Performance Surfaces and Optimum Points
|
|
- Brent Whitehead
- 5 years ago
- Views:
Transcription
1 CSC Neural Networks Performance Surfaces and Optimum Points 1
2 Entrance Performance learning is another important class of learning law. Network parameters are adjusted to optimize the performance of the network. Investigating the performance surface. Determine conditions for the existence of minima and maxima of the performance surface. 2
3 Performance Learning Several different learning laws that fall under the category of performance learning. During a training process network parameters are adjusted to optimize the performance of a network. Performance index a quantitative measure which measures the network performance. Small when the network performs well. Large when the network performs poorly. 3
4 Optimization Process Two steps: Define a performance index Search the parameter space in order to reduce the performance index. 4
5 Taylor Expansion Lets assume that F(x) is the performance index and x is the scalar parameter we are adjusting. Assume also F(x) is analytic (all of its derivatives exist) Then the Taylor s expansion about the point x * 5
6 Example 6
7 Plot of Approximation All three approximations are accurate if x is close to x * = 0. However, as x moves farther away from x * only the higher order approximations are accurate. 7
8 Vector Case The neural network performance index will not be a functions of a scalar x. It will be a function of all the network parameters (weights and biases) 8
9 Matrix Form The gradient and the Hessian are very important to understand the behaviour of a performance surface. 9
10 Directional Derivatives 10
11 Example The direction p is orthogonal to the gradient vector. 11
12 Plots Contour Plot (a series of curves along which the function value remains constant) The maximum derivative occurs in the direction of the gradient. The zero derivative is in the direction orthogonal to the gradient (tangent to the contour line). 12
13 Minima 13
14 Scalar Example 14
15 Necessary and Sufficient Conditions A necessary condition of a statement must be satisfied for the statement to be true. Formally, a statement P is a necessary condition of a statement Q if Q implies P. A sufficient condition is one that, if satisfied, assures the statement's truth. Formally, a statement P is a sufficient condition of a statement Q if P implies Q. E.g. Being a mammal (P) is necessary but not sufficient to being human (Q) 15
16 Stationary Points First order, necessary (but not sufficient) condition for x* to be a local minimum point. Any point that satisfies the above equation are called stationary points. 16
17 Positive Definite Matrix Positive Definite Matrix A is positive definite matrix if If all eigenvalues are positive, then the matrix is positive definite. Positive Semidefinite Matrix A is positive semidefinite matrix if If all eigenvalues are non-negative, then the matrix is positive semidefinite. 17
18 Necessary and Sufficient Conditions for a Minimum Sufficient condition (for a strong minimum) Hessian matrix must be positive definite. Necessary condition Hessian matrix must be positive semidefinite. E.g x * 0 = x F( x) = = = x 2x 0 2 F ( x) + x 2 F( x) 12x = = x = 0 2 x= 0 Eigen values λ = 0, 2 Hessian is positive semidefinite We cannot guarantee it is a minimum point, but we have not eliminated it as a possible minimum point. 18
19 Another Example 19
20 Quadratic Function h constant vector 20
21 Properties (1) If the eigenvalues of the Hessian matrix are all positive, the function will have a single strong minimum. If the eigenvalues of the Hessian matrix are all negative, the function will have a single strong maximum. If some eigenvalues are positive and others are negative, the function will have a single saddle point. 21
22 Properties (2) If the eigenvalues are all nonnegative, but some eigenvalues are zero, then the function will either have a weak minimum, or will have no stationary point. If the eigenvalues are all nonpositive, but some eigenvalues are zero, then the function will either have a weak maximum, or will have no stationary point. 22
23 Properties (3) If d is nonzero and A is invertible, then the stationary point of the function is x* = -A -1 d 23
Functions of Several Variables
Functions of Several Variables The Unconstrained Minimization Problem where In n dimensions the unconstrained problem is stated as f() x variables. minimize f()x x, is a scalar objective function of vector
More informationIntroduction to gradient descent
6-1: Introduction to gradient descent Prof. J.C. Kao, UCLA Introduction to gradient descent Derivation and intuitions Hessian 6-2: Introduction to gradient descent Prof. J.C. Kao, UCLA Introduction Our
More informationGradient Descent. Dr. Xiaowei Huang
Gradient Descent Dr. Xiaowei Huang https://cgi.csc.liv.ac.uk/~xiaowei/ Up to now, Three machine learning algorithms: decision tree learning k-nn linear regression only optimization objectives are discussed,
More informationLinear Algebra And Its Applications Chapter 6. Positive Definite Matrix
Linear Algebra And Its Applications Chapter 6. Positive Definite Matrix KAIST wit Lab 2012. 07. 10 남성호 Introduction The signs of the eigenvalues can be important. The signs can also be related to the minima,
More informationMATH 5720: Unconstrained Optimization Hung Phan, UMass Lowell September 13, 2018
MATH 57: Unconstrained Optimization Hung Phan, UMass Lowell September 13, 18 1 Global and Local Optima Let a function f : S R be defined on a set S R n Definition 1 (minimizers and maximizers) (i) x S
More informationMath 5BI: Problem Set 6 Gradient dynamical systems
Math 5BI: Problem Set 6 Gradient dynamical systems April 25, 2007 Recall that if f(x) = f(x 1, x 2,..., x n ) is a smooth function of n variables, the gradient of f is the vector field f(x) = ( f)(x 1,
More informationNeural Network Training
Neural Network Training Sargur Srihari Topics in Network Training 0. Neural network parameters Probabilistic problem formulation Specifying the activation and error functions for Regression Binary classification
More informationVasil Khalidov & Miles Hansard. C.M. Bishop s PRML: Chapter 5; Neural Networks
C.M. Bishop s PRML: Chapter 5; Neural Networks Introduction The aim is, as before, to find useful decompositions of the target variable; t(x) = y(x, w) + ɛ(x) (3.7) t(x n ) and x n are the observations,
More informationECE580 Exam 1 October 4, Please do not write on the back of the exam pages. Extra paper is available from the instructor.
ECE580 Exam 1 October 4, 2012 1 Name: Solution Score: /100 You must show ALL of your work for full credit. This exam is closed-book. Calculators may NOT be used. Please leave fractions as fractions, etc.
More informationOptimality Conditions
Chapter 2 Optimality Conditions 2.1 Global and Local Minima for Unconstrained Problems When a minimization problem does not have any constraints, the problem is to find the minimum of the objective function.
More informationLecture 2 - Unconstrained Optimization Definition[Global Minimum and Maximum]Let f : S R be defined on a set S R n. Then
Lecture 2 - Unconstrained Optimization Definition[Global Minimum and Maximum]Let f : S R be defined on a set S R n. Then 1. x S is a global minimum point of f over S if f (x) f (x ) for any x S. 2. x S
More informationHW3 - Due 02/06. Each answer must be mathematically justified. Don t forget your name. 1 2, A = 2 2
HW3 - Due 02/06 Each answer must be mathematically justified Don t forget your name Problem 1 Find a 2 2 matrix B such that B 3 = A, where A = 2 2 If A was diagonal, it would be easy: we would just take
More informationCE 191: Civil and Environmental Engineering Systems Analysis. LEC 05 : Optimality Conditions
CE 191: Civil and Environmental Engineering Systems Analysis LEC : Optimality Conditions Professor Scott Moura Civil & Environmental Engineering University of California, Berkeley Fall 214 Prof. Moura
More informationOptimization Methods
Optimization Methods Decision making Examples: determining which ingredients and in what quantities to add to a mixture being made so that it will meet specifications on its composition allocating available
More informationMath (P)refresher Lecture 8: Unconstrained Optimization
Math (P)refresher Lecture 8: Unconstrained Optimization September 2006 Today s Topics : Quadratic Forms Definiteness of Quadratic Forms Maxima and Minima in R n First Order Conditions Second Order Conditions
More informationECE580 Fall 2015 Solution to Midterm Exam 1 October 23, Please leave fractions as fractions, but simplify them, etc.
ECE580 Fall 2015 Solution to Midterm Exam 1 October 23, 2015 1 Name: Solution Score: /100 This exam is closed-book. You must show ALL of your work for full credit. Please read the questions carefully.
More information1 Overview. 2 A Characterization of Convex Functions. 2.1 First-order Taylor approximation. AM 221: Advanced Optimization Spring 2016
AM 221: Advanced Optimization Spring 2016 Prof. Yaron Singer Lecture 8 February 22nd 1 Overview In the previous lecture we saw characterizations of optimality in linear optimization, and we reviewed the
More informationNumerisches Rechnen. (für Informatiker) M. Grepl P. Esser & G. Welper & L. Zhang. Institut für Geometrie und Praktische Mathematik RWTH Aachen
Numerisches Rechnen (für Informatiker) M. Grepl P. Esser & G. Welper & L. Zhang Institut für Geometrie und Praktische Mathematik RWTH Aachen Wintersemester 2011/12 IGPM, RWTH Aachen Numerisches Rechnen
More informationLecture Unconstrained optimization. In this lecture we will study the unconstrained problem. minimize f(x), (2.1)
Lecture 2 In this lecture we will study the unconstrained problem minimize f(x), (2.1) where x R n. Optimality conditions aim to identify properties that potential minimizers need to satisfy in relation
More information1 Newton s Method. Suppose we want to solve: x R. At x = x, f (x) can be approximated by:
Newton s Method Suppose we want to solve: (P:) min f (x) At x = x, f (x) can be approximated by: n x R. f (x) h(x) := f ( x)+ f ( x) T (x x)+ (x x) t H ( x)(x x), 2 which is the quadratic Taylor expansion
More informationUnconstrained Optimization
1 / 36 Unconstrained Optimization ME598/494 Lecture Max Yi Ren Department of Mechanical Engineering, Arizona State University February 2, 2015 2 / 36 3 / 36 4 / 36 5 / 36 1. preliminaries 1.1 local approximation
More informationTranspose & Dot Product
Transpose & Dot Product Def: The transpose of an m n matrix A is the n m matrix A T whose columns are the rows of A. So: The columns of A T are the rows of A. The rows of A T are the columns of A. Example:
More informationSIMPLE MULTIVARIATE OPTIMIZATION
SIMPLE MULTIVARIATE OPTIMIZATION 1. DEFINITION OF LOCAL MAXIMA AND LOCAL MINIMA 1.1. Functions of variables. Let f(, x ) be defined on a region D in R containing the point (a, b). Then a: f(a, b) is a
More informationTranspose & Dot Product
Transpose & Dot Product Def: The transpose of an m n matrix A is the n m matrix A T whose columns are the rows of A. So: The columns of A T are the rows of A. The rows of A T are the columns of A. Example:
More informationREVIEW OF DIFFERENTIAL CALCULUS
REVIEW OF DIFFERENTIAL CALCULUS DONU ARAPURA 1. Limits and continuity To simplify the statements, we will often stick to two variables, but everything holds with any number of variables. Let f(x, y) be
More informationLecture Notes: Geometric Considerations in Unconstrained Optimization
Lecture Notes: Geometric Considerations in Unconstrained Optimization James T. Allison February 15, 2006 The primary objectives of this lecture on unconstrained optimization are to: Establish connections
More informationNon-Convex Optimization in Machine Learning. Jan Mrkos AIC
Non-Convex Optimization in Machine Learning Jan Mrkos AIC The Plan 1. Introduction 2. Non convexity 3. (Some) optimization approaches 4. Speed and stuff? Neural net universal approximation Theorem (1989):
More informationThe Derivative. Appendix B. B.1 The Derivative of f. Mappings from IR to IR
Appendix B The Derivative B.1 The Derivative of f In this chapter, we give a short summary of the derivative. Specifically, we want to compare/contrast how the derivative appears for functions whose domain
More informationOR MSc Maths Revision Course
OR MSc Maths Revision Course Tom Byrne School of Mathematics University of Edinburgh t.m.byrne@sms.ed.ac.uk 15 September 2017 General Information Today JCMB Lecture Theatre A, 09:30-12:30 Mathematics revision
More informationCHAPTER 2: QUADRATIC PROGRAMMING
CHAPTER 2: QUADRATIC PROGRAMMING Overview Quadratic programming (QP) problems are characterized by objective functions that are quadratic in the design variables, and linear constraints. In this sense,
More informationAn Iterative Descent Method
Conjugate Gradient: An Iterative Descent Method The Plan Review Iterative Descent Conjugate Gradient Review : Iterative Descent Iterative Descent is an unconstrained optimization process x (k+1) = x (k)
More informationAppendix A Taylor Approximations and Definite Matrices
Appendix A Taylor Approximations and Definite Matrices Taylor approximations provide an easy way to approximate a function as a polynomial, using the derivatives of the function. We know, from elementary
More informationSeptember Math Course: First Order Derivative
September Math Course: First Order Derivative Arina Nikandrova Functions Function y = f (x), where x is either be a scalar or a vector of several variables (x,..., x n ), can be thought of as a rule which
More information10-725/36-725: Convex Optimization Prerequisite Topics
10-725/36-725: Convex Optimization Prerequisite Topics February 3, 2015 This is meant to be a brief, informal refresher of some topics that will form building blocks in this course. The content of the
More informationFirst and Second Order Training Algorithms for Artificial Neural Networks to Detect the Cardiac State
First Second Order Training Algorithms for Artificial Neural Networks to Detect the Cardiac State Sanjit K. Dash Department of ECE Raajdhani Engineering College, Bhubaneswar, Odisha, India G. Sasibhushana
More informationMATH2070 Optimisation
MATH2070 Optimisation Nonlinear optimisation with constraints Semester 2, 2012 Lecturer: I.W. Guo Lecture slides courtesy of J.R. Wishart Review The full nonlinear optimisation problem with equality constraints
More informationARE211, Fall 2005 CONTENTS. 5. Characteristics of Functions Surjective, Injective and Bijective functions. 5.2.
ARE211, Fall 2005 LECTURE #18: THU, NOV 3, 2005 PRINT DATE: NOVEMBER 22, 2005 (COMPSTAT2) CONTENTS 5. Characteristics of Functions. 1 5.1. Surjective, Injective and Bijective functions 1 5.2. Homotheticity
More informationA A x i x j i j (i, j) (j, i) Let. Compute the value of for and
7.2 - Quadratic Forms quadratic form on is a function defined on whose value at a vector in can be computed by an expression of the form, where is an symmetric matrix. The matrix R n Q R n x R n Q(x) =
More informationGradient Descent. Sargur Srihari
Gradient Descent Sargur srihari@cedar.buffalo.edu 1 Topics Simple Gradient Descent/Ascent Difficulties with Simple Gradient Descent Line Search Brent s Method Conjugate Gradient Descent Weight vectors
More informationLecture 6 Positive Definite Matrices
Linear Algebra Lecture 6 Positive Definite Matrices Prof. Chun-Hung Liu Dept. of Electrical and Computer Engineering National Chiao Tung University Spring 2017 2017/6/8 Lecture 6: Positive Definite Matrices
More informationNonlinear Optimization for Optimal Control
Nonlinear Optimization for Optimal Control Pieter Abbeel UC Berkeley EECS Many slides and figures adapted from Stephen Boyd [optional] Boyd and Vandenberghe, Convex Optimization, Chapters 9 11 [optional]
More informationChapter 8 Gradient Methods
Chapter 8 Gradient Methods An Introduction to Optimization Spring, 2014 Wei-Ta Chu 1 Introduction Recall that a level set of a function is the set of points satisfying for some constant. Thus, a point
More informationTangent spaces, normals and extrema
Chapter 3 Tangent spaces, normals and extrema If S is a surface in 3-space, with a point a S where S looks smooth, i.e., without any fold or cusp or self-crossing, we can intuitively define the tangent
More informationSuppose that the approximate solutions of Eq. (1) satisfy the condition (3). Then (1) if η = 0 in the algorithm Trust Region, then lim inf.
Maria Cameron 1. Trust Region Methods At every iteration the trust region methods generate a model m k (p), choose a trust region, and solve the constraint optimization problem of finding the minimum of
More informationAdvanced Techniques for Mobile Robotics Least Squares. Wolfram Burgard, Cyrill Stachniss, Kai Arras, Maren Bennewitz
Advanced Techniques for Mobile Robotics Least Squares Wolfram Burgard, Cyrill Stachniss, Kai Arras, Maren Bennewitz Problem Given a system described by a set of n observation functions {f i (x)} i=1:n
More information5 Handling Constraints
5 Handling Constraints Engineering design optimization problems are very rarely unconstrained. Moreover, the constraints that appear in these problems are typically nonlinear. This motivates our interest
More informationPreliminary draft only: please check for final version
ARE211, Fall2012 CALCULUS4: THU, OCT 11, 2012 PRINTED: AUGUST 22, 2012 (LEC# 15) Contents 3. Univariate and Multivariate Differentiation (cont) 1 3.6. Taylor s Theorem (cont) 2 3.7. Applying Taylor theory:
More informationChapter 2 BASIC PRINCIPLES. 2.1 Introduction. 2.2 Gradient Information
Chapter 2 BASIC PRINCIPLES 2.1 Introduction Nonlinear programming is based on a collection of definitions, theorems, and principles that must be clearly understood if the available nonlinear programming
More informationA summary of Deep Learning without Poor Local Minima
A summary of Deep Learning without Poor Local Minima by Kenji Kawaguchi MIT oral presentation at NIPS 2016 Learning Supervised (or Predictive) learning Learn a mapping from inputs x to outputs y, given
More informationThe Error Surface of the XOR Network: The finite stationary Points
The Error Surface of the 2-2- XOR Network: The finite stationary Points Ida G. Sprinkhuizen-Kuyper Egbert J.W. Boers Abstract We investigated the error surface of the XOR problem with a 2-2- network with
More informationGeometry optimization
Geometry optimization Trygve Helgaker Centre for Theoretical and Computational Chemistry Department of Chemistry, University of Oslo, Norway European Summer School in Quantum Chemistry (ESQC) 211 Torre
More informationProduction Possibility Frontier
Division of the Humanities and Social Sciences Production Possibility Frontier KC Border v 20151111::1410 This is a very simple model of the production possibilities of an economy, which was formulated
More informationHow to Escape Saddle Points Efficiently? Praneeth Netrapalli Microsoft Research India
How to Escape Saddle Points Efficiently? Praneeth Netrapalli Microsoft Research India Chi Jin UC Berkeley Michael I. Jordan UC Berkeley Rong Ge Duke Univ. Sham M. Kakade U Washington Nonconvex optimization
More informationDeep Learning. Authors: I. Goodfellow, Y. Bengio, A. Courville. Chapter 4: Numerical Computation. Lecture slides edited by C. Yim. C.
Chapter 4: Numerical Computation Deep Learning Authors: I. Goodfellow, Y. Bengio, A. Courville Lecture slides edited by 1 Chapter 4: Numerical Computation 4.1 Overflow and Underflow 4.2 Poor Conditioning
More informationNonlinearOptimization
1/35 NonlinearOptimization Pavel Kordík Department of Computer Systems Faculty of Information Technology Czech Technical University in Prague Jiří Kašpar, Pavel Tvrdík, 2011 Unconstrained nonlinear optimization,
More informationSupport Vector Machines
Support Vector Machines Le Song Machine Learning I CSE 6740, Fall 2013 Naïve Bayes classifier Still use Bayes decision rule for classification P y x = P x y P y P x But assume p x y = 1 is fully factorized
More informationUnconstrained optimization
Chapter 4 Unconstrained optimization An unconstrained optimization problem takes the form min x Rnf(x) (4.1) for a target functional (also called objective function) f : R n R. In this chapter and throughout
More informationCourse Notes for EE227C (Spring 2018): Convex Optimization and Approximation
Course Notes for EE227C (Spring 2018): Convex Optimization and Approximation Instructor: Moritz Hardt Email: hardt+ee227c@berkeley.edu Graduate Instructor: Max Simchowitz Email: msimchow+ee227c@berkeley.edu
More information8. Diagonalization.
8. Diagonalization 8.1. Matrix Representations of Linear Transformations Matrix of A Linear Operator with Respect to A Basis We know that every linear transformation T: R n R m has an associated standard
More informationLecture 10. Neural networks and optimization. Machine Learning and Data Mining November Nando de Freitas UBC. Nonlinear Supervised Learning
Lecture 0 Neural networks and optimization Machine Learning and Data Mining November 2009 UBC Gradient Searching for a good solution can be interpreted as looking for a minimum of some error (loss) function
More informationParameter Norm Penalties. Sargur N. Srihari
Parameter Norm Penalties Sargur N. srihari@cedar.buffalo.edu 1 Regularization Strategies 1. Parameter Norm Penalties 2. Norm Penalties as Constrained Optimization 3. Regularization and Underconstrained
More informationNONLINEAR. (Hillier & Lieberman Introduction to Operations Research, 8 th edition)
NONLINEAR PROGRAMMING (Hillier & Lieberman Introduction to Operations Research, 8 th edition) Nonlinear Programming g Linear programming has a fundamental role in OR. In linear programming all its functions
More informationNon-convex optimization. Issam Laradji
Non-convex optimization Issam Laradji Strongly Convex Objective function f(x) x Strongly Convex Objective function Assumptions Gradient Lipschitz continuous f(x) Strongly convex x Strongly Convex Objective
More informationLec3p1, ORF363/COS323
Lec3 Page 1 Lec3p1, ORF363/COS323 This lecture: Optimization problems - basic notation and terminology Unconstrained optimization The Fermat-Weber problem Least squares First and second order necessary
More informationConjugate gradient algorithm for training neural networks
. Introduction Recall that in the steepest-descent neural network training algorithm, consecutive line-search directions are orthogonal, such that, where, gwt [ ( + ) ] denotes E[ w( t + ) ], the gradient
More informationOptimization Tutorial 1. Basic Gradient Descent
E0 270 Machine Learning Jan 16, 2015 Optimization Tutorial 1 Basic Gradient Descent Lecture by Harikrishna Narasimhan Note: This tutorial shall assume background in elementary calculus and linear algebra.
More informationLecture 3: Basics of set-constrained and unconstrained optimization
Lecture 3: Basics of set-constrained and unconstrained optimization (Chap 6 from textbook) Xiaoqun Zhang Shanghai Jiao Tong University Last updated: October 9, 2018 Optimization basics Outline Optimization
More informationMultivariate Newton Minimanization
Multivariate Newton Minimanization Optymalizacja syntezy biosurfaktantu Rhamnolipid Rhamnolipids are naturally occuring glycolipid produced commercially by the Pseudomonas aeruginosa species of bacteria.
More informationHere each term has degree 2 (the sum of exponents is 2 for all summands). A quadratic form of three variables looks as
Reading [SB], Ch. 16.1-16.3, p. 375-393 1 Quadratic Forms A quadratic function f : R R has the form f(x) = a x. Generalization of this notion to two variables is the quadratic form Q(x 1, x ) = a 11 x
More informationPractical Optimization: Basic Multidimensional Gradient Methods
Practical Optimization: Basic Multidimensional Gradient Methods László Kozma Lkozma@cis.hut.fi Helsinki University of Technology S-88.4221 Postgraduate Seminar on Signal Processing 22. 10. 2008 Contents
More informationEAD 115. Numerical Solution of Engineering and Scientific Problems. David M. Rocke Department of Applied Science
EAD 115 Numerical Solution of Engineering and Scientific Problems David M. Rocke Department of Applied Science Multidimensional Unconstrained Optimization Suppose we have a function f() of more than one
More informationN. L. P. NONLINEAR PROGRAMMING (NLP) deals with optimization models with at least one nonlinear function. NLP. Optimization. Models of following form:
0.1 N. L. P. Katta G. Murty, IOE 611 Lecture slides Introductory Lecture NONLINEAR PROGRAMMING (NLP) deals with optimization models with at least one nonlinear function. NLP does not include everything
More informationReview of Classical Optimization
Part II Review of Classical Optimization Multidisciplinary Design Optimization of Aircrafts 51 2 Deterministic Methods 2.1 One-Dimensional Unconstrained Minimization 2.1.1 Motivation Most practical optimization
More information, b = 0. (2) 1 2 The eigenvectors of A corresponding to the eigenvalues λ 1 = 1, λ 2 = 3 are
Quadratic forms We consider the quadratic function f : R 2 R defined by f(x) = 2 xt Ax b T x with x = (x, x 2 ) T, () where A R 2 2 is symmetric and b R 2. We will see that, depending on the eigenvalues
More informationQuadratic Programming
Quadratic Programming Outline Linearly constrained minimization Linear equality constraints Linear inequality constraints Quadratic objective function 2 SideBar: Matrix Spaces Four fundamental subspaces
More informationNumerical Optimization
Numerical Optimization Unit 2: Multivariable optimization problems Che-Rung Lee Scribe: February 28, 2011 (UNIT 2) Numerical Optimization February 28, 2011 1 / 17 Partial derivative of a two variable function
More informationLecture 7: CS395T Numerical Optimization for Graphics and AI Trust Region Methods
Lecture 7: CS395T Numerical Optimization for Graphics and AI Trust Region Methods Qixing Huang The University of Texas at Austin huangqx@cs.utexas.edu 1 Disclaimer This note is adapted from Section 4 of
More informationMath 1553, Introduction to Linear Algebra
Learning goals articulate what students are expected to be able to do in a course that can be measured. This course has course-level learning goals that pertain to the entire course, and section-level
More informationCh4: Method of Steepest Descent
Ch4: Method of Steepest Descent The method of steepest descent is recursive in the sense that starting from some initial (arbitrary) value for the tap-weight vector, it improves with the increased number
More informationEAD 115. Numerical Solution of Engineering and Scientific Problems. David M. Rocke Department of Applied Science
EAD 115 Numerical Solution of Engineering and Scientific Problems David M. Rocke Department of Applied Science Taylor s Theorem Can often approximate a function by a polynomial The error in the approximation
More informationMAT 419 Lecture Notes Transcribed by Eowyn Cenek 6/1/2012
(Homework 1: Chapter 1: Exercises 1-7, 9, 11, 19, due Monday June 11th See also the course website for lectures, assignments, etc) Note: today s lecture is primarily about definitions Lots of definitions
More informationMath 302 Outcome Statements Winter 2013
Math 302 Outcome Statements Winter 2013 1 Rectangular Space Coordinates; Vectors in the Three-Dimensional Space (a) Cartesian coordinates of a point (b) sphere (c) symmetry about a point, a line, and a
More informationNonparametric Bayesian Methods (Gaussian Processes)
[70240413 Statistical Machine Learning, Spring, 2015] Nonparametric Bayesian Methods (Gaussian Processes) Jun Zhu dcszj@mail.tsinghua.edu.cn http://bigml.cs.tsinghua.edu.cn/~jun State Key Lab of Intelligent
More informationECE 680 Modern Automatic Control. Gradient and Newton s Methods A Review
ECE 680Modern Automatic Control p. 1/1 ECE 680 Modern Automatic Control Gradient and Newton s Methods A Review Stan Żak October 25, 2011 ECE 680Modern Automatic Control p. 2/1 Review of the Gradient Properties
More informationAM 205: lecture 18. Last time: optimization methods Today: conditions for optimality
AM 205: lecture 18 Last time: optimization methods Today: conditions for optimality Existence of Global Minimum For example: f (x, y) = x 2 + y 2 is coercive on R 2 (global min. at (0, 0)) f (x) = x 3
More informationCalculus 2502A - Advanced Calculus I Fall : Local minima and maxima
Calculus 50A - Advanced Calculus I Fall 014 14.7: Local minima and maxima Martin Frankland November 17, 014 In these notes, we discuss the problem of finding the local minima and maxima of a function.
More informationMatrix Algebra, part 2
Matrix Algebra, part 2 Ming-Ching Luoh 2005.9.12 1 / 38 Diagonalization and Spectral Decomposition of a Matrix Optimization 2 / 38 Diagonalization and Spectral Decomposition of a Matrix Also called Eigenvalues
More informationNon-Convex Optimization. CS6787 Lecture 7 Fall 2017
Non-Convex Optimization CS6787 Lecture 7 Fall 2017 First some words about grading I sent out a bunch of grades on the course management system Everyone should have all their grades in Not including paper
More information7. Response Surface Methodology (Ch.10. Regression Modeling Ch. 11. Response Surface Methodology)
7. Response Surface Methodology (Ch.10. Regression Modeling Ch. 11. Response Surface Methodology) Hae-Jin Choi School of Mechanical Engineering, Chung-Ang University 1 Introduction Response surface methodology,
More informationnonrobust estimation The n measurement vectors taken together give the vector X R N. The unknown parameter vector is P R M.
Introduction to nonlinear LS estimation R. I. Hartley and A. Zisserman: Multiple View Geometry in Computer Vision. Cambridge University Press, 2ed., 2004. After Chapter 5 and Appendix 6. We will use x
More informationOptimization for neural networks
0 - : Optimization for neural networks Prof. J.C. Kao, UCLA Optimization for neural networks We previously introduced the principle of gradient descent. Now we will discuss specific modifications we make
More informationMULTIVARIABLE CALCULUS, LINEAR ALGEBRA, AND DIFFERENTIAL EQUATIONS
T H I R D E D I T I O N MULTIVARIABLE CALCULUS, LINEAR ALGEBRA, AND DIFFERENTIAL EQUATIONS STANLEY I. GROSSMAN University of Montana and University College London SAUNDERS COLLEGE PUBLISHING HARCOURT BRACE
More informationx +3y 2t = 1 2x +y +z +t = 2 3x y +z t = 7 2x +6y +z +t = a
UCM Final Exam, 05/8/014 Solutions 1 Given the parameter a R, consider the following linear system x +y t = 1 x +y +z +t = x y +z t = 7 x +6y +z +t = a (a (6 points Discuss the system depending on the
More informationDay 3 Lecture 3. Optimizing deep networks
Day 3 Lecture 3 Optimizing deep networks Convex optimization A function is convex if for all α [0,1]: f(x) Tangent line Examples Quadratics 2-norms Properties Local minimum is global minimum x Gradient
More informationMA202 Calculus III Fall, 2009 Laboratory Exploration 2: Gradients, Directional Derivatives, Stationary Points
MA0 Calculus III Fall, 009 Laboratory Exploration : Gradients, Directional Derivatives, Stationary Points Introduction: This lab deals with several topics from Chapter. Some of the problems should be done
More informationECE580 Solution to Problem Set 3: Applications of the FONC, SONC, and SOSC
ECE580 Spring 2016 Solution to Problem Set 3 February 8, 2016 1 ECE580 Solution to Problem Set 3: Applications of the FONC, SONC, and SOSC These problems are from the textbook by Chong and Zak, 4th edition,
More informationMath 2204 Multivariable Calculus Chapter 14: Partial Derivatives Sec. 14.7: Maximum and Minimum Values
Math 2204 Multivariable Calculus Chapter 14: Partial Derivatives Sec. 14.7: Maximum and Minimum Values I. Review from 1225 A. Definitions 1. Local Extreme Values (Relative) a. A function f has a local
More informationComputational Optimization. Convexity and Unconstrained Optimization 1/29/08 and 2/1(revised)
Computational Optimization Convexity and Unconstrained Optimization 1/9/08 and /1(revised) Convex Sets A set S is convex if the line segment joining any two points in the set is also in the set, i.e.,
More informationVector Derivatives and the Gradient
ECE 275AB Lecture 10 Fall 2008 V1.1 c K. Kreutz-Delgado, UC San Diego p. 1/1 Lecture 10 ECE 275A Vector Derivatives and the Gradient ECE 275AB Lecture 10 Fall 2008 V1.1 c K. Kreutz-Delgado, UC San Diego
More informationarxiv: v2 [math.oc] 5 May 2018
The Impact of Local Geometry and Batch Size on Stochastic Gradient Descent for Nonconvex Problems Viva Patel a a Department of Statistics, University of Chicago, Illinois, USA arxiv:1709.04718v2 [math.oc]
More information