COMP 551 – Applied Machine Learning Lecture 16: Deep Learning
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Training Autoencoders by Alternating Minimization
Under review as a conference paper at ICLR 2018 TRAINING AUTOENCODERS BY ALTERNATING MINI- MIZATION Anonymous authors Paper under double-blind review ABSTRACT We present DANTE, a novel method for training neural networks, in particular autoencoders, using the alternating minimization principle. DANTE provides a distinct perspective in lieu of traditional gradient-based backpropagation techniques commonly used to train deep networks. It utilizes an adaptation of quasi-convex optimization techniques to cast autoencoder training as a bi-quasi-convex optimiza- tion problem. We show that for autoencoder configurations with both differentiable (e.g. sigmoid) and non-differentiable (e.g. ReLU) activation functions, we can perform the alternations very effectively. DANTE effortlessly extends to networks with multiple hidden layers and varying network configurations. In experiments on standard datasets, autoencoders trained using the proposed method were found to be very promising and competitive to traditional backpropagation techniques, both in terms of quality of solution, as well as training speed. 1 INTRODUCTION For much of the recent march of deep learning, gradient-based backpropagation methods, e.g. Stochastic Gradient Descent (SGD) and its variants, have been the mainstay of practitioners. The use of these methods, especially on vast amounts of data, has led to unprecedented progress in several areas of artificial intelligence. On one hand, the intense focus on these techniques has led to an intimate understanding of hardware requirements and code optimizations needed to execute these routines on large datasets in a scalable manner. Today, myriad off-the-shelf and highly optimized packages exist that can churn reasonably large datasets on GPU architectures with relatively mild human involvement and little bootstrap effort. -
Data Science (ML-DL-Ai)
Data science (ML-DL-ai) Statistics Multiple Regression Model Building and Evaluation Introduction to Statistics Model post fitting for Inference Types of Statistics Examining Residuals Analytics Methodology and Problem- Regression Assumptions Solving Framework Identifying Influential Observations Populations and samples Detecting Collinearity Parameter and Statistics Uses of variable: Dependent and Categorical Data Analysis Independent variable Describing categorical Data Types of Variable: Continuous and One-way frequency tables categorical variable Association Cross Tabulation Tables Descriptive Statistics Test of Association Probability Theory and Distributions Logistic Regression Model Building Picturing your Data Multiple Logistic Regression and Histogram Interpretation Normal Distribution Skewness, Kurtosis Model Building and scoring for Outlier detection Prediction Introduction to predictive modelling Inferential Statistics Building predictive model Scoring Predictive Model Hypothesis Testing Introduction to Machine Learning and Analytics Analysis of variance (ANOVA) Two sample t-Test Introduction to Machine Learning F-test What is Machine Learning? One-way ANOVA Fundamental of Machine Learning ANOVA hypothesis Key Concepts and an example of ML ANOVA Model Supervised Learning Two-way ANOVA Unsupervised Learning Regression Linear Regression with one variable Exploratory data analysis Model Representation Hypothesis testing for correlation Cost Function Outliers, Types of Relationship, -
Machine Learning V1.1
An Introduction to Machine Learning v1.1 E. J. Sagra Agenda ● Why is Machine Learning in the News again? ● ArtificiaI Intelligence vs Machine Learning vs Deep Learning ● Artificial Intelligence ● Machine Learning & Data Science ● Machine Learning ● Data ● Machine Learning - By The Steps ● Tasks that Machine Learning solves ○ Classification ○ Cluster Analysis ○ Regression ○ Ranking ○ Generation Agenda (cont...) ● Model Training ○ Supervised Learning ○ Unsupervised Learning ○ Reinforcement Learning ● Reinforcement Learning - Going Deeper ○ Simple Example ○ The Bellman Equation ○ Deterministic vs. Non-Deterministic Search ○ Markov Decision Process (MDP) ○ Living Penalty ● Machine Learning - Decision Trees ● Machine Learning - Augmented Random Search (ARS) Why is Machine Learning In The News Again? Processing capabilities General ● GPU’s etc have reached level where Machine ● Tools / Languages / Automation Learning / Deep Learning practical ● Need for Data Science no longer limited to ● Cloud computing allows even individuals the tech giants capability to create / train complex models on ● Education is behind in creating Data vast data sets Scientists ● Organizing data is hard. Organizations Memory (Hard Drive (now SSD) as well RAM) challenged ● Speed / capacity increasing ● High demand due to lack of qualified talent ● Cost decreasing Data ● Volume of Data ● Access to vast public data sets ArtificiaI Intelligence vs Machine Learning vs Deep Learning Artificial Intelligence is the all-encompassing concept that initially erupted Followed by Machine Learning that thrived later Finally Deep Learning is escalating the advances of Artificial Intelligence to another level Artificial Intelligence Artificial intelligence (AI) is perhaps the most vaguely understood field of data science. The main idea behind building AI is to use pattern recognition and machine learning to build an agent able to think and reason as humans do (or approach this ability). -
Dynamic Feature Scaling for Online Learning of Binary Classifiers
Dynamic Feature Scaling for Online Learning of Binary Classifiers Danushka Bollegala University of Liverpool, Liverpool, United Kingdom May 30, 2017 Abstract Scaling feature values is an important step in numerous machine learning tasks. Different features can have different value ranges and some form of a feature scal- ing is often required in order to learn an accurate classifier. However, feature scaling is conducted as a preprocessing task prior to learning. This is problematic in an online setting because of two reasons. First, it might not be possible to accu- rately determine the value range of a feature at the initial stages of learning when we have observed only a handful of training instances. Second, the distribution of data can change over time, which render obsolete any feature scaling that we perform in a pre-processing step. We propose a simple but an effective method to dynamically scale features at train time, thereby quickly adapting to any changes in the data stream. We compare the proposed dynamic feature scaling method against more complex methods for estimating scaling parameters using several benchmark datasets for classification. Our proposed feature scaling method consistently out- performs more complex methods on all of the benchmark datasets and improves classification accuracy of a state-of-the-art online classification algorithm. 1 Introduction Machine learning algorithms require train and test instances to be represented using a set of features. For example, in supervised document classification [9], a document is often represented as a vector of its words and the value of a feature is set to the num- ber of times the word corresponding to the feature occurs in that document. -
Capacity and Trainability in Recurrent Neural Networks
Published as a conference paper at ICLR 2017 CAPACITY AND TRAINABILITY IN RECURRENT NEURAL NETWORKS Jasmine Collins,∗ Jascha Sohl-Dickstein & David Sussillo Google Brain Google Inc. Mountain View, CA 94043, USA {jlcollins, jaschasd, sussillo}@google.com ABSTRACT Two potential bottlenecks on the expressiveness of recurrent neural networks (RNNs) are their ability to store information about the task in their parameters, and to store information about the input history in their units. We show experimentally that all common RNN architectures achieve nearly the same per-task and per-unit capacity bounds with careful training, for a variety of tasks and stacking depths. They can store an amount of task information which is linear in the number of parameters, and is approximately 5 bits per parameter. They can additionally store approximately one real number from their input history per hidden unit. We further find that for several tasks it is the per-task parameter capacity bound that determines performance. These results suggest that many previous results comparing RNN architectures are driven primarily by differences in training effectiveness, rather than differences in capacity. Supporting this observation, we compare training difficulty for several architectures, and show that vanilla RNNs are far more difficult to train, yet have slightly higher capacity. Finally, we propose two novel RNN architectures, one of which is easier to train than the LSTM or GRU for deeply stacked architectures. 1 INTRODUCTION Research and application of recurrent neural networks (RNNs) have seen explosive growth over the last few years, (Martens & Sutskever, 2011; Graves et al., 2009), and RNNs have become the central component for some very successful model classes and application domains in deep learning (speech recognition (Amodei et al., 2015), seq2seq (Sutskever et al., 2014), neural machine translation (Bahdanau et al., 2014), the DRAW model (Gregor et al., 2015), educational applications (Piech et al., 2015), and scientific discovery (Mante et al., 2013)). -
Deep Learning Based Computer Generated Face Identification Using
applied sciences Article Deep Learning Based Computer Generated Face Identification Using Convolutional Neural Network L. Minh Dang 1, Syed Ibrahim Hassan 1, Suhyeon Im 1, Jaecheol Lee 2, Sujin Lee 1 and Hyeonjoon Moon 1,* 1 Department of Computer Science and Engineering, Sejong University, Seoul 143-747, Korea; [email protected] (L.M.D.); [email protected] (S.I.H.); [email protected] (S.I.); [email protected] (S.L.) 2 Department of Information Communication Engineering, Sungkyul University, Seoul 143-747, Korea; [email protected] * Correspondence: [email protected] Received: 30 October 2018; Accepted: 10 December 2018; Published: 13 December 2018 Abstract: Generative adversarial networks (GANs) describe an emerging generative model which has made impressive progress in the last few years in generating photorealistic facial images. As the result, it has become more and more difficult to differentiate between computer-generated and real face images, even with the human’s eyes. If the generated images are used with the intent to mislead and deceive readers, it would probably cause severe ethical, moral, and legal issues. Moreover, it is challenging to collect a dataset for computer-generated face identification that is large enough for research purposes because the number of realistic computer-generated images is still limited and scattered on the internet. Thus, a development of a novel decision support system for analyzing and detecting computer-generated face images generated by the GAN network is crucial. In this paper, we propose a customized convolutional neural network, namely CGFace, which is specifically designed for the computer-generated face detection task by customizing the number of convolutional layers, so it performs well in detecting computer-generated face images. -
Matrix Calculus
Appendix D Matrix Calculus From too much study, and from extreme passion, cometh madnesse. Isaac Newton [205, §5] − D.1 Gradient, Directional derivative, Taylor series D.1.1 Gradients Gradient of a differentiable real function f(x) : RK R with respect to its vector argument is defined uniquely in terms of partial derivatives→ ∂f(x) ∂x1 ∂f(x) , ∂x2 RK f(x) . (2053) ∇ . ∈ . ∂f(x) ∂xK while the second-order gradient of the twice differentiable real function with respect to its vector argument is traditionally called the Hessian; 2 2 2 ∂ f(x) ∂ f(x) ∂ f(x) 2 ∂x1 ∂x1∂x2 ··· ∂x1∂xK 2 2 2 ∂ f(x) ∂ f(x) ∂ f(x) 2 2 K f(x) , ∂x2∂x1 ∂x2 ··· ∂x2∂xK S (2054) ∇ . ∈ . .. 2 2 2 ∂ f(x) ∂ f(x) ∂ f(x) 2 ∂xK ∂x1 ∂xK ∂x2 ∂x ··· K interpreted ∂f(x) ∂f(x) 2 ∂ ∂ 2 ∂ f(x) ∂x1 ∂x2 ∂ f(x) = = = (2055) ∂x1∂x2 ³∂x2 ´ ³∂x1 ´ ∂x2∂x1 Dattorro, Convex Optimization Euclidean Distance Geometry, Mεβoo, 2005, v2020.02.29. 599 600 APPENDIX D. MATRIX CALCULUS The gradient of vector-valued function v(x) : R RN on real domain is a row vector → v(x) , ∂v1(x) ∂v2(x) ∂vN (x) RN (2056) ∇ ∂x ∂x ··· ∂x ∈ h i while the second-order gradient is 2 2 2 2 , ∂ v1(x) ∂ v2(x) ∂ vN (x) RN v(x) 2 2 2 (2057) ∇ ∂x ∂x ··· ∂x ∈ h i Gradient of vector-valued function h(x) : RK RN on vector domain is → ∂h1(x) ∂h2(x) ∂hN (x) ∂x1 ∂x1 ··· ∂x1 ∂h1(x) ∂h2(x) ∂hN (x) h(x) , ∂x2 ∂x2 ··· ∂x2 ∇ . -
Training and Testing of a Single-Layer LSTM Network for Near-Future Solar Forecasting
applied sciences Conference Report Training and Testing of a Single-Layer LSTM Network for Near-Future Solar Forecasting Naylani Halpern-Wight 1,2,*,†, Maria Konstantinou 1, Alexandros G. Charalambides 1 and Angèle Reinders 2,3 1 Chemical Engineering Department, Cyprus University of Technology, Lemesos 3036, Cyprus; [email protected] (M.K.); [email protected] (A.G.C.) 2 Energy Technology Group, Department of Mechanical Engineering, Eindhoven University of Technology, 5612 AE Eindhoven, The Netherlands; [email protected] 3 Department of Design, Production and Management, Faculty of Engineering Technology, University of Twente, 7522 NB Enschede, The Netherlands * Correspondence: [email protected] or [email protected] † Current address: Archiepiskopou Kyprianou 30, Limassol 3036, Cyprus. Received: 30 June 2020; Accepted: 31 July 2020; Published: 25 August 2020 Abstract: Increasing integration of renewable energy sources, like solar photovoltaic (PV), necessitates the development of power forecasting tools to predict power fluctuations caused by weather. With trustworthy and accurate solar power forecasting models, grid operators could easily determine when other dispatchable sources of backup power may be needed to account for fluctuations in PV power plants. Additionally, PV customers and designers would feel secure knowing how much energy to expect from their PV systems on an hourly, daily, monthly, or yearly basis. The PROGNOSIS project, based at the Cyprus University of Technology, is developing a tool for intra-hour solar irradiance forecasting. This article presents the design, training, and testing of a single-layer long-short-term-memory (LSTM) artificial neural network for intra-hour power forecasting of a single PV system in Cyprus. -
Training Dnns: Tricks
Training DNNs: Tricks Ju Sun Computer Science & Engineering University of Minnesota, Twin Cities March 5, 2020 1 / 33 Recap: last lecture Training DNNs m 1 X min ` (yi; DNNW (xi)) +Ω( W ) W m i=1 { What methods? Mini-batch stochastic optimization due to large m * SGD (with momentum), Adagrad, RMSprop, Adam * diminishing LR (1/t, exp delay, staircase delay) { Where to start? * Xavier init., Kaiming init., orthogonal init. { When to stop? * early stopping: stop when validation error doesn't improve This lecture: additional tricks/heuristics that improve { convergence speed { task-specific (e.g., classification, regression, generation) performance 2 / 33 Outline Data Normalization Regularization Hyperparameter search, data augmentation Suggested reading 3 / 33 Why scaling matters? : 1 Pm | Consider a ML objective: minw f (w) = m i=1 ` (w xi; yi), e.g., 1 Pm | 2 { Least-squares (LS): minw m i=1 kyi − w xik2 h i 1 Pm | w|xi { Logistic regression: minw − m i=1 yiw xi − log 1 + e 1 Pm | 2 { Shallow NN prediction: minw m i=1 kyi − σ (w xi)k2 1 Pm 0 | Gradient: rwf = m i=1 ` (w xi; yi) xi. { What happens when coordinates (i.e., features) of xi have different orders of magnitude? Partial derivatives have different orders of magnitudes =) slow convergence of vanilla GD (recall why adaptive grad methods) 2 1 Pm 00 | | Hessian: rwf = m i=1 ` (w xi; yi) xixi . | { Suppose the off-diagonal elements of xixi are relatively small (e.g., when features are \independent"). { What happens when coordinates (i.e., features) of xi have different orders 2 of magnitude? Conditioning of rwf is bad, i.e., f is elongated 4 / 33 Fix the scaling: first idea Normalization: make each feature zero-mean and unit variance, i.e., make each row of X = [x1;:::; xm] zero-mean and unit variance, i.e. -
Differentiability in Several Variables: Summary of Basic Concepts
DIFFERENTIABILITY IN SEVERAL VARIABLES: SUMMARY OF BASIC CONCEPTS 3 3 1. Partial derivatives If f : R ! R is an arbitrary function and a = (x0; y0; z0) 2 R , then @f f(a + tj) ¡ f(a) f(x ; y + t; z ) ¡ f(x ; y ; z ) (1) (a) := lim = lim 0 0 0 0 0 0 @y t!0 t t!0 t etc.. provided the limit exists. 2 @f f(t;0)¡f(0;0) Example 1. for f : R ! R, @x (0; 0) = limt!0 t . Example 2. Let f : R2 ! R given by ( 2 x y ; (x; y) 6= (0; 0) f(x; y) = x2+y2 0; (x; y) = (0; 0) Then: @f f(t;0)¡f(0;0) 0 ² @x (0; 0) = limt!0 t = limt!0 t = 0 @f f(0;t)¡f(0;0) ² @y (0; 0) = limt!0 t = 0 Note: away from (0; 0), where f is the quotient of differentiable functions (with non-zero denominator) one can apply the usual rules of derivation: @f 2xy(x2 + y2) ¡ 2x3y (x; y) = ; for (x; y) 6= (0; 0) @x (x2 + y2)2 2 2 2. Directional derivatives. If f : R ! R is a map, a = (x0; y0) 2 R and v = ®i + ¯j is a vector in R2, then by definition f(a + tv) ¡ f(a) f(x0 + t®; y0 + t¯) ¡ f(x0; y0) (2) @vf(a) := lim = lim t!0 t t!0 t Example 3. Let f the function from the Example 2 above. Then for v = ®i + ¯j a unit vector (i.e. -
Lipschitz Recurrent Neural Networks
Published as a conference paper at ICLR 2021 LIPSCHITZ RECURRENT NEURAL NETWORKS N. Benjamin Erichson Omri Azencot Alejandro Queiruga ICSI and UC Berkeley Ben-Gurion University Google Research [email protected] [email protected] [email protected] Liam Hodgkinson Michael W. Mahoney ICSI and UC Berkeley ICSI and UC Berkeley [email protected] [email protected] ABSTRACT Viewing recurrent neural networks (RNNs) as continuous-time dynamical sys- tems, we propose a recurrent unit that describes the hidden state’s evolution with two parts: a well-understood linear component plus a Lipschitz nonlinearity. This particular functional form facilitates stability analysis of the long-term behavior of the recurrent unit using tools from nonlinear systems theory. In turn, this en- ables architectural design decisions before experimentation. Sufficient conditions for global stability of the recurrent unit are obtained, motivating a novel scheme for constructing hidden-to-hidden matrices. Our experiments demonstrate that the Lipschitz RNN can outperform existing recurrent units on a range of bench- mark tasks, including computer vision, language modeling and speech prediction tasks. Finally, through Hessian-based analysis we demonstrate that our Lipschitz recurrent unit is more robust with respect to input and parameter perturbations as compared to other continuous-time RNNs. 1 INTRODUCTION Many interesting problems exhibit temporal structures that can be modeled with recurrent neural networks (RNNs), including problems in robotics, system identification, natural language process- ing, and machine learning control. In contrast to feed-forward neural networks, RNNs consist of one or more recurrent units that are designed to have dynamical (recurrent) properties, thereby enabling them to acquire some form of internal memory. -
Neural Networks
School of Computer Science 10-601B Introduction to Machine Learning Neural Networks Readings: Matt Gormley Bishop Ch. 5 Lecture 15 Murphy Ch. 16.5, Ch. 28 October 19, 2016 Mitchell Ch. 4 1 Reminders 2 Outline • Logistic Regression (Recap) • Neural Networks • Backpropagation 3 RECALL: LOGISTIC REGRESSION 4 Using gradient ascent for linear Recall… classifiers Key idea behind today’s lecture: 1. Define a linear classifier (logistic regression) 2. Define an objective function (likelihood) 3. Optimize it with gradient descent to learn parameters 4. Predict the class with highest probability under the model 5 Using gradient ascent for linear Recall… classifiers This decision function isn’t Use a differentiable function differentiable: instead: 1 T pθ(y =1t)= h(t)=sign(θ t) | 1+2tT( θT t) − sign(x) 1 logistic(u) ≡ −u 1+ e 6 Using gradient ascent for linear Recall… classifiers This decision function isn’t Use a differentiable function differentiable: instead: 1 T pθ(y =1t)= h(t)=sign(θ t) | 1+2tT( θT t) − sign(x) 1 logistic(u) ≡ −u 1+ e 7 Recall… Logistic Regression Data: Inputs are continuous vectors of length K. Outputs are discrete. (i) (i) N K = t ,y where t R and y 0, 1 D { }i=1 ∈ ∈ { } Model: Logistic function applied to dot product of parameters with input vector. 1 pθ(y =1t)= | 1+2tT( θT t) − Learning: finds the parameters that minimize some objective function. θ∗ = argmin J(θ) θ Prediction: Output is the most probable class. yˆ = `;Kt pθ(y t) y 0,1 | ∈{ } 8 NEURAL NETWORKS 9 Learning highly non-linear functions f: X à Y l f might be non-linear