Generic descent algorithm Generalization to multiple dimensions Problems of descent methods, possible improvements Fixes Local minima
|
|
|
- Allyson Charles
- 6 years ago
- Views:
Transcription
1 1 Lecture 10: descent methods Generic descent algorithm Generalization to multiple dimensions Problems of descent methods, possible improvements Fixes Local minima Gradient descent (reminder) Minimum of a function is found by following the slope of the function f f(x) guess f(m) m x
2 2 Gradient descent (illustration) f f(x) next step guess f(m) m x Gradient descent (illustration) f f(x) guess new gradient next step f(m) m x
3 3 Gradient descent (illustration) f f(x) guess next step f(m) m x Gradient descent (illustration) f f(x) guess f(m) stop m x
4 4 Gradient descent: algorithm guess Gradient descent: algorithm Direction: downhill
5 5 Gradient descent: algorithm step Gradient descent: algorithm Now you are here
6 6 Gradient descent: algorithm Stop when close from minimum Gradient descent: algorithm guess = x direction = -f (x) step = h > 0 x:=x hf (x) f (x)~0
7 7 Example of 2D gradient: pic of the MATLAB demo Illustration of the gradient in 2D Example of 2D gradient: pic of the MATLAB demo Illustration of the gradient in 2D
8 8 Example of 2D gradient: pic of the MATLAB demo Illustration of the gradient in 2D Example of 2D gradient: pic of the MATLAB demo Definition of the gradient in 2D This is just a genaralization of the derivative in two dimensions. This can be generalized to any dimension.
9 9 Example of 2D gradient: pic of the MATLAB demo Illustration of the gradient in 2D Example of 2D gradient: pic of the MATLAB demo Gradient descent works in 2D
10 10 Generalization to multiple dimensions guess Generalization to multiple dimensions Direction: downhill
11 11 Generalization to multiple dimensions step Generalization to multiple dimensions Now you are here
12 12 Generalization to multiple dimensions Stop when close from minimum Generalization to multiple dimensions guess = x direction = -f(x) step = h > 0 x:=x h Vf (x) Vf (x)~0
13 13 Multiple dimensions Everything that you have seen with derivatives can be generalized with the gradient. For the descent method, f (x) can be replaced by In two dimensions, and by in N dimensions. Example of 2D gradient: MATLAB demo The cost to buy a portfolio is: Stock N Stock i Stock 2 Stock 1 If you want to minimize the price to buy your portfolio, you need to compute the gradient of its price:
14 14 Problem 1: choice of the step f When updating the current computation: - small steps: inefficient - large steps: potentially bad results f(x) guess f(m) stop m Too many steps: takes too long to converge x Problem 1: choice of the step f When updating the current computation: - small steps: inefficient - large steps: potentially bad results f(x) Next point (went too far) f(m) m Current point x
15 15 Problem 2: «ping pong effect» [S. Boyd, L. Vandenberghe, Convex Convex Optimization lect. Notes, Stanford Univ ] Problem 2: «ping pong effect» [S. Boyd, L. Vandenberghe, Convex Convex Optimization lect. Notes, Stanford Univ ]
16 16 Problem 2: (other norm dependent issues) [S. Boyd, L. Vandenberghe, Convex Convex Optimization lect. Notes, Stanford Univ ] Problem 3: stopping criterion Intuitive criterion: In multiple dimensions: Or equivalently Rarely used in practice. More about this in EE227A (convex optimization, Prof. L. El Ghaoui).
17 17 Fixes Several methods exist to address this problem - Line search methods, in particular - Backtracking line search - Exact line search - Normalized steepest descent - Newton steps Fundamental problem of the method: local minima Local minima: pic of the MATLAB demo The iterations of the algorithm converge to a local minimum
18 18 Local minima: pic of the MATLAB demo View of the algorithm is «myopic»
Gradient descent (illustration) Gradient descent (illustration) Gradient descent (illustration) Gradient descent (illustration)
1 Lecture 10: descent ethods Gradient descent (reinder) Miniu o a unction is ound by ollowing the slope o the unction Generic descent algorith Generalization to ultiple diensions Probles o descent ethods,
Lecture 12: convergence. Derivative (one variable)
Lecture 12: convergence More about multivariable calculus Descent methods Backtracking line search More about convexity (first and second order) Newton step Example 1: linear programming (one var., one
CPSC 340: Machine Learning and Data Mining. Robust Regression Fall 2015
CPSC 340: Machine Learning and Data Mining Robust Regression Fall 2015 Admin Can you see Assignment 1 grades on UBC connect? Auditors, don t worry about it. You should already be working on Assignment
Iterative Methods for Solving Linear Problems
Iterative Methods for Solving Linear Problems When problems become too large (too many data points, too many model parameters), SVD and related approaches become impractical. Iterative Methods for Solving
Lecture 12: Feasible direction methods
Lecture 12 Lecture 12: Feasible direction methods Kin Cheong Sou December 2, 2013 TMA947 Lecture 12 Lecture 12: Feasible direction methods 1 / 1 Feasible-direction methods, I Intro Consider the problem
Gradient Descent. Wed Sept 20th, James McInenrey Adapted from slides by Francisco J. R. Ruiz
Gradient Descent Wed Sept 20th, 2017 James McInenrey Adapted from slides by Francisco J. R. Ruiz Housekeeping A few clarifications of and adjustments to the course schedule: No more breaks at the midpoint
Modern Methods of Data Analysis - WS 07/08
Modern Methods of Data Analysis Lecture XV (04.02.08) Contents: Function Minimization (see E. Lohrmann & V. Blobel) Optimization Problem Set of n independent variables Sometimes in addition some constraints
Chapter Multidimensional Gradient Method
Chapter 09.04 Multidimensional Gradient Method After reading this chapter, you should be able to: 1. Understand how multi-dimensional gradient methods are different from direct search methods. Understand
25. NLP algorithms. ˆ Overview. ˆ Local methods. ˆ Constrained optimization. ˆ Global methods. ˆ Black-box methods.
CS/ECE/ISyE 524 Introduction to Optimization Spring 2017 18 25. NLP algorithms ˆ Overview ˆ Local methods ˆ Constrained optimization ˆ Global methods ˆ Black-box methods ˆ Course wrap-up Laurent Lessard
Today. Golden section, discussion of error Newton s method. Newton s method, steepest descent, conjugate gradient
Optimization Last time Root finding: definition, motivation Algorithms: Bisection, false position, secant, Newton-Raphson Convergence & tradeoffs Example applications of Newton s method Root finding in
Lecture 25 Nonlinear Programming. November 9, 2009
Nonlinear Programming November 9, 2009 Outline Nonlinear Programming Another example of NLP problem What makes these problems complex Scalar Function Unconstrained Problem Local and global optima: definition,
Lecture 4. Convexity Robust cost functions Optimizing non-convex functions. 3B1B Optimization Michaelmas 2017 A. Zisserman
Lecture 4 3B1B Optimization Michaelmas 2017 A. Zisserman Convexity Robust cost functions Optimizing non-convex functions grid search branch and bound simulated annealing evolutionary optimization The Optimization
Introduction to unconstrained optimization - derivative-free methods
Introduction to unconstrained optimization - derivative-free methods Jussi Hakanen Post-doctoral researcher Office: AgC426.3 jussi.hakanen@jyu.fi Learning outcomes To understand the basic principles of
Constrained Optimization COS 323
Constrained Optimization COS 323 Last time Introduction to optimization objective function, variables, [constraints] 1-dimensional methods Golden section, discussion of error Newton s method Multi-dimensional
Linear Regression Optimization
Gradient Descent Linear Regression Optimization Goal: Find w that minimizes f(w) f(w) = Xw y 2 2 Closed form solution exists Gradient Descent is iterative (Intuition: go downhill!) n w * w Scalar objective:
Introduction to optimization methods and line search
Introduction to optimization methods and line search Jussi Hakanen Post-doctoral researcher jussi.hakanen@jyu.fi How to find optimal solutions? Trial and error widely used in practice, not efficient and
INTRODUCTION TO LINEAR AND NONLINEAR PROGRAMMING
INTRODUCTION TO LINEAR AND NONLINEAR PROGRAMMING DAVID G. LUENBERGER Stanford University TT ADDISON-WESLEY PUBLISHING COMPANY Reading, Massachusetts Menlo Park, California London Don Mills, Ontario CONTENTS
16. LECTURE 16. I understand how to find the rate of change in any direction. I understand in what direction the maximum rate of change happens.
6. LETURE 6 Objectives I understand how to find the rate of change in any direction. I understand in what direction the maximum rate of change happens. So far, we ve learned the definition of the gradient
Numerical Methods Lecture 7 - Optimization
Numerical Methods Lecture 7 - Optimization Topics: numerical optimization - Newton again - Random search - Golden Section Search READING : text pgs. 331-349 Optimization - motivation What? Locating where
Optimization. 1. Optimization. by Prof. Seungchul Lee Industrial AI Lab POSTECH. Table of Contents
Optimization by Prof. Seungchul Lee Industrial AI Lab http://isystems.unist.ac.kr/ POSTECH Table of Contents I. 1. Optimization II. 2. Solving Optimization Problems III. 3. How do we Find x f(x) = 0 IV.
A Brief Overview of Optimization Problems. Steven G. Johnson MIT course , Fall 2008
A Brief Overview of Optimization Problems Steven G. Johnson MIT course 18.335, Fall 2008 Why optimization? In some sense, all engineering design is optimization: choosing design parameters to improve some
Module 1 Lecture Notes 2. Optimization Problem and Model Formulation
Optimization Methods: Introduction and Basic concepts 1 Module 1 Lecture Notes 2 Optimization Problem and Model Formulation Introduction In the previous lecture we studied the evolution of optimization
61A Lecture 6. Friday, September 7
61A Lecture 6 Friday, September 7 Lambda Expressions >>> ten = 10 An expression: this one evaluates to a number >>> square = x * x Also an expression: evaluates to a function >>> square = lambda x: x *
AM 221: Advanced Optimization Spring 2016
AM 221: Advanced Optimization Spring 2016 Prof Yaron Singer Lecture 3 February 1st 1 Overview In our previous lecture we presented fundamental results from convex analysis and in particular the separating
INF Biologically inspired computing Lecture 1: Marsland chapter 9.1, Optimization and Search Jim Tørresen
INF3490 - Biologically inspired computing Lecture 1: Marsland chapter 9.1, 9.4-9.6 2017 Optimization and Search Jim Tørresen Optimization and Search 2 Optimization and Search Methods (selection) 1. Exhaustive
Contents. I Basics 1. Copyright by SIAM. Unauthorized reproduction of this article is prohibited.
page v Preface xiii I Basics 1 1 Optimization Models 3 1.1 Introduction... 3 1.2 Optimization: An Informal Introduction... 4 1.3 Linear Equations... 7 1.4 Linear Optimization... 10 Exercises... 12 1.5
Optimization and least squares. Prof. Noah Snavely CS1114
Optimization and least squares Prof. Noah Snavely CS1114 http://cs1114.cs.cornell.edu Administrivia A5 Part 1 due tomorrow by 5pm (please sign up for a demo slot) Part 2 will be due in two weeks (4/17)
Machine Learning for Signal Processing Lecture 4: Optimization
Machine Learning for Signal Processing Lecture 4: Optimization 13 Sep 2015 Instructor: Bhiksha Raj (slides largely by Najim Dehak, JHU) 11-755/18-797 1 Index 1. The problem of optimization 2. Direct optimization
Visualizing Images. Lecture 2: Intensity Surfaces and Gradients. Images as Surfaces. Bridging the Gap. Examples. Examples
Visualizing Images Recall two ways of visualizing an image Lecture : Intensity Surfaces and Gradients Intensity pattern d array of numbers We see it at this level Computer works at this level Bridging
6. Linear Discriminant Functions
6. Linear Discriminant Functions Linear Discriminant Functions Assumption: we know the proper forms for the discriminant functions, and use the samples to estimate the values of parameters of the classifier
Convex Optimization CMU-10725
Convex Optimization CMU-10725 Conjugate Direction Methods Barnabás Póczos & Ryan Tibshirani Conjugate Direction Methods 2 Books to Read David G. Luenberger, Yinyu Ye: Linear and Nonlinear Programming Nesterov:
Solution Methods Numerical Algorithms
Solution Methods Numerical Algorithms Evelien van der Hurk DTU Managment Engineering Class Exercises From Last Time 2 DTU Management Engineering 42111: Static and Dynamic Optimization (6) 09/10/2017 Class
Optimization. Industrial AI Lab.
Optimization Industrial AI Lab. Optimization An important tool in 1) Engineering problem solving and 2) Decision science People optimize Nature optimizes 2 Optimization People optimize (source: http://nautil.us/blog/to-save-drowning-people-ask-yourself-what-would-light-do)
Today. Gradient descent for minimization of functions of real variables. Multi-dimensional scaling. Self-organizing maps
Today Gradient descent for minimization of functions of real variables. Multi-dimensional scaling Self-organizing maps Gradient Descent Derivatives Consider function f(x) : R R. The derivative w.r.t. x
Linear Regression & Gradient Descent
Linear Regression & Gradient Descent These slides were assembled by Byron Boots, with grateful acknowledgement to Eric Eaton and the many others who made their course materials freely available online.
Computer Vision Group Prof. Daniel Cremers. 4a. Inference in Graphical Models
Group Prof. Daniel Cremers 4a. Inference in Graphical Models Inference on a Chain (Rep.) The first values of µ α and µ β are: The partition function can be computed at any node: Overall, we have O(NK 2
Multivariate Numerical Optimization
Jianxin Wei March 1, 2013 Outline 1 Graphics for Function of Two Variables 2 Nelder-Mead Simplex Method 3 Steepest Descent Method 4 Newton s Method 5 Quasi-Newton s Method 6 Built-in R Function 7 Linear
Robert Collins CSE486, Penn State. Lecture 2: Intensity Surfaces and Gradients
Lecture 2: Intensity Surfaces and Gradients Visualizing Images Recall two ways of visualizing an image Intensity pattern 2d array of numbers We see it at this level Computer works at this level Bridging
A Brief Overview of Optimization Problems. Steven G. Johnson MIT course , Fall 2008
A Brief Overview of Optimization Problems Steven G. Johnson MIT course 18.335, Fall 2008 Why optimization? In some sense, all engineering design is optimization: choosing design parameters to improve some
Convex Optimization MLSS 2015
Convex Optimization MLSS 2015 Constantine Caramanis The University of Texas at Austin The Optimization Problem minimize : f (x) subject to : x X. The Optimization Problem minimize : f (x) subject to :
Introduction to Modern Control Systems
Introduction to Modern Control Systems Convex Optimization, Duality and Linear Matrix Inequalities Kostas Margellos University of Oxford AIMS CDT 2016-17 Introduction to Modern Control Systems November
Machine Learning Basics: Stochastic Gradient Descent. Sargur N. Srihari
Machine Learning Basics: Stochastic Gradient Descent Sargur N. srihari@cedar.buffalo.edu 1 Topics 1. Learning Algorithms 2. Capacity, Overfitting and Underfitting 3. Hyperparameters and Validation Sets
Gradient Descent. Michail Michailidis & Patrick Maiden
Gradient Descent Michail Michailidis & Patrick Maiden Outline Mo4va4on Gradient Descent Algorithm Issues & Alterna4ves Stochas4c Gradient Descent Parallel Gradient Descent HOGWILD! Mo4va4on It is good
Lecture 18: March 23
0-725/36-725: Convex Optimization Spring 205 Lecturer: Ryan Tibshirani Lecture 8: March 23 Scribes: James Duyck Note: LaTeX template courtesy of UC Berkeley EECS dept. Disclaimer: These notes have not
Image Registration Lecture 4: First Examples
Image Registration Lecture 4: First Examples Prof. Charlene Tsai Outline Example Intensity-based registration SSD error function Image mapping Function minimization: Gradient descent Derivative calculation
Optimization for Machine Learning
Optimization for Machine Learning (Problems; Algorithms - C) SUVRIT SRA Massachusetts Institute of Technology PKU Summer School on Data Science (July 2017) Course materials http://suvrit.de/teaching.html
EE/AA 578: Convex Optimization
EE/AA 578: Convex Optimization Instructor: Maryam Fazel University of Washington Fall 2016 1. Introduction EE/AA 578, Univ of Washington, Fall 2016 course logistics mathematical optimization least-squares;
Proximal operator and methods
Proximal operator and methods Master 2 Data Science, Univ. Paris Saclay Robert M. Gower Optimization Sum of Terms A Datum Function Finite Sum Training Problem The Training Problem Convergence GD I Theorem
15. Cutting plane and ellipsoid methods
EE 546, Univ of Washington, Spring 2012 15. Cutting plane and ellipsoid methods localization methods cutting-plane oracle examples of cutting plane methods ellipsoid method convergence proof inequality
Efficient Optimization for L -problems using Pseudoconvexity
Efficient Optimization for L -problems using Pseudoconvexity Carl Olsson calle@maths.lth.se Anders P. Eriksson anderspe@maths.lth.se Centre for Mathematical Sciences Lund University, Sweden Fredrik Kahl
Machine Learning Classifiers and Boosting
Machine Learning Classifiers and Boosting Reading Ch 18.6-18.12, 20.1-20.3.2 Outline Different types of learning problems Different types of learning algorithms Supervised learning Decision trees Naïve
Machine Learning / Jan 27, 2010
Revisiting Logistic Regression & Naïve Bayes Aarti Singh Machine Learning 10-701/15-781 Jan 27, 2010 Generative and Discriminative Classifiers Training classifiers involves learning a mapping f: X -> Y,
Convex Optimization. Lijun Zhang Modification of
Convex Optimization Lijun Zhang zlj@nju.edu.cn http://cs.nju.edu.cn/zlj Modification of http://stanford.edu/~boyd/cvxbook/bv_cvxslides.pdf Outline Introduction Convex Sets & Functions Convex Optimization
Adaptive Filters Algorithms (Part 2)
Adaptive Filters Algorithms (Part 2) Gerhard Schmidt Christian-Albrechts-Universität zu Kiel Faculty of Engineering Electrical Engineering and Information Technology Digital Signal Processing and System
Lecture 15: Log Barrier Method
10-725/36-725: Convex Optimization Spring 2015 Lecturer: Ryan Tibshirani Lecture 15: Log Barrier Method Scribes: Pradeep Dasigi, Mohammad Gowayyed Note: LaTeX template courtesy of UC Berkeley EECS dept.
7 OPTIMIZATION 46. Contributing to savings versus achieving enjoyment from purchases made now;
7 OPTIMIZATION 46 7 OPTIMIZATION The engineer is continually faced with non-trivial decisions, and discerning the best among alternatives is one of the most useful and general tasks that one can master.
Convex Optimization. Erick Delage, and Ashutosh Saxena. October 20, (a) (b) (c)
Convex Optimization (for CS229) Erick Delage, and Ashutosh Saxena October 20, 2006 1 Convex Sets Definition: A set G R n is convex if every pair of point (x, y) G, the segment beteen x and y is in A. More
Computational Optimization. Constrained Optimization Algorithms
Computational Optimization Constrained Optimization Algorithms Same basic algorithms Repeat Determine descent direction Determine step size Take a step Until Optimal But now must consider feasibility,
A Brief Look at Optimization
A Brief Look at Optimization CSC 412/2506 Tutorial David Madras January 18, 2018 Slides adapted from last year s version Overview Introduction Classes of optimization problems Linear programming Steepest
A Comparative Study of Frequency-domain Finite Element Updating Approaches Using Different Optimization Procedures
A Comparative Study of Frequency-domain Finite Element Updating Approaches Using Different Optimization Procedures Xinjun DONG 1, Yang WANG 1* 1 School of Civil and Environmental Engineering, Georgia Institute
Lecture 6 - Multivariate numerical optimization
Lecture 6 - Multivariate numerical optimization Björn Andersson (w/ Jianxin Wei) Department of Statistics, Uppsala University February 13, 2014 1 / 36 Table of Contents 1 Plotting functions of two variables
Lecture 17: Neural Networks and Deep Learning. Instructor: Saravanan Thirumuruganathan
Lecture 17: Neural Networks and Deep Learning Instructor: Saravanan Thirumuruganathan Outline Perceptron Neural Networks Deep Learning Convolutional Neural Networks Recurrent Neural Networks Auto Encoders
Lecture P6: Recursion
Overview Lecture P6: Recursion What is recursion? When one function calls ITSELF directly or indirectly. Why learn recursion? Powerful programming tool to solve a problem by breaking it up into one (or
Ellipsoid Algorithm :Algorithms in the Real World. Ellipsoid Algorithm. Reduction from general case
Ellipsoid Algorithm 15-853:Algorithms in the Real World Linear and Integer Programming II Ellipsoid algorithm Interior point methods First polynomial-time algorithm for linear programming (Khachian 79)
David G. Luenberger Yinyu Ye. Linear and Nonlinear. Programming. Fourth Edition. ö Springer
David G. Luenberger Yinyu Ye Linear and Nonlinear Programming Fourth Edition ö Springer Contents 1 Introduction 1 1.1 Optimization 1 1.2 Types of Problems 2 1.3 Size of Problems 5 1.4 Iterative Algorithms
REGRESSION ANALYSIS : LINEAR BY MAUAJAMA FIRDAUS & TULIKA SAHA
REGRESSION ANALYSIS : LINEAR BY MAUAJAMA FIRDAUS & TULIKA SAHA MACHINE LEARNING It is the science of getting computer to learn without being explicitly programmed. Machine learning is an area of artificial
Lec13p1, ORF363/COS323
Lec13 Page 1 Lec13p1, ORF363/COS323 This lecture: Semidefinite programming (SDP) Definition and basic properties Review of positive semidefinite matrices SDP duality SDP relaxations for nonconvex optimization
Convex Optimization / Homework 2, due Oct 3
Convex Optimization 0-725/36-725 Homework 2, due Oct 3 Instructions: You must complete Problems 3 and either Problem 4 or Problem 5 (your choice between the two) When you submit the homework, upload a
Perceptron: This is convolution!
Perceptron: This is convolution! v v v Shared weights v Filter = local perceptron. Also called kernel. By pooling responses at different locations, we gain robustness to the exact spatial location of image
Symbolic Subdifferentiation in Python
Symbolic Subdifferentiation in Python Maurizio Caló and Jaehyun Park EE 364B Project Final Report Stanford University, Spring 2010-11 June 2, 2011 1 Introduction 1.1 Subgradient-PY We implemented a Python
Programming, numerics and optimization
Programming, numerics and optimization Lecture C-4: Constrained optimization Łukasz Jankowski ljank@ippt.pan.pl Institute of Fundamental Technological Research Room 4.32, Phone +22.8261281 ext. 428 June
Introduction to Optimization
Introduction to Optimization Second Order Optimization Methods Marc Toussaint U Stuttgart Planned Outline Gradient-based optimization (1st order methods) plain grad., steepest descent, conjugate grad.,
IE521 Convex Optimization Introduction
IE521 Convex Optimization Introduction Instructor: Niao He Jan 18, 2017 1 About Me Assistant Professor, UIUC, 2016 Ph.D. in Operations Research, M.S. in Computational Sci. & Eng. Georgia Tech, 2010 2015
COMPUTATIONAL INTELLIGENCE (INTRODUCTION TO MACHINE LEARNING) SS18. Lecture 2: Linear Regression Gradient Descent Non-linear basis functions
COMPUTATIONAL INTELLIGENCE (INTRODUCTION TO MACHINE LEARNING) SS18 Lecture 2: Linear Regression Gradient Descent Non-linear basis functions LINEAR REGRESSION MOTIVATION Why Linear Regression? Simplest
Characterizing Improving Directions Unconstrained Optimization
Final Review IE417 In the Beginning... In the beginning, Weierstrass's theorem said that a continuous function achieves a minimum on a compact set. Using this, we showed that for a convex set S and y not
Iterative Algorithms I: Elementary Iterative Methods and the Conjugate Gradient Algorithms
Iterative Algorithms I: Elementary Iterative Methods and the Conjugate Gradient Algorithms By:- Nitin Kamra Indian Institute of Technology, Delhi Advisor:- Prof. Ulrich Reude 1. Introduction to Linear
Introduction. Optimization
Introduction to Optimization Amy Langville SAMSI Undergraduate Workshop N.C. State University SAMSI 6/1/05 GOAL: minimize f(x 1, x 2, x 3, x 4, x 5 ) = x 2 1.5x 2x 3 + x 4 /x 5 PRIZE: $1 million # of independent
Outline for today s lecture. Informed Search. Informed Search II. Review: Properties of greedy best-first search. Review: Greedy best-first search:
Outline for today s lecture Informed Search II Informed Search Optimal informed search: A* (AIMA 3.5.2) Creating good heuristic functions Hill Climbing 2 Review: Greedy best-first search: f(n): estimated
HST.582J / 6.555J / J Biomedical Signal and Image Processing Spring 2007
MIT OpenCourseWare http://ocw.mit.edu HST.582J / 6.555J / 16.456J Biomedical Signal and Image Processing Spring 2007 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
DS Machine Learning and Data Mining I. Alina Oprea Associate Professor, CCIS Northeastern University
DS 4400 Machine Learning and Data Mining I Alina Oprea Associate Professor, CCIS Northeastern University September 20 2018 Review Solution for multiple linear regression can be computed in closed form
Energy Minimization -Non-Derivative Methods -First Derivative Methods. Background Image Courtesy: 3dciencia.com visual life sciences
Energy Minimization -Non-Derivative Methods -First Derivative Methods Background Image Courtesy: 3dciencia.com visual life sciences Introduction Contents Criteria to start minimization Energy Minimization
Beyond Classical Search: Local Search. CMPSCI 383 September 23, 2011
Beyond Classical Search: Local Search CMPSCI 383 September 23, 2011 1 Today s lecture Local Search Hill-climbing Simulated annealing Local beam search Genetic algorithms Genetic programming Local search
OPPA European Social Fund Prague & EU: We invest in your future.
OPPA European Social Fund Prague & EU: We invest in your future. Patch tracking based on comparing its piels 1 Tomáš Svoboda, svoboda@cmp.felk.cvut.cz Czech Technical University in Prague, Center for Machine
2. Linear Regression and Gradient Descent
Pattern Recognition And Machine Learning - EPFL - Fall 2015 Emtiyaz Khan, Timur Bagautdinov, Carlos Becker, Ilija Bogunovic & Ksenia Konyushkova 2. Linear Regression and Gradient Descent 2.1 Goals The
CPSC 340: Machine Learning and Data Mining. Logistic Regression Fall 2016
CPSC 340: Machine Learning and Data Mining Logistic Regression Fall 2016 Admin Assignment 1: Marks visible on UBC Connect. Assignment 2: Solution posted after class. Assignment 3: Due Wednesday (at any
ITERATION AND RECURSION 3
ITERATION AND RECURSION 3 COMPUTER SCIENCE 61A June 26, 2012 1 Newton s Method Newton s method is an algorithm that is widely used to compute the zeros of functions. It can be used to approximate a root
Algoritmi di Ottimizzazione: Parte A Aspetti generali e algoritmi classici
Identificazione e Controllo Intelligente Algoritmi di Ottimizzazione: Parte A Aspetti generali e algoritmi classici David Naso A.A. 2006-2007 Identificazione e Controllo Intelligente 1 Search and optimization
Machine Learning Lecture-1
Machine Learning Lecture-1 Programming Club Indian Institute of Technology Kanpur pclubiitk@gmail.com February 25, 2016 Programming Club (IIT Kanpur) ML-1 February 25, 2016 1 / 18 Acknowledgement This
Applying SnapVX to Real-World Problems
Applying SnapVX to Real-World Problems David Hallac Stanford University Goal Other resources (MLOSS Paper, website, documentation,...) describe the math/software side of SnapVX This presentation is meant
The exam is closed book, closed notes except your one-page (two-sided) cheat sheet.
CS 189 Spring 2015 Introduction to Machine Learning Final You have 2 hours 50 minutes for the exam. The exam is closed book, closed notes except your one-page (two-sided) cheat sheet. No calculators or
Department of Mathematics Oleg Burdakov of 30 October Consider the following linear programming problem (LP):
Linköping University Optimization TAOP3(0) Department of Mathematics Examination Oleg Burdakov of 30 October 03 Assignment Consider the following linear programming problem (LP): max z = x + x s.t. x x
Gradient, Newton and conjugate direction methods for unconstrained nonlinear optimization
Gradient, Newton and conjugate direction methods for unconstrained nonlinear optimization Consider the gradient method (steepest descent), with exact unidimensional search, the Newton method and the conjugate
CS 179 Lecture 16. Logistic Regression & Parallel SGD
CS 179 Lecture 16 Logistic Regression & Parallel SGD 1 Outline logistic regression (stochastic) gradient descent parallelizing SGD for neural nets (with emphasis on Google s distributed neural net implementation)
LECTURE NOTES Non-Linear Programming
CEE 6110 David Rosenberg p. 1 Learning Objectives LECTURE NOTES Non-Linear Programming 1. Write out the non-linear model formulation 2. Describe the difficulties of solving a non-linear programming model
Cost Functions in Machine Learning
Cost Functions in Machine Learning Kevin Swingler Motivation Given some data that reflects measurements from the environment We want to build a model that reflects certain statistics about that data Something
Problem Set #3 (With Corrections)
IC-32 Optimal Control Winter 2006 Benoît Chachuat ME C2 401, Ph: 33844, benoit.chachuat@epfl.ch Problem Set #3 (With Corrections) 1. Consider the following NLP problem: min x IR 3 f(x) := x2 1 + x 1 x
And Now to Something Completely Different: Finding Roots of Real Valued Functions
And Now to Something Completely Different: Finding Roots of Real Valued Functions Four other Oysters followed them, And yet another four; And thick and fast they came at last, And more, and more, and more{
Assignment 2 Master Solution. Part (a) /ESD.77 Spring 2004 Multidisciplinary System Design Optimization. (a1) Decomposition
16.888/ESD.77 Spring 004 Multidisciplinary System Design Optimization Assignment Master Solution Part (a) (a1) Decomposition The following system of equations must be solved: xx x + = 0 (1) 1 3 x + 3x
MATH2070: LAB 4: Newton s method
MATH2070: LAB 4: Newton s method 1 Introduction Introduction Exercise 1 Stopping Tests Exercise 2 Failure Exercise 3 Introduction to Newton s Method Exercise 4 Writing Matlab code for functions Exercise
Numerical Optimization
Numerical Optimization Quantitative Macroeconomics Raül Santaeulàlia-Llopis MOVE-UAB and Barcelona GSE Fall 2018 Raül Santaeulàlia-Llopis (MOVE-UAB,BGSE) QM: Numerical Optimization Fall 2018 1 / 46 1 Introduction