Stochastic Gradient Descent As Approximate Bayesian Inference
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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. -
Learning to Learn by Gradient Descent by Gradient Descent
Learning to learn by gradient descent by gradient descent Marcin Andrychowicz1, Misha Denil1, Sergio Gómez Colmenarejo1, Matthew W. Hoffman1, David Pfau1, Tom Schaul1, Brendan Shillingford1,2, Nando de Freitas1,2,3 1Google DeepMind 2University of Oxford 3Canadian Institute for Advanced Research [email protected] {mdenil,sergomez,mwhoffman,pfau,schaul}@google.com [email protected], [email protected] Abstract The move from hand-designed features to learned features in machine learning has been wildly successful. In spite of this, optimization algorithms are still designed by hand. In this paper we show how the design of an optimization algorithm can be cast as a learning problem, allowing the algorithm to learn to exploit structure in the problems of interest in an automatic way. Our learned algorithms, implemented by LSTMs, outperform generic, hand-designed competitors on the tasks for which they are trained, and also generalize well to new tasks with similar structure. We demonstrate this on a number of tasks, including simple convex problems, training neural networks, and styling images with neural art. 1 Introduction Frequently, tasks in machine learning can be expressed as the problem of optimizing an objective function f(✓) defined over some domain ✓ ⇥. The goal in this case is to find the minimizer 2 ✓⇤ = arg min✓ ⇥ f(✓). While any method capable of minimizing this objective function can be applied, the standard2 approach for differentiable functions is some form of gradient descent, resulting in a sequence of updates ✓ = ✓ ↵ f(✓ ) . t+1 t − tr t The performance of vanilla gradient descent, however, is hampered by the fact that it only makes use of gradients and ignores second-order information. -
Machine Learning Or Econometrics for Credit Scoring: Let’S Get the Best of Both Worlds Elena Dumitrescu, Sullivan Hué, Christophe Hurlin, Sessi Tokpavi
Machine Learning or Econometrics for Credit Scoring: Let’s Get the Best of Both Worlds Elena Dumitrescu, Sullivan Hué, Christophe Hurlin, Sessi Tokpavi To cite this version: Elena Dumitrescu, Sullivan Hué, Christophe Hurlin, Sessi Tokpavi. Machine Learning or Econometrics for Credit Scoring: Let’s Get the Best of Both Worlds. 2020. hal-02507499v2 HAL Id: hal-02507499 https://hal.archives-ouvertes.fr/hal-02507499v2 Preprint submitted on 2 Nov 2020 (v2), last revised 15 Jan 2021 (v3) HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Machine Learning or Econometrics for Credit Scoring: Let's Get the Best of Both Worlds∗ Elena, Dumitrescu,y Sullivan, Hu´e,z Christophe, Hurlin,x Sessi, Tokpavi{ October 18, 2020 Abstract In the context of credit scoring, ensemble methods based on decision trees, such as the random forest method, provide better classification performance than standard logistic regression models. However, logistic regression remains the benchmark in the credit risk industry mainly because the lack of interpretability of ensemble methods is incompatible with the requirements of financial regulators. In this paper, we pro- pose to obtain the best of both worlds by introducing a high-performance and inter- pretable credit scoring method called penalised logistic tree regression (PLTR), which uses information from decision trees to improve the performance of logistic regression. -
Training Neural Networks Without Gradients: a Scalable ADMM Approach
Training Neural Networks Without Gradients: A Scalable ADMM Approach Gavin Taylor1 [email protected] Ryan Burmeister1 Zheng Xu2 [email protected] Bharat Singh2 [email protected] Ankit Patel3 [email protected] Tom Goldstein2 [email protected] 1United States Naval Academy, Annapolis, MD USA 2University of Maryland, College Park, MD USA 3Rice University, Houston, TX USA Abstract many parameters. Because big datasets provide results that With the growing importance of large network (often dramatically) outperform the prior state-of-the-art in models and enormous training datasets, GPUs many machine learning tasks, researchers are willing to have become increasingly necessary to train neu- purchase specialized hardware such as GPUs, and commit ral networks. This is largely because conven- large amounts of time to training models and tuning hyper- tional optimization algorithms rely on stochastic parameters. gradient methods that don’t scale well to large Gradient-based training methods have several properties numbers of cores in a cluster setting. Further- that contribute to this need for specialized hardware. First, more, the convergence of all gradient methods, while large amounts of data can be shared amongst many including batch methods, suffers from common cores, existing optimization methods suffer when paral- problems like saturation effects, poor condition- lelized. Second, training neural nets requires optimizing ing, and saddle points. This paper explores an highly non-convex objectives that exhibit saddle points, unconventional training method that uses alter- poor conditioning, and vanishing gradients, all of which nating direction methods and Bregman iteration slow down gradient-based methods such as stochastic gra- to train networks without gradient descent steps. -
Training Deep Networks with Stochastic Gradient Normalized by Layerwise Adaptive Second Moments
Training Deep Networks with Stochastic Gradient Normalized by Layerwise Adaptive Second Moments Boris Ginsburg 1 Patrice Castonguay 1 Oleksii Hrinchuk 1 Oleksii Kuchaiev 1 Ryan Leary 1 Vitaly Lavrukhin 1 Jason Li 1 Huyen Nguyen 1 Yang Zhang 1 Jonathan M. Cohen 1 Abstract We started with Adam, and then (1) replaced the element- wise second moment with the layer-wise moment, (2) com- We propose NovoGrad, an adaptive stochastic puted the first moment using gradients normalized by layer- gradient descent method with layer-wise gradient wise second moment, (3) decoupled weight decay (WD) normalization and decoupled weight decay. In from normalized gradients (similar to AdamW). our experiments on neural networks for image classification, speech recognition, machine trans- The resulting algorithm, NovoGrad, combines SGD’s and lation, and language modeling, it performs on par Adam’s strengths. We applied NovoGrad to a variety of or better than well-tuned SGD with momentum, large scale problems — image classification, neural machine Adam, and AdamW. Additionally, NovoGrad (1) translation, language modeling, and speech recognition — is robust to the choice of learning rate and weight and found that in all cases, it performs as well or better than initialization, (2) works well in a large batch set- Adam/AdamW and SGD with momentum. ting, and (3) has half the memory footprint of Adam. 2. Related Work NovoGrad belongs to the family of Stochastic Normalized 1. Introduction Gradient Descent (SNGD) optimizers (Nesterov, 1984; Hazan et al., 2015). SNGD uses only the direction of the The most popular algorithms for training Neural Networks stochastic gradient gt to update the weights wt: (NNs) are Stochastic Gradient Descent (SGD) with mo- gt mentum (Polyak, 1964; Sutskever et al., 2013) and Adam wt+1 = wt − λt · (Kingma & Ba, 2015). -
CSE 152: Computer Vision Manmohan Chandraker
CSE 152: Computer Vision Manmohan Chandraker Lecture 15: Optimization in CNNs Recap Engineered against learned features Label Convolutional filters are trained in a Dense supervised manner by back-propagating classification error Dense Dense Convolution + pool Label Convolution + pool Classifier Convolution + pool Pooling Convolution + pool Feature extraction Convolution + pool Image Image Jia-Bin Huang and Derek Hoiem, UIUC Two-layer perceptron network Slide credit: Pieter Abeel and Dan Klein Neural networks Non-linearity Activation functions Multi-layer neural network From fully connected to convolutional networks next layer image Convolutional layer Slide: Lazebnik Spatial filtering is convolution Convolutional Neural Networks [Slides credit: Efstratios Gavves] 2D spatial filters Filters over the whole image Weight sharing Insight: Images have similar features at various spatial locations! Key operations in a CNN Feature maps Spatial pooling Non-linearity Convolution (Learned) . Input Image Input Feature Map Source: R. Fergus, Y. LeCun Slide: Lazebnik Convolution as a feature extractor Key operations in a CNN Feature maps Rectified Linear Unit (ReLU) Spatial pooling Non-linearity Convolution (Learned) Input Image Source: R. Fergus, Y. LeCun Slide: Lazebnik Key operations in a CNN Feature maps Spatial pooling Max Non-linearity Convolution (Learned) Input Image Source: R. Fergus, Y. LeCun Slide: Lazebnik Pooling operations • Aggregate multiple values into a single value • Invariance to small transformations • Keep only most important information for next layer • Reduces the size of the next layer • Fewer parameters, faster computations • Observe larger receptive field in next layer • Hierarchically extract more abstract features Key operations in a CNN Feature maps Spatial pooling Non-linearity Convolution (Learned) . Input Image Input Feature Map Source: R. -
18.650 (F16) Lecture 10: Generalized Linear Models (Glms)
Statistics for Applications Chapter 10: Generalized Linear Models (GLMs) 1/52 Linear model A linear model assumes Y X (µ(X), σ2I), | ∼ N And ⊤ IE(Y X) = µ(X) = X β, | 2/52 Components of a linear model The two components (that we are going to relax) are 1. Random component: the response variable Y X is continuous | and normally distributed with mean µ = µ(X) = IE(Y X). | 2. Link: between the random and covariates X = (X(1),X(2), ,X(p))⊤: µ(X) = X⊤β. · · · 3/52 Generalization A generalized linear model (GLM) generalizes normal linear regression models in the following directions. 1. Random component: Y some exponential family distribution ∼ 2. Link: between the random and covariates: ⊤ g µ(X) = X β where g called link function� and� µ = IE(Y X). | 4/52 Example 1: Disease Occuring Rate In the early stages of a disease epidemic, the rate at which new cases occur can often increase exponentially through time. Hence, if µi is the expected number of new cases on day ti, a model of the form µi = γ exp(δti) seems appropriate. ◮ Such a model can be turned into GLM form, by using a log link so that log(µi) = log(γ) + δti = β0 + β1ti. ◮ Since this is a count, the Poisson distribution (with expected value µi) is probably a reasonable distribution to try. 5/52 Example 2: Prey Capture Rate(1) The rate of capture of preys, yi, by a hunting animal, tends to increase with increasing density of prey, xi, but to eventually level off, when the predator is catching as much as it can cope with. -
Automated Source Code Generation and Auto-Completion Using Deep Learning: Comparing and Discussing Current Language Model-Related Approaches
Article Automated Source Code Generation and Auto-Completion Using Deep Learning: Comparing and Discussing Current Language Model-Related Approaches Juan Cruz-Benito 1,* , Sanjay Vishwakarma 2,†, Francisco Martin-Fernandez 1 and Ismael Faro 1 1 IBM Quantum, IBM T.J. Watson Research Center, Yorktown Heights, NY 10598, USA; [email protected] (F.M.-F.); [email protected] (I.F.) 2 Electrical and Computer Engineering, Carnegie Mellon University, Mountain View, CA 94035, USA; [email protected] * Correspondence: [email protected] † Intern at IBM Quantum at the time of writing this paper. Abstract: In recent years, the use of deep learning in language models has gained much attention. Some research projects claim that they can generate text that can be interpreted as human writ- ing, enabling new possibilities in many application areas. Among the different areas related to language processing, one of the most notable in applying this type of modeling is programming languages. For years, the machine learning community has been researching this software engi- neering area, pursuing goals like applying different approaches to auto-complete, generate, fix, or evaluate code programmed by humans. Considering the increasing popularity of the deep learning- enabled language models approach, we found a lack of empirical papers that compare different deep learning architectures to create and use language models based on programming code. This paper compares different neural network architectures like Average Stochastic Gradient Descent (ASGD) Weight-Dropped LSTMs (AWD-LSTMs), AWD-Quasi-Recurrent Neural Networks (QRNNs), and Citation: Cruz-Benito, J.; Transformer while using transfer learning and different forms of tokenization to see how they behave Vishwakarma, S.; Martin- Fernandez, F.; Faro, I. -
GEE: a Gradient-Based Explainable Variational Autoencoder for Network Anomaly Detection
GEE: A Gradient-based Explainable Variational Autoencoder for Network Anomaly Detection Quoc Phong Nguyen Kar Wai Lim Dinil Mon Divakaran National University of Singapore National University of Singapore Trustwave [email protected] [email protected] [email protected] Kian Hsiang Low Mun Choon Chan National University of Singapore National University of Singapore [email protected] [email protected] Abstract—This paper looks into the problem of detecting number of factors, such as end-user behavior, customer busi- network anomalies by analyzing NetFlow records. While many nesses (e.g., banking, retail), applications, location, time of the previous works have used statistical models and machine learning day, and are expected to evolve with time. Such diversity and techniques in a supervised way, such solutions have the limitations that they require large amount of labeled data for training and dynamism limits the utility of rule-based detection systems. are unlikely to detect zero-day attacks. Existing anomaly detection Next, as capturing, storing and processing raw traffic from solutions also do not provide an easy way to explain or identify such high capacity networks is not practical, Internet routers attacks in the anomalous traffic. To address these limitations, today have the capability to extract and export meta data such we develop and present GEE, a framework for detecting and explaining anomalies in network traffic. GEE comprises of two as NetFlow records [3]. With NetFlow, the amount of infor- components: (i) Variational Autoencoder (VAE) — an unsuper- mation captured is brought down by orders of magnitude (in vised deep-learning technique for detecting anomalies, and (ii) a comparison to raw packet capture), not only because a NetFlow gradient-based fingerprinting technique for explaining anomalies. -
Optimization and Gradient Descent INFO-4604, Applied Machine Learning University of Colorado Boulder
Optimization and Gradient Descent INFO-4604, Applied Machine Learning University of Colorado Boulder September 11, 2018 Prof. Michael Paul Prediction Functions Remember: a prediction function is the function that predicts what the output should be, given the input. Prediction Functions Linear regression: f(x) = wTx + b Linear classification (perceptron): f(x) = 1, wTx + b ≥ 0 -1, wTx + b < 0 Need to learn what w should be! Learning Parameters Goal is to learn to minimize error • Ideally: true error • Instead: training error The loss function gives the training error when using parameters w, denoted L(w). • Also called cost function • More general: objective function (in general objective could be to minimize or maximize; with loss/cost functions, we want to minimize) Learning Parameters Goal is to minimize loss function. How do we minimize a function? Let’s review some math. Rate of Change The slope of a line is also called the rate of change of the line. y = ½x + 1 “rise” “run” Rate of Change For nonlinear functions, the “rise over run” formula gives you the average rate of change between two points Average slope from x=-1 to x=0 is: 2 f(x) = x -1 Rate of Change There is also a concept of rate of change at individual points (rather than two points) Slope at x=-1 is: f(x) = x2 -2 Rate of Change The slope at a point is called the derivative at that point Intuition: f(x) = x2 Measure the slope between two points that are really close together Rate of Change The slope at a point is called the derivative at that point Intuition: Measure the -
Maximum Likelihood from Incomplete Data Via the EM Algorithm A. P. Dempster; N. M. Laird; D. B. Rubin Journal of the Royal Stati
Maximum Likelihood from Incomplete Data via the EM Algorithm A. P. Dempster; N. M. Laird; D. B. Rubin Journal of the Royal Statistical Society. Series B (Methodological), Vol. 39, No. 1. (1977), pp. 1-38. Stable URL: http://links.jstor.org/sici?sici=0035-9246%281977%2939%3A1%3C1%3AMLFIDV%3E2.0.CO%3B2-Z Journal of the Royal Statistical Society. Series B (Methodological) is currently published by Royal Statistical Society. Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/about/terms.html. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/journals/rss.html. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is an independent not-for-profit organization dedicated to and preserving a digital archive of scholarly journals. For more information regarding JSTOR, please contact [email protected]. http://www.jstor.org Fri Apr 6 01:07:17 2007 Maximum Likelihood from Incomplete Data via the EM Algorithm By A. P. DEMPSTER,N. M. LAIRDand D. B. RDIN Harvard University and Educational Testing Service [Read before the ROYALSTATISTICAL SOCIETY at a meeting organized by the RESEARCH SECTIONon Wednesday, December 8th, 1976, Professor S. -
Gradient Descent (PDF)
18.657: Mathematics of Machine Learning Lecturer: Philippe Rigollet Lecture 11 Scribe: Kevin Li Oct. 14, 2015 2. CONVEX OPTIMIZATION FOR MACHINE LEARNING In this lecture, we will cover the basics of convex optimization as it applies to machine learning. There is much more to this topic than will be covered in this class so you may be interested in the following books. Convex Optimization by Boyd and Vandenberghe Lecture notes on Convex Optimization by Nesterov Convex Optimization: Algorithms and Complexity by Bubeck Online Convex Optimization by Hazan The last two are drafts and can be obtained online. 2.1 Convex Problems A convex problem is an optimization problem of the form min f(x) where f andC are x2C convex. First, we will debunk the idea that convex problems are easy by showing that virtually all optimization problems can be written as a convex problem. We can rewrite an optimization problem as follows. min f(x) , min t , min t X 2X t≥f(x);x2X (x;t)2epi(f) where the epigraph of a function is defined by epi(f)=f(x; t) 2X 2 ×IR : t ≥f(x)g Figure 1: An example of an epigraph. Source: https://en.wikipedia.org/wiki/Epigraph_(mathematics) 1 Now we observe that for linear functions, min c>x = min c>x x2D x2conv(D) where the convex hull is defined N N X X conv(D) = fy : 9 N 2 Z+; x1; : : : ; xN 2 D; αi ≥ 0; αi = 1; y = αixig i=1 i=1 To prove this, we know that the left side is a least as big as the right side since D ⊂ conv(D).