factor

  • Chien-Chih Wang, Kent Loong Tan, Chun-Ting Chen, Yu-Hsiang Lin, S. Sathiya Keerthi, Dhruv Mahajan, S. Sundararajan, Chih-Jen Lin, Distributed Newton Methods for Deep Neural Networks

  • Minhyung Cho, Jaehyung Lee, Riemannian approach to batch normalization

  • Nilesh Tripuraneni, Nicolas Flammarion, Francis Bach, Michael I. Jordan, Averaging Stochastic Gradient Descent on Riemannian Manifolds

  • Teng Zhang, Yi Yang, Robust PCA by Manifold Optimization

  • Weijie Su, Yuancheng Zhu, Statistical Inference for Online Learning and Stochastic Approximation via Hierarchical Incremental Gradient Descent

  • Cong Ma, Kaizheng Wang, Yuejie Chi, Yuxin Chen, Implicit Regularization in Nonconvex Statistical Estimation: Gradient Descent Converges Linearly for Phase Retrieval, Matrix Completion and Blind Deconvolution

  • Jason D. Lee, Ioannis Panageas, Georgios Piliouras, Max Simchowitz, Michael I. Jordan, Benjamin Recht, First-order Methods Almost Always Avoid Saddle Points

  • Ji Chen, Xiaodong Li, Memory-efficient Kernel PCA via Partial Matrix Sampling and Nonconvex Optimization: a Model-free Analysis of Local Minima

  • Prateek Jain, Purushottam Kar, Non-convex Optimization for Machine Learning

  • Y. Chen and M. Wang. Stochastic Primal-Dual Methods and Sample Complexity of Reinforcement Learning

  • S. Yang, M. Wang, E. Fang. Multi-Level Stochastic Gradient Methods for Nested Composition Optimization

  • Raghu Bollapragada, Dheevatsa Mudigere, Jorge Nocedal, Hao-Jun Michael Shi, Ping Tak Peter Tang, A Progressive Batching L-BFGS Method for Machine Learning

  • Raghu Bollapragada, Richard Byrd, Jorge Nocedal, Adaptive Sampling Strategies for Stochastic Optimization

  • Albert S. Berahas, Raghu Bollapragada, Jorge Nocedal, An Investigation of Newton-Sketch and Subsampled Newton Methods

  • Song Mei, Theodor Misiakiewicz, Andrea Montanari, Roberto I. Oliveira, Solving SDPs for synchronization and MaxCut problems via the Grothendieck inequality

  • Nathan Halko, PerGunnar Martinsson, and Joel Tropp, SIA, Finding structure with randomness: Probabilistic algorithms

  • Rong Ge, Chi Jin, Sham M. Kakade, Praneeth Netrapalli, Aaron Sidford, Efficient Algorithms for Large-scale Generalized Eigenvector Computation and Canonical Correlation Analysis

  • A. Y. Aravkin, J. V. Burke, D. Drusvyatskiy, M. P. Friedlander, S. Roy, Level-set methods for convex optimization

  • Ruoyu Sun, Zhi-Quan Luo, Guaranteed Matrix Completion via Non-convex Factorization

  • Rong Ge, Furong Huang, Chi Jin, Yang Yuan, Escaping From Saddle Points  —  Online Stochastic Gradient for Tensor Decomposition

  • Rong Ge, Jason D. Lee, Tengyu Ma, Matrix Completion has No Spurious Local Minimum

  • Rong Ge, Chi Jin, Yi Zheng, No Spurious Local Minima in Nonconvex Low Rank Problems: A Unified Geometric Analysis

  • Afonso S. Bandeira, Christopher Kennedy, Amit Singer, Approximating the Little Grothendieck Problem over the Orthogonal and Unitary Groups

  • A Variational Perspective on Accelerated Methods in Optimization, Andre Wibisono, Ashia C. Wilson, Michael I. Jordan

  • Less than a Single Pass: Stochastically Controlled Stochastic Gradient Method, Lihua Lei, Michael I. Jordan

  • Exact and Inexact Subsampled Newton Methods for Optimization, Raghu Bollapragada, Richard Byrd, Jorge Nocedal

  • Finding Approximate Local Minima for Nonconvex Optimization in Linear Time, Naman Agarwal, Zeyuan Allen-Zhu, Brian Bullins, Elad Hazan, Tengyu Ma

  • Trust Region Policy Optimization, John Schulman, Sergey Levine, Philipp Moritz, Michael I. Jordan, Pieter Abbeel