Solutions. Part I Logistic regression backpropagation with a single training example
|
|
- Benjamin Hodge
- 5 years ago
- Views:
Transcription
1 Solutions Part I Logistic regression backpropagation with a single training example In this part, you are using the Stochastic Gradient Optimizer to train your Logistic Regression. Consequently, the gradients leading to the parameter updates are computed on a single training example. a) Forward propagation equations Before getting into the details of backpropagation, let s spend a few minutes on the forward pass. For one training example x = (x 1, x 2,..., x n) of dimension n, the forward propagation is: z = wx + b ŷ = a = σ(z) L = (ylog(ŷ) + (1 y) log(1 ŷ)) b) Dimensions of the variables in the forward propagation equations It s important to note the shapes of the quantities in the previous equations: x = (n,1), w = (1,n), b = (1,1), z = (1,1), a = (1,1), L is a scalar. c) Backpropagation equations Training our model means updating our weights and biases, W and b, using the gradient of the loss with respect to these parameters. At every step, we need to calculate : To do this, we will apply the chain rule. So we need to calculate the following derivatives :
2 We will calculate those derivatives to get an expression of and. Why did we choose X.T rather than X? We can have a look at the following dimensions without forgetting that the dimensions of the derivative of a term are the same as the dimensions of the term. (1, n) (1, n) (1, 1) (1, n) Then :
3 Part II Backpropagation for a batch of m training examples In this part, you are using a Batch Gradient Optimization to train your Logistic Regression. Consequently, the gradients leading to the parameter updates are computed on the entire batch of m training examples. a) Write down the forward propagation equations leading to J. b) Analyze the dimensions of all the variables in your forward propagation equations. c) Write down the backpropagation equations to compute. a) Forward propagation equations Before getting into the details of backpropagation, let s study the forward pass. For a batch of m training examples, each of dimension n, the forward propagation is: z = w X + b a = σ(z) J = m L (i) (1) (2) where L is the binary cross entropy loss L (i) = y (i) log(a (i) ) + ( 1 y (i) )log(1 a (i) ) b) Dimensions of the variables in the forward propagation equations It s important to note the shapes of the quantities in equations (1) and (2). 1 n w = R, X = R n m, b = R 1 m, but is really of shape 1 1 and broadcasted to z = R 1 m and a = R 1 m and J is a scalar. 1 m c) Backpropagation equations To train our model, we need to update our weights and biases w and b, using the gradient of the loss with respect to these parameters. In other words, we need to calculate b and. To do this, we will apply the chain rule. a z We can write as a z a The first step is to calculate. a z
4 a = m (i) = m (i) (i) a (y log(a ) + ( 1 y (i) )log(1 a (i) )) = m y ( (i) + (1 y (i) ) 1 ) (i) 1 a (i) a(i) a(i) a (i) = a (i) (1 a (i) ) which is the derivative of the sigmoid function. z (i) and Putting this together, W = m z (i) z (i) W y = ( (i) 1 ) )a (1 ) (1 ) 1 )a z (i) a (i) ( y (i) 1 (i) 1 a a (i) = y (i) a (i) + ( y (i) (i) = a (i) y (i) (i) z (i) = Therefore, (wx i + b ) = wxi = w j X n 1 n 1 ji ) X = m (i) (a y (i) w j ji To evaluate this derivative, we will find the derivative with respect to each element of W. p = m (a (i) y (i) ) n 1 X z (i) = p p p p w j ji (wx + b ) = wx = w j X ji = X pi Where X p is a row vector corresponding to p p n 1 the p th row of the X matrix. = m (a (i) y (i) )Xpi To get we simply stack all these derivatives up, row wise. This can efficiently be written in matrix form as: A )X = ( Y T z Following a very similar procedure, and noting that (i) b = 1 A ).1 Where 1 is a column vector of 1 s. = ( Y Part III Revisiting Backpropagation There are several possible ways to obtain an optimal set of weights/parameters for a neural network. The naive approach would consist in randomly generating a new set of weights at each iteration step. An improved method would use local information of the loss function (e.g the gradient) to pick a better guess in the next iteration. Does backpropagation compute a numerical or analytical value of the gradients in a neural network? (Answer on Menti)
5 1. You are given the following neural network and your goal is to compute. a[1] 1 a) What other derivatives do you need to compute before finding [1]? b) What values do you need to cache during the forward propagation in order to compute? A: a) You need to compute the intermediary derivatives,,,,, ŷ ŷ z 1 z 1 a i a i z j z i [1] b) d z [L] = d a [L] [L] g (z ) d W [L] [L] [L 1] = d z a d b [L] = d z [L] [L] d a [L 1] = W [L]T dz d z [L] = W [L+1]T [L+1] [L] dz g (z ) Backpropagation example on a univariate scalar function (e.g. f : R R ): Let s suppose that you have built a model that uses the following loss function: L = (ˆ y y) 2 where ŷ = tanh [ σ (wx ) 2 + b ] Assume that all the above variables are scalars. Using backpropagation, calculate. ( ŷ 2 ) z ( ŷ 2 ) z z x 2 A: (y ) (y ) = 2 ˆ y ŷ = 2 ˆ y 1 (y ) = 2 ˆ y 1 (1 ) where z = σ (wx + b )
6 n 3. Backpropagation example on a multivariate scalar function (e.g. f : R R ): Let s suppose that you have built a model that uses the following loss function: L = y log(y) ˆ where ŷ = R elu (wt x + b ) n a) Assume x R. What s the shape of w? n 1 A: w R b) Using backpropagation, obtain. A: We will derive for...n: i log(y) ˆ 1 ŷ = y (z) (z) where, if i i = y ŷ = y 1 i ŷ f z = y 1 i ŷ f x i f (z) = 1 z > 0, f (z) = 0 otherwise (where z = w T x + b.) n m 4. Backpropagation applied to scalar-matrix functions ( f : R R ): The final case that is worth exploring is: L = ŷ y 2 2 where ŷ = σ (x) W 1 m a) Assume that ŷ R and x R. What is the shape of W? 1 n A: W is an nxm matrix. (Note that the shapes of y and x differ from what you are used to in the class notations.) b) Using backpropagation, calculate. x A: = 2(y ˆ y ) ŷ y z (1 ) where x W = 2ˆ W T z z = σ (x)
Neural Networks: Backpropagation
Neural Networks: Backpropagation Seung-Hoon Na 1 1 Department of Computer Science Chonbuk National University 2018.10.25 eung-hoon Na (Chonbuk National University) Neural Networks: Backpropagation 2018.10.25
More informationECE521 Lecture 7/8. Logistic Regression
ECE521 Lecture 7/8 Logistic Regression Outline Logistic regression (Continue) A single neuron Learning neural networks Multi-class classification 2 Logistic regression The output of a logistic regression
More informationCSC321 Lecture 6: Backpropagation
CSC321 Lecture 6: Backpropagation Roger Grosse Roger Grosse CSC321 Lecture 6: Backpropagation 1 / 21 Overview We ve seen that multilayer neural networks are powerful. But how can we actually learn them?
More informationMultilayer Perceptron
Outline Hong Chang Institute of Computing Technology, Chinese Academy of Sciences Machine Learning Methods (Fall 2012) Outline Outline I 1 Introduction 2 Single Perceptron 3 Boolean Function Learning 4
More informationNeural Networks. Yan Shao Department of Linguistics and Philology, Uppsala University 7 December 2016
Neural Networks Yan Shao Department of Linguistics and Philology, Uppsala University 7 December 2016 Outline Part 1 Introduction Feedforward Neural Networks Stochastic Gradient Descent Computational Graph
More informationMachine Learning: Chenhao Tan University of Colorado Boulder LECTURE 16
Machine Learning: Chenhao Tan University of Colorado Boulder LECTURE 16 Slides adapted from Jordan Boyd-Graber, Justin Johnson, Andrej Karpathy, Chris Ketelsen, Fei-Fei Li, Mike Mozer, Michael Nielson
More informationNeural Networks, Computation Graphs. CMSC 470 Marine Carpuat
Neural Networks, Computation Graphs CMSC 470 Marine Carpuat Binary Classification with a Multi-layer Perceptron φ A = 1 φ site = 1 φ located = 1 φ Maizuru = 1 φ, = 2 φ in = 1 φ Kyoto = 1 φ priest = 0 φ
More informationCSC 411 Lecture 10: Neural Networks
CSC 411 Lecture 10: Neural Networks Roger Grosse, Amir-massoud Farahmand, and Juan Carrasquilla University of Toronto UofT CSC 411: 10-Neural Networks 1 / 35 Inspiration: The Brain Our brain has 10 11
More informationLogistic Regression & Neural Networks
Logistic Regression & Neural Networks CMSC 723 / LING 723 / INST 725 Marine Carpuat Slides credit: Graham Neubig, Jacob Eisenstein Logistic Regression Perceptron & Probabilities What if we want a probability
More informationError Backpropagation
Error Backpropagation Sargur Srihari 1 Topics in Error Backpropagation Terminology of backpropagation 1. Evaluation of Error function derivatives 2. Error Backpropagation algorithm 3. A simple example
More informationIntro to Neural Networks and Deep Learning
Intro to Neural Networks and Deep Learning Jack Lanchantin Dr. Yanjun Qi UVA CS 6316 1 Neurons 1-Layer Neural Network Multi-layer Neural Network Loss Functions Backpropagation Nonlinearity Functions NNs
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 informationMachine Learning (CSE 446): Neural Networks
Machine Learning (CSE 446): Neural Networks Noah Smith c 2017 University of Washington nasmith@cs.washington.edu November 6, 2017 1 / 22 Admin No Wednesday office hours for Noah; no lecture Friday. 2 /
More informationLecture 13 Back-propagation
Lecture 13 Bac-propagation 02 March 2016 Taylor B. Arnold Yale Statistics STAT 365/665 1/21 Notes: Problem set 4 is due this Friday Problem set 5 is due a wee from Monday (for those of you with a midterm
More informationy(x n, w) t n 2. (1)
Network training: Training a neural network involves determining the weight parameter vector w that minimizes a cost function. Given a training set comprising a set of input vector {x n }, n = 1,...N,
More informationCSE 190 Fall 2015 Midterm DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO START!!!!
CSE 190 Fall 2015 Midterm DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO START!!!! November 18, 2015 THE EXAM IS CLOSED BOOK. Once the exam has started, SORRY, NO TALKING!!! No, you can t even say see ya
More informationMachine Learning Basics III
Machine Learning Basics III Benjamin Roth CIS LMU München Benjamin Roth (CIS LMU München) Machine Learning Basics III 1 / 62 Outline 1 Classification Logistic Regression 2 Gradient Based Optimization Gradient
More informationECE521 Lectures 9 Fully Connected Neural Networks
ECE521 Lectures 9 Fully Connected Neural Networks Outline Multi-class classification Learning multi-layer neural networks 2 Measuring distance in probability space We learnt that the squared L2 distance
More informationApprentissage, réseaux de neurones et modèles graphiques (RCP209) Neural Networks and Deep Learning
Apprentissage, réseaux de neurones et modèles graphiques (RCP209) Neural Networks and Deep Learning Nicolas Thome Prenom.Nom@cnam.fr http://cedric.cnam.fr/vertigo/cours/ml2/ Département Informatique Conservatoire
More informationNeural Networks and Deep Learning
Neural Networks and Deep Learning Professor Ameet Talwalkar November 12, 2015 Professor Ameet Talwalkar Neural Networks and Deep Learning November 12, 2015 1 / 16 Outline 1 Review of last lecture AdaBoost
More informationNeural Networks. CSE 6363 Machine Learning Vassilis Athitsos Computer Science and Engineering Department University of Texas at Arlington
Neural Networks CSE 6363 Machine Learning Vassilis Athitsos Computer Science and Engineering Department University of Texas at Arlington 1 Perceptrons x 0 = 1 x 1 x 2 z = h w T x Output: z x D A perceptron
More informationArtificial Neural Networks. MGS Lecture 2
Artificial Neural Networks MGS 2018 - Lecture 2 OVERVIEW Biological Neural Networks Cell Topology: Input, Output, and Hidden Layers Functional description Cost functions Training ANNs Back-Propagation
More informationCSC321 Lecture 4: Learning a Classifier
CSC321 Lecture 4: Learning a Classifier Roger Grosse Roger Grosse CSC321 Lecture 4: Learning a Classifier 1 / 31 Overview Last time: binary classification, perceptron algorithm Limitations of the perceptron
More information1 What a Neural Network Computes
Neural Networks 1 What a Neural Network Computes To begin with, we will discuss fully connected feed-forward neural networks, also known as multilayer perceptrons. A feedforward neural network consists
More informationDeep Feedforward Networks
Deep Feedforward Networks Liu Yang March 30, 2017 Liu Yang Short title March 30, 2017 1 / 24 Overview 1 Background A general introduction Example 2 Gradient based learning Cost functions Output Units 3
More informationIntroduction Neural Networks - Architecture Network Training Small Example - ZIP Codes Summary. Neural Networks - I. Henrik I Christensen
Neural Networks - I Henrik I Christensen Robotics & Intelligent Machines @ GT Georgia Institute of Technology, Atlanta, GA 30332-0280 hic@cc.gatech.edu Henrik I Christensen (RIM@GT) Neural Networks 1 /
More informationReading Group on Deep Learning Session 1
Reading Group on Deep Learning Session 1 Stephane Lathuiliere & Pablo Mesejo 2 June 2016 1/31 Contents Introduction to Artificial Neural Networks to understand, and to be able to efficiently use, the popular
More informationCSC321 Lecture 4: Learning a Classifier
CSC321 Lecture 4: Learning a Classifier Roger Grosse Roger Grosse CSC321 Lecture 4: Learning a Classifier 1 / 28 Overview Last time: binary classification, perceptron algorithm Limitations of the perceptron
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 informationCS 6501: Deep Learning for Computer Graphics. Basics of Neural Networks. Connelly Barnes
CS 6501: Deep Learning for Computer Graphics Basics of Neural Networks Connelly Barnes Overview Simple neural networks Perceptron Feedforward neural networks Multilayer perceptron and properties Autoencoders
More informationCSC242: Intro to AI. Lecture 21
CSC242: Intro to AI Lecture 21 Administrivia Project 4 (homeworks 18 & 19) due Mon Apr 16 11:59PM Posters Apr 24 and 26 You need an idea! You need to present it nicely on 2-wide by 4-high landscape pages
More informationCS 453X: Class 20. Jacob Whitehill
CS 3X: Class 20 Jacob Whitehill More on training neural networks Training neural networks While training neural networks by hand is (arguably) fun, it is completely impractical except for toy examples.
More informationRegression and Classification" with Linear Models" CMPSCI 383 Nov 15, 2011!
Regression and Classification" with Linear Models" CMPSCI 383 Nov 15, 2011! 1 Todayʼs topics" Learning from Examples: brief review! Univariate Linear Regression! Batch gradient descent! Stochastic gradient
More informationNeural networks COMS 4771
Neural networks COMS 4771 1. Logistic regression Logistic regression Suppose X = R d and Y = {0, 1}. A logistic regression model is a statistical model where the conditional probability function has a
More informationComments. Assignment 3 code released. Thought questions 3 due this week. Mini-project: hopefully you have started. implement classification algorithms
Neural networks Comments Assignment 3 code released implement classification algorithms use kernels for census dataset Thought questions 3 due this week Mini-project: hopefully you have started 2 Example:
More informationA Tutorial On Backward Propagation Through Time (BPTT) In The Gated Recurrent Unit (GRU) RNN
A Tutorial On Backward Propagation Through Time (BPTT In The Gated Recurrent Unit (GRU RNN Minchen Li Department of Computer Science The University of British Columbia minchenl@cs.ubc.ca Abstract In this
More informationCOMP 551 Applied Machine Learning Lecture 14: Neural Networks
COMP 551 Applied Machine Learning Lecture 14: Neural Networks Instructor: Ryan Lowe (ryan.lowe@mail.mcgill.ca) Slides mostly by: Class web page: www.cs.mcgill.ca/~hvanho2/comp551 Unless otherwise noted,
More informationNeural Networks Learning the network: Backprop , Fall 2018 Lecture 4
Neural Networks Learning the network: Backprop 11-785, Fall 2018 Lecture 4 1 Recap: The MLP can represent any function The MLP can be constructed to represent anything But how do we construct it? 2 Recap:
More information8-1: Backpropagation Prof. J.C. Kao, UCLA. Backpropagation. Chain rule for the derivatives Backpropagation graphs Examples
8-1: Backpropagation Prof. J.C. Kao, UCLA Backpropagation Chain rule for the derivatives Backpropagation graphs Examples 8-2: Backpropagation Prof. J.C. Kao, UCLA Motivation for backpropagation To do gradient
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 informationCSCI567 Machine Learning (Fall 2018)
CSCI567 Machine Learning (Fall 2018) Prof. Haipeng Luo U of Southern California Sep 12, 2018 September 12, 2018 1 / 49 Administration GitHub repos are setup (ask TA Chi Zhang for any issues) HW 1 is due
More informationLecture 2: Modular Learning
Lecture 2: Modular Learning Deep Learning @ UvA MODULAR LEARNING - PAGE 1 Announcement o C. Maddison is coming to the Deep Vision Seminars on the 21 st of Sep Author of concrete distribution, co-author
More information17 Neural Networks NEURAL NETWORKS. x XOR 1. x Jonathan Richard Shewchuk
94 Jonathan Richard Shewchuk 7 Neural Networks NEURAL NETWORKS Can do both classification & regression. [They tie together several ideas from the course: perceptrons, logistic regression, ensembles of
More informationFeedforward Neural Nets and Backpropagation
Feedforward Neural Nets and Backpropagation Julie Nutini University of British Columbia MLRG September 28 th, 2016 1 / 23 Supervised Learning Roadmap Supervised Learning: Assume that we are given the features
More informationLecture 4: Types of errors. Bayesian regression models. Logistic regression
Lecture 4: Types of errors. Bayesian regression models. Logistic regression A Bayesian interpretation of regularization Bayesian vs maximum likelihood fitting more generally COMP-652 and ECSE-68, Lecture
More informationLecture 2: Learning with neural networks
Lecture 2: Learning with neural networks Deep Learning @ UvA LEARNING WITH NEURAL NETWORKS - PAGE 1 Lecture Overview o Machine Learning Paradigm for Neural Networks o The Backpropagation algorithm for
More informationLecture 5: Logistic Regression. Neural Networks
Lecture 5: Logistic Regression. Neural Networks Logistic regression Comparison with generative models Feed-forward neural networks Backpropagation Tricks for training neural networks COMP-652, Lecture
More information4. Multilayer Perceptrons
4. Multilayer Perceptrons This is a supervised error-correction learning algorithm. 1 4.1 Introduction A multilayer feedforward network consists of an input layer, one or more hidden layers, and an output
More informationInput layer. Weight matrix [ ] Output layer
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Electrical Engineering and Computer Science 6.034 Artificial Intelligence, Fall 2003 Recitation 10, November 4 th & 5 th 2003 Learning by perceptrons
More informationArtificial Neural Networks 2
CSC2515 Machine Learning Sam Roweis Artificial Neural s 2 We saw neural nets for classification. Same idea for regression. ANNs are just adaptive basis regression machines of the form: y k = j w kj σ(b
More informationNeural Networks in Structured Prediction. November 17, 2015
Neural Networks in Structured Prediction November 17, 2015 HWs and Paper Last homework is going to be posted soon Neural net NER tagging model This is a new structured model Paper - Thursday after Thanksgiving
More informationStatistical Machine Learning from Data
January 17, 2006 Samy Bengio Statistical Machine Learning from Data 1 Statistical Machine Learning from Data Multi-Layer Perceptrons Samy Bengio IDIAP Research Institute, Martigny, Switzerland, and Ecole
More informationBackpropagation Neural Net
Backpropagation Neural Net As is the case with most neural networks, the aim of Backpropagation is to train the net to achieve a balance between the ability to respond correctly to the input patterns that
More informationLecture 3 Feedforward Networks and Backpropagation
Lecture 3 Feedforward Networks and Backpropagation CMSC 35246: Deep Learning Shubhendu Trivedi & Risi Kondor University of Chicago April 3, 2017 Things we will look at today Recap of Logistic Regression
More informationDeep Learning book, by Ian Goodfellow, Yoshua Bengio and Aaron Courville
Deep Learning book, by Ian Goodfellow, Yoshua Bengio and Aaron Courville Chapter 6 :Deep Feedforward Networks Benoit Massé Dionyssos Kounades-Bastian Benoit Massé, Dionyssos Kounades-Bastian Deep Feedforward
More informationError Functions & Linear Regression (1)
Error Functions & Linear Regression (1) John Kelleher & Brian Mac Namee Machine Learning @ DIT Overview 1 Introduction Overview 2 Univariate Linear Regression Linear Regression Analytical Solution Gradient
More informationLecture 6: Backpropagation
Lecture 6: Backpropagation Roger Grosse 1 Introduction So far, we ve seen how to train shallow models, where the predictions are computed as a linear function of the inputs. We ve also observed that deeper
More informationCS545 Contents XVI. Adaptive Control. Reading Assignment for Next Class. u Model Reference Adaptive Control. u Self-Tuning Regulators
CS545 Contents XVI Adaptive Control u Model Reference Adaptive Control u Self-Tuning Regulators u Linear Regression u Recursive Least Squares u Gradient Descent u Feedback-Error Learning Reading Assignment
More informationTopics in AI (CPSC 532L): Multimodal Learning with Vision, Language and Sound. Lecture 3: Introduction to Deep Learning (continued)
Topics in AI (CPSC 532L): Multimodal Learning with Vision, Language and Sound Lecture 3: Introduction to Deep Learning (continued) Course Logistics - Update on course registrations - 6 seats left now -
More informationHow to do backpropagation in a brain
How to do backpropagation in a brain Geoffrey Hinton Canadian Institute for Advanced Research & University of Toronto & Google Inc. Prelude I will start with three slides explaining a popular type of deep
More informationNeural Networks with Applications to Vision and Language. Feedforward Networks. Marco Kuhlmann
Neural Networks with Applications to Vision and Language Feedforward Networks Marco Kuhlmann Feedforward networks Linear separability x 2 x 2 0 1 0 1 0 0 x 1 1 0 x 1 linearly separable not linearly separable
More informationLecture 17: Neural Networks and Deep Learning
UVA CS 6316 / CS 4501-004 Machine Learning Fall 2016 Lecture 17: Neural Networks and Deep Learning Jack Lanchantin Dr. Yanjun Qi 1 Neurons 1-Layer Neural Network Multi-layer Neural Network Loss Functions
More informationMark Gales October y (x) x 1. x 2 y (x) Inputs. Outputs. x d. y (x) Second Output layer layer. layer.
University of Cambridge Engineering Part IIB & EIST Part II Paper I0: Advanced Pattern Processing Handouts 4 & 5: Multi-Layer Perceptron: Introduction and Training x y (x) Inputs x 2 y (x) 2 Outputs x
More informationLearning Deep Architectures for AI. Part I - Vijay Chakilam
Learning Deep Architectures for AI - Yoshua Bengio Part I - Vijay Chakilam Chapter 0: Preliminaries Neural Network Models The basic idea behind the neural network approach is to model the response as a
More informationLogistic Regression. Stochastic Gradient Descent
Tutorial 8 CPSC 340 Logistic Regression Stochastic Gradient Descent Logistic Regression Model A discriminative probabilistic model for classification e.g. spam filtering Let x R d be input and y { 1, 1}
More informationStatistical Machine Learning
Statistical Machine Learning Lecture 9 Numerical optimization and deep learning Niklas Wahlström Division of Systems and Control Department of Information Technology Uppsala University niklas.wahlstrom@it.uu.se
More informationRecurrent Neural Networks with Flexible Gates using Kernel Activation Functions
2018 IEEE International Workshop on Machine Learning for Signal Processing (MLSP 18) Recurrent Neural Networks with Flexible Gates using Kernel Activation Functions Authors: S. Scardapane, S. Van Vaerenbergh,
More informationMore on Neural Networks
More on Neural Networks Yujia Yan Fall 2018 Outline Linear Regression y = Wx + b (1) Linear Regression y = Wx + b (1) Polynomial Regression y = Wφ(x) + b (2) where φ(x) gives the polynomial basis, e.g.,
More informationCheng Soon Ong & Christian Walder. Canberra February June 2018
Cheng Soon Ong & Christian Walder Research Group and College of Engineering and Computer Science Canberra February June 2018 Outlines Overview Introduction Linear Algebra Probability Linear Regression
More informationLecture 4 Backpropagation
Lecture 4 Backpropagation CMSC 35246: Deep Learning Shubhendu Trivedi & Risi Kondor University of Chicago April 5, 2017 Things we will look at today More Backpropagation Still more backpropagation Quiz
More information18.6 Regression and Classification with Linear Models
18.6 Regression and Classification with Linear Models 352 The hypothesis space of linear functions of continuous-valued inputs has been used for hundreds of years A univariate linear function (a straight
More informationLogistic Regression Review Fall 2012 Recitation. September 25, 2012 TA: Selen Uguroglu
Logistic Regression Review 10-601 Fall 2012 Recitation September 25, 2012 TA: Selen Uguroglu!1 Outline Decision Theory Logistic regression Goal Loss function Inference Gradient Descent!2 Training Data
More informationLecture 3 Feedforward Networks and Backpropagation
Lecture 3 Feedforward Networks and Backpropagation CMSC 35246: Deep Learning Shubhendu Trivedi & Risi Kondor University of Chicago April 3, 2017 Things we will look at today Recap of Logistic Regression
More informationDeep Feedforward Networks. Seung-Hoon Na Chonbuk National University
Deep Feedforward Networks Seung-Hoon Na Chonbuk National University Neural Network: Types Feedforward neural networks (FNN) = Deep feedforward networks = multilayer perceptrons (MLP) No feedback connections
More informationIntroduction to Neural Networks
CUONG TUAN NGUYEN SEIJI HOTTA MASAKI NAKAGAWA Tokyo University of Agriculture and Technology Copyright by Nguyen, Hotta and Nakagawa 1 Pattern classification Which category of an input? Example: Character
More informationAdvanced Machine Learning
Advanced Machine Learning Lecture 4: Deep Learning Essentials Pierre Geurts, Gilles Louppe, Louis Wehenkel 1 / 52 Outline Goal: explain and motivate the basic constructs of neural networks. From linear
More informationMachine Learning Lecture 10
Machine Learning Lecture 10 Neural Networks 26.11.2018 Bastian Leibe RWTH Aachen http://www.vision.rwth-aachen.de leibe@vision.rwth-aachen.de Today s Topic Deep Learning 2 Course Outline Fundamentals Bayes
More informationDATA MINING AND MACHINE LEARNING. Lecture 4: Linear models for regression and classification Lecturer: Simone Scardapane
DATA MINING AND MACHINE LEARNING Lecture 4: Linear models for regression and classification Lecturer: Simone Scardapane Academic Year 2016/2017 Table of contents Linear models for regression Regularized
More informationLearning Neural Networks
Learning Neural Networks Neural Networks can represent complex decision boundaries Variable size. Any boolean function can be represented. Hidden units can be interpreted as new features Deterministic
More informationClassification goals: Make 1 guess about the label (Top-1 error) Make 5 guesses about the label (Top-5 error) No Bounding Box
ImageNet Classification with Deep Convolutional Neural Networks Alex Krizhevsky, Ilya Sutskever, Geoffrey E. Hinton Motivation Classification goals: Make 1 guess about the label (Top-1 error) Make 5 guesses
More informationClassification. Sandro Cumani. Politecnico di Torino
Politecnico di Torino Outline Generative model: Gaussian classifier (Linear) discriminative model: logistic regression (Non linear) discriminative model: neural networks Gaussian Classifier We want to
More information6.036: Midterm, Spring Solutions
6.036: Midterm, Spring 2018 Solutions This is a closed book exam. Calculators not permitted. The problems are not necessarily in any order of difficulty. Record all your answers in the places provided.
More informationNeural Networks: Backpropagation
Neural Networks: Backpropagation Machine Learning Fall 2017 Based on slides and material from Geoffrey Hinton, Richard Socher, Dan Roth, Yoav Goldberg, Shai Shalev-Shwartz and Shai Ben-David, and others
More informationECE521: Inference Algorithms and Machine Learning University of Toronto. Assignment 1: k-nn and Linear Regression
ECE521: Inference Algorithms and Machine Learning University of Toronto Assignment 1: k-nn and Linear Regression TA: Use Piazza for Q&A Due date: Feb 7 midnight, 2017 Electronic submission to: ece521ta@gmailcom
More informationArtificial Neural Networks
Introduction ANN in Action Final Observations Application: Poverty Detection Artificial Neural Networks Alvaro J. Riascos Villegas University of los Andes and Quantil July 6 2018 Artificial Neural Networks
More informationCOMP9444 Neural Networks and Deep Learning 11. Boltzmann Machines. COMP9444 c Alan Blair, 2017
COMP9444 Neural Networks and Deep Learning 11. Boltzmann Machines COMP9444 17s2 Boltzmann Machines 1 Outline Content Addressable Memory Hopfield Network Generative Models Boltzmann Machine Restricted Boltzmann
More informationBACKPROPAGATION. Neural network training optimization problem. Deriving backpropagation
BACKPROPAGATION Neural network training optimization problem min J(w) w The application of gradient descent to this problem is called backpropagation. Backpropagation is gradient descent applied to J(w)
More informationTopic 3: Neural Networks
CS 4850/6850: Introduction to Machine Learning Fall 2018 Topic 3: Neural Networks Instructor: Daniel L. Pimentel-Alarcón c Copyright 2018 3.1 Introduction Neural networks are arguably the main reason why
More informationNeural Nets Supervised learning
6.034 Artificial Intelligence Big idea: Learning as acquiring a function on feature vectors Background Nearest Neighbors Identification Trees Neural Nets Neural Nets Supervised learning y s(z) w w 0 w
More informationOnline Videos FERPA. Sign waiver or sit on the sides or in the back. Off camera question time before and after lecture. Questions?
Online Videos FERPA Sign waiver or sit on the sides or in the back Off camera question time before and after lecture Questions? Lecture 1, Slide 1 CS224d Deep NLP Lecture 4: Word Window Classification
More informationYou submitted this quiz on Wed 16 Apr :18 PM IST. You got a score of 5.00 out of 5.00.
Feedback IX. Neural Networks: Learning Help You submitted this quiz on Wed 16 Apr 2014 10:18 PM IST. You got a score of 5.00 out of 5.00. Question 1 You are training a three layer neural network and would
More informationCSC 578 Neural Networks and Deep Learning
CSC 578 Neural Networks and Deep Learning Fall 2018/19 3. Improving Neural Networks (Some figures adapted from NNDL book) 1 Various Approaches to Improve Neural Networks 1. Cost functions Quadratic Cross
More informationSupervised Learning. George Konidaris
Supervised Learning George Konidaris gdk@cs.brown.edu Fall 2017 Machine Learning Subfield of AI concerned with learning from data. Broadly, using: Experience To Improve Performance On Some Task (Tom Mitchell,
More informationMultilayer Neural Networks. (sometimes called Multilayer Perceptrons or MLPs)
Multilayer Neural Networks (sometimes called Multilayer Perceptrons or MLPs) Linear separability Hyperplane In 2D: w x + w 2 x 2 + w 0 = 0 Feature x 2 = w w 2 x w 0 w 2 Feature 2 A perceptron can separate
More informationNeural Network Tutorial & Application in Nuclear Physics. Weiguang Jiang ( 蒋炜光 ) UTK / ORNL
Neural Network Tutorial & Application in Nuclear Physics Weiguang Jiang ( 蒋炜光 ) UTK / ORNL Machine Learning Logistic Regression Gaussian Processes Neural Network Support vector machine Random Forest Genetic
More informationCOGS Q250 Fall Homework 7: Learning in Neural Networks Due: 9:00am, Friday 2nd November.
COGS Q250 Fall 2012 Homework 7: Learning in Neural Networks Due: 9:00am, Friday 2nd November. For the first two questions of the homework you will need to understand the learning algorithm using the delta
More informationLecture 5 Neural models for NLP
CS546: Machine Learning in NLP (Spring 2018) http://courses.engr.illinois.edu/cs546/ Lecture 5 Neural models for NLP Julia Hockenmaier juliahmr@illinois.edu 3324 Siebel Center Office hours: Tue/Thu 2pm-3pm
More informationMachine Learning. Lecture 04: Logistic and Softmax Regression. Nevin L. Zhang
Machine Learning Lecture 04: Logistic and Softmax Regression Nevin L. Zhang lzhang@cse.ust.hk Department of Computer Science and Engineering The Hong Kong University of Science and Technology This set
More informationMultilayer Neural Networks. (sometimes called Multilayer Perceptrons or MLPs)
Multilayer Neural Networks (sometimes called Multilayer Perceptrons or MLPs) Linear separability Hyperplane In 2D: w 1 x 1 + w 2 x 2 + w 0 = 0 Feature 1 x 2 = w 1 w 2 x 1 w 0 w 2 Feature 2 A perceptron
More informationNeural Networks (Part 1) Goals for the lecture
Neural Networks (Part ) Mark Craven and David Page Computer Sciences 760 Spring 208 www.biostat.wisc.edu/~craven/cs760/ Some of the slides in these lectures have been adapted/borrowed from materials developed
More information