Science, Irving K. Barber Faculty of (Okanagan)
Computer Science, Mathematics, Physics and Statistics, Department of (Okanagan)
DSpace
UBCO
Sarkar, Debangsha Kusum
Master of Science - MSc
University of British Columbia
Active learning solves machine learning problems where acquiring labels for the data is costly. A representative subset helps active learning by selecting the most useful subset of a complete data set and querying the labels of the examples present in it. Various sampling strategies exist for the construction of this type of representative subset. However, the state-of-the-art approaches for subset sampling aim to construct a representative subset where the cost of optimization on the subset is comparable to that of the complete data set. They do not attempt to capture the underlying distribution of the data, which we believe is the most important part of representative sampling. This thesis proposes an adaptation of the sigma point sampling (SPS) technique from unscented transformation (UT) for subset sampling. Unscented transformation (UT) has shown to be very effective in non-linear transformation modeling in object tracking and robotics. We show in this thesis that when combined with the Gaussian mixture model, sigma points can estimate the true statistics of an unknown distribution of data with very few samples. The sigma point sampling approach being parameterized gives better control over the sampling process than the non-parameterized sampling techniques. We use the representative subset for active learning on an existing data set. As a practical application of this novel sampling technique, we use it for Active Transfer Learning (ATL) on autoclave processing data to optimize the manufacturing of high-quality aerospace composite parts. The sampling technique is tested on both classification and regression problems in a pool-based active learning scenario. After comparing our approach to the state-of-the-art sampling techniques, We show that sigma point sampling either outperforms or matches their performance. We also test the sensitivity of the parameters in the sigma point sampling. The success of this sampling technique is very significant for future developments of representative sampling on big data sets. The application of our representative subset is not limited to active learning and transfer learning. There is further potential to extend this approach to parallelizing computation on big data sets among multiple clusters.
Improved Sampling Strategy forRepresentative Set ConstructionbyDebangsha Kusum Sarkar,MDS., The University of British Columbia, 2019A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF SCIENCEinTHE COLLEGE OF GRADUATE STUDIES(Computer Science)THE UNIVERSITY OF BRITISH COLUMBIA(Okanagan)January 2022© Debangsha Kusum Sarkar, 2022The following individuals certify that they have read, and recommendto the College of Graduate Studies for acceptance, a thesis/dissertation en-titled:Improved Sampling Strategy for Representative Set Constructionsubmitted by Debangsha Kusum Sarkar in partial fulfillment of the re-quirements of the degree of Master of Science.Dr. Apurva Narayan, Irving K. Barber Faculty of ScienceSupervisorDr. Ramon Lawrence, Irving K. Barber Faculty of ScienceSupervisory Committee MemberDr. Jeffrey Andrews, Irving K. Barber Faculty of ScienceSupervisory Committee MemberDr. Mahmudur Fatmi, School of EngineeringUniversity Examinerii“The great thing about predictions is that you can be wrong.”— Dr. Chris Wiggins (Columbia University)iiAbstractActive learning solves machine learning problems where acquiring labels forthe data is costly. A representative subset helps active learning by selectingthe most useful subset of a complete data set and querying the labels of theexamples present in it. Various sampling strategies exist for the construc-tion of this type of representative subset. However, the state-of-the-art ap-proaches for subset sampling aim to construct a representative subset wherethe cost of optimization on the subset is comparable to that of the com-plete data set. They do not attempt to capture the underlying distributionof the data, which we believe is the most important part of representativesampling. This thesis proposes an adaptation of the sigma point sampling(SPS) technique from unscented transformation (UT) for subset sampling.Unscented transformation (UT) has shown to be very effective in non-lineartransformation modeling in object tracking and robotics.We show in this thesis that when combined with the Gaussian mixturemodel, sigma points can estimate the true statistics of an unknown distri-bution of data with very few samples. The sigma point sampling approachbeing parameterized, gives better control over the sampling process than thenon-parameterized sampling techniques. We use the representative subsetfor active learning on an existing data set. As a practical application ofthis novel sampling technique, we use it for Active Transfer Learning (ATL)on autoclave processing data to optimize the manufacturing of high-qualityaerospace composite parts. The sampling technique is tested on both clas-sification and regression problems in a pool-based active learning scenario.After comparing our approach to the state-of-the-art sampling tech-niques, We show that sigma point sampling either outperforms or matchestheir performance. We also test the sensitivity of the parameters in thesigma point sampling. The success of this sampling technique is very signif-icant for future developments of representative sampling on big data sets.The application of our representative subset is not limited to active learningand transfer learning. There is further potential to extend this approach toparallelizing computation on big data sets among multiple clusters.iiiLay SummaryTraditional machine learning models are data-hungry. Training supervisedmodels require a massive amount of labeled data. However, in some indus-tries acquiring labels might be costly even though unlabeled data is cheaplyavailable. Active learning solves this problem by querying the labels ofthe most informative unlabeled training examples and training a supervisedmodel with a small amount of data. In this thesis, we study the construc-tion of representative subsets which can be used to train an active learningmodel. A representative subset is a summarized version of the completedata set. We propose a novel approach for representative subset construc-tion that preserves the local data distribution in a data set. Our approachgenerates sigma points, and they are calculated based on the local statisticsof the data set, which is acquired through Gaussian mixtures. We train theactive learner on the sigma points instead of the data points.ivPrefaceThe experiments in the thesis outlined in Chapter 5 were conducted by theauthor Debangsha Sarkar. The experiments in section 5.1 were designed bythe author. The data used in this experimental setup was prepared by theauthor.The experimental design described in section 5.2 was adopted from Ramezankhaniet al. [43]. Milad Ramezankhani helped the author in the preparation ofthe composite manufacturing data set used in this section. A detailed de-scription of the data set is in Chapter 4.Finally, all the experimental plots and figures, manuscript compositionwere done by the author under the careful supervision of the thesis supervisorDr. Apurva Narayan.vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixList of Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiList of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . xiiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . xivDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Passive vs Active Machine Learning . . . . . . . . . . . . . . 21.1.1 Passive Learning . . . . . . . . . . . . . . . . . . . . . 21.1.2 Active Learning . . . . . . . . . . . . . . . . . . . . . . 41.2 Scalability in Unsupervised Learning . . . . . . . . . . . . . . 51.3 Thesis Proposal . . . . . . . . . . . . . . . . . . . . . . . . . . 51.4 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . 82 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.1 Problem Statement in Pool-based Active Learning . . . . . . 92.2 Uncertainty Sampling . . . . . . . . . . . . . . . . . . . . . . 92.3 Query-By-Committee . . . . . . . . . . . . . . . . . . . . . . 10viTABLE OF CONTENTS2.4 Representative Sampling . . . . . . . . . . . . . . . . . . . . . 112.4.1 Coreset . . . . . . . . . . . . . . . . . . . . . . . . . . 112.4.2 K-means Based Coreset . . . . . . . . . . . . . . . . . 122.5 Unscented Transform . . . . . . . . . . . . . . . . . . . . . . . 133 Sigma Point Sampling, a Strategy for Improved ActiveLearning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.1 Variational Autoencoder (VAE) . . . . . . . . . . . . . . . . . 153.2 Sigma Point Sampling on Latent Space . . . . . . . . . . . . . 174 Applications of Sigma Point Sampling in Industry 4.0 . . . 214.1 Transfer Learning on Composites Manufacturing Data . . . . 224.1.1 Transfer Learning with Sigma Point Sampling . . . . . 244.1.2 Composite Material data set . . . . . . . . . . . . . . 254.2 Regression with Sigma Point Sampling on Composites Man-ufacturing Data . . . . . . . . . . . . . . . . . . . . . . . . . . 275 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285.1 Sigma Point Sampling on MNIST data set . . . . . . . . . . . 285.1.1 Experimental Setup . . . . . . . . . . . . . . . . . . . 285.1.2 Experimental Results . . . . . . . . . . . . . . . . . . 335.1.3 Comparison of different sampling techniques . . . . . . 335.1.4 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . 345.2 Transfer Learning with Sigma Point Sampling on CompositeManufacturing data set . . . . . . . . . . . . . . . . . . . . . 415.2.1 Experimental results . . . . . . . . . . . . . . . . . . . 425.3 Regression Learning with Sigma Point Sampling on Compos-ite Manufacturing data set . . . . . . . . . . . . . . . . . . . . 475.3.1 Experimental results . . . . . . . . . . . . . . . . . . . 486 Conclusion and Future Work . . . . . . . . . . . . . . . . . . . 49Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51viiList of Tables4.1 Observed variables used in predictive modeling of compositematerials data set. . . . . . . . . . . . . . . . . . . . . . . . . 265.1 Labels used for classification problems with their correspond-ing data set sizes used for training and testing. . . . . . . . . 295.2 Labels used for classification problems with their correspond-ing data set sizes used for training and testing. . . . . . . . . 47viiiList of Figures1.1 Sub-figure (a) shows unlabeled data. (b) shows a rough es-timate of the groups present in the data achieved through avery efficient algorithm e.g. initialization step of k-means++.(c) shows the selected samples from each group . . . . . . . . 62.1 Efficient prediction of unknown distribution of a non-linearfunction y = f(x) using Unscented Transform. For 2-dimensionaldata it uses (2× 2 + 1) = 5 sigma points in x [63]. . . . . . . 143.1 The data set distribution D lies in a high dimensional x-spacewhile the latent space is relatively simpler and low dimen-sional (figure shows a spherical latent space to emphasize thesimplicity of latent space and the complexity of the observedvariable space). The generator model estimates the joint dis-tribution pθ(x, z), which for a isotropic Gaussian distributioncan be factorized into pθ(x, z) = pθ(z)pθ(x|z). pθ(z) is theprior distribution which we assume to be pθ(z) ∼ N(z; 0, I)and pθ(x|z) is the stochastic decoder. The stochastic encoderqφ(z|x) approximates the true posterior distribution pθ(z|x).[26] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.2 Representative querying using sigma point sampling. Sigmapoints X are highlighted in red. xj ∈ U are unlabeled datain the encoded latent space. A Sigma point takes label fromtheir nearest neighbor.The (Xj , yj) pair is added to the la-beled pool for training active learner. . . . . . . . . . . . . . . 184.1 Autoclave curing process in aerospace manufacturing. [43] . . 224.2 Transfer Learning with Sigma Point Sampling. . . . . . . . . 244.3 Two-step autoclave cure cycle.[45] . . . . . . . . . . . . . . . 265.1 Accuracy comparison of sigma point sampling vs. other rep-resentative sampling methods . . . . . . . . . . . . . . . . . . 30ixLIST OF FIGURES5.2 Accuracy comparison of sigma point sampling vs. maximumuncertainty sampling, margin sampling and entropy sampling 315.3 Accuracy comparison of sigma point sampling vs. consensusentropy sampling, maximum disagreement sampling and voteentropy sampling of Query by committee. . . . . . . . . . . . 325.4 Effect of α on the spread of sigma points. As the value ofα increases the sigma points spread more across a Gaussiandistribution. The spread of the sigma points affects theirrepresentative quality. . . . . . . . . . . . . . . . . . . . . . . 355.5 Spread of the sigma points in the 2 dimensional latent spaceof MNIST data set (with digits 0,3, and 9) for different α values. 365.6 Spread of the sigma points inside each Gaussian mixturecomponent and throughout the 2 dimensional latent spaceof MNIST data set (with digits 0,3, and 9) for different αvalues. In the regions where the latent space data is sparsethe Gaussian components have higher covariance. . . . . . . . 375.7 Representation of the 2 dimensional latent space of MNISTdata set (with digits 0,3, and 9) with the sigma points fordifferent α values. In the regions where the latent space datais sparse, the higher covariance of the Gaussian componentscauses the sigma points to be more spread out than the denseregions. By changing the value of α, which is a scaled covari-ance parameter for sigma points, we can control the spread ofthe sigma points inside the Gaussian components even furtherto get a satisfactory representation. . . . . . . . . . . . . . . . 385.8 Effect of α on multi-class classification accuracy . . . . . . . . 395.9 Comparison of performance metrics with increasing numberof training examples in AS4/8552 for SPS vs. random sampling 425.10 Comparison of accuracy in Random vs Transfer Learning(TL) vs Active Learning (AL) vs ATL on 20 points . . . . . . 445.11 Comparison of accuracy in Random vs TL vs AL vs ATL on100 points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455.12 Comparison of accuracy in Random vs TL vs AL vs ATL on200 points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465.13 Performance metrics Mean Absolute Error (MAE), Mean SquaredError (MSE) andR2 Score on test data set for Regression withSigma Point Sampling (SPS) . . . . . . . . . . . . . . . . . . 48xList of Equations1.1 Lower Bound in PAC Learning . . . . . . . . . . . . . . . . . . . . 32.1 Least Confident Sampling . . . . . . . . . . . . . . . . . . . . . . . 102.2 Margin Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.3 Entropy Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.4 Voting Entropy in Query by Committee (QBC) . . . . . . . . . . . 102.5 Cost of Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . 112.6 Coreset Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.7 Cost of Optimization in Uniform Sampling on Complete data set . 122.8 Cost of Optimization in Uniform Sampling on Coreset . . . . . . . 122.9 Sigma Point Calculation . . . . . . . . . . . . . . . . . . . . . . . . 133.1 Variational Lower Bound in Variational Auto Encoder (VAE) . . . 173.2 Mixture of Gaussians . . . . . . . . . . . . . . . . . . . . . . . . . . 183.3 Sigma Point Calculation with Gaussian Mixture Models . . . . . . 19xiList of AbbreviationsAL Active Learning. x, xi, 4, 44–47, 49ATL Active Transfer Learning. iii, x, xi, 42–47, 49, 50ELBO Evidence Lower Bound. xi, 15GMM Gaussian Mixture Model. xi, 5, 7, 17, 30, 34, 49, 50HTC Heat Transfer Co-efficient. xi, 25, 26KL Kullback Leibler. xi, 10KLD Kullback Leibler Divergence. xiMAE Mean Absolute Error. x, xi, 27, 47, 48ML Machine Learning. xi, 1, 50MRCC Manufacturer’s Recommended Curing Cycle. xi, 21, 27MSE Mean Squared Error. x, xi, 27, 47, 48OVR One Versus Rest. xi, 28PAC Probably Approximately Correct. xi, 3, 12, 50PDF Probability Density Function. xi, 15QBC Query by Committee. xi, 10, 32, 33RBF Radial Basis Function. xi, 28ReLU Rectified Linear Unit. xi, 29xiiList of AbbreviationsSELU Scaled Exponential Linear Unit. xi, 41SPKF Sigma Point Kalman Filters. xi, 13SPS Sigma Point Sampling. x, xi, 2, 28, 42, 43, 48–50SPSATL Sigma Point Sampling based Active Transfer Learning. xi, 49SVM Support Vector Machine. xi, 28, 29, 31TL Transfer Learning. x, xi, 22, 24, 25, 44–47, 49, 50UKF Unscented Kalman Filters. xi, 7, 13, 49UT Unscented Transformation. xi, 13VAE Variational Auto Encoder. xi, 12, 15, 17, 18, 29, 50xiiiAcknowledgementsI would like to sincerely acknowledge the endless patience of my supervisor,Dr. Apurva Narayan. I am grateful to him for always motivating me towork smarter especially when I made clumsy mistakes and providing mewith numerous research opportunities. I would like to thank Dr. RamonLawrence for giving me the opportunity to be a part of this wonderful aca-demic journey.I enjoyed working with other students on many projects and was able tolearn a lot from them. I would like to thank Milad Ramezankhani from theMechanical Engineering department, for providing me with the industrialdata set for my research and guiding me through the composite materialmanufacturing process. I also appreciate Pradeep Mahato for helping mewith many aspects of research.All the experiments in my research was made possible by the computa-tional resources provided by UBC ARC and Compute Canada.xivDedicationI would like to dedicate this work to my parents who were extremely under-standing and always motivated me to pursue my dreams.xvChapter 1IntroductionMachine Learning (ML) in computer science is the study of computer pro-grams that automatically learn from examples and use their learnt experi-ence to improve their performance in specific tasks. In the last decade, wehave seen an explosion of machine learning applications in different problemsof computer science and other disciplines of research. Machine learning hasbeen successful in risk analysis in financial sectors [9, 15, 19], disease diag-nosis in health-care [32, 55], quality control in manufacturing [39, 69], andmany more. There are three distinct categories of machine learning meth-ods, namely supervised learning, unsupervised learning and reinforcementlearning.• Supervised learning: In this learning category, d-dimensional datax ∈ Rd and its corresponding labels y is fed to the learner as pair(xi, yi)Ni=1. The pairs are used to train the learner and predict thelabels for new data. If the labels y are real numbers the learning taskis called regression and if the labels y are ordinal the task is calledclassification.• Unsupervised learning: In this learning category, only d-dimensionaldata x ∈ Rd is fed to the learner without the labels. The unlabeleddata is used to find the groups in the data. Clustering is one applica-tion of unsupervised learning.• Reinforcement learning: In this machine learning category, an in-telligent agent receives a sequence of acceptable states and its corre-sponding rewards (st, rt) for each time step t. The learner predictsa policy pi(a, s) = P (at|st) that moves the state to st+1 by an actionat and a reward rt+1 associated with the transition (st, at, st+1). Thegoal of the learner is to predict a policy that maximizes the cumulativereward over time [22]. Reinforcement learning does not require labeleddata.The machine learning systems are set to optimize one or more specificobjective(s) by learning from data ingested into it. Hence, these systems11.1. Passive vs Active Machine Learningare called data-driven systems [42, 59]. In recent years, as opposed to hand-crafted rule-based expert systems, machine learning models have shown tobe more effective and more efficient in most cases [37, 49]. However, onecaveat is the amount of data machine learning models require to achievesuch performance.In the current times, data collection is effortless in most industries, anddata storage is very cheap. However, data labelling is still costly and time-consuming. For example, labelling medical images for diagnosis requiresmedical professionals to hand label images. Labeling industrial sensor datafor fault detection requires passing the data through costly software [58].This restricts the availability of the labelled data for machine learning appli-cations. Apart from the cost associated with labeling the data, training bothsupervised and unsupervised models on massive data sets is time-consumingand computationally expensive. It means that neither supervised nor unsu-pervised learning algorithms are scalable when applied on a huge number ofexamples, even though a considerably large data set is necessary for betterlearning. To solve this problem, in this thesis, we propose a novel data sum-marization technique called the SPS, which can help in training the machinelearners with very few examples and still achieve significant performance.1.1 Passive vs Active Machine Learning1.1.1 Passive LearningComputation learning theory deals with the general principles that dictatethe final hypothesis space proposed by the machine learning models. It theo-rizes the characteristics of machine learners to measure the quality of learntexperience from examples. These characteristics include the convergenceof the learner to the correct hypothesis, the sample complexity (number ofexamples) and computation complexity required to learn a concept class(the distribution function associated with a particular class). If we considera traditional supervised learning task, labeled samples (xi, yi)Ni=1 with iidxi ∼ D and yi = c∗(xi) are given, where D is the unknown distribution ofthe data and C∗ is the unknown concept class which maps all the yis to thexis. We can frame this task as a computation learning theoretic problemwhere the goal of the learner is to approximate the unknown distributionD and output a hypothesis h : X → y for a collection of unseen data Xthat approximates the concept class c∗. The error of hypothesis is the mis-match between the output hypothesis and the concept class when they areapplied on the input data xi. For a predicted hypothesis h ∈ H, where21.1. Passive vs Active Machine LearningH is all the possible hypotheses, the error of hypothesis is the probabilityP (h(xi) 6= c∗(xi)|i ∈ N). The objective of the learner is to minimize theerror of hypothesis h for all examples i.e. minx L = P (h(x) 6= c∗(x)|i ∈ N).Since the learner approximates the concept class and error of hypothesis forthe unseen data, the learner is called approximately correct if Li ≤  onan example for some  > 0. One assumption here is that the train and testexamples are chosen from the same independent and identical distribution(iid). Considering this assumption that the examples are randomly chosenfrom iid, there might be some cases where the hypothesis will fail to satisfythe condition of being approximately correct in other words in some casesthe hypothesis will fail to be consistent with the examples. For a parameter0 < δ < 1, if the hypothesis is approximately correct (the hypothesis errorL < ) at least 1− δ fraction of times (or approximately incorrect at most δfraction of times), the hypothesis is called probably approximately cor-rect. In Probably Approximately Correct (PAC) learning [62] approach,the machine learner requires a lot of training examples to output a consis-tent hypothesis. In fact, the total number of training examples requiredfor a hypothesis to be consistent with an error  and probability (1 − δ) isexpressed in equation 1.1 [21]:m ≥ 1(ln|H|+ ln1δ)(1.1)The number of examples required (m) to guarantee consistent hypothesisis called the sample complexity. Haussler et al. [21] explains several prob-lems with PAC learning. For example, a PAC learning model to convergeto a correct hypothesis requires the examples to be noise-free. The additionof noisy data hinders the capability of the learner to discriminate betweencorrect hypotheses and incorrect hypotheses. The author also mentions thatcalculating the sample complexity for even a relatively simple learning algo-rithm is very difficult because the total number of hypotheses |H| is difficultto estimate. In such cases, we are forced to insert the upper bound of hy-pothesis space in equation (1.1) which inflates the sampling complexity ofthe learner. We can underline two significant drawbacks of PAC learningconcerning sampling complexity then,• For most learners, the calculated sampling complexity overestimatesthe error of hypothesis [21].• In real data sets, noisy data and randomly choosing noisy examplesfor training hinder learning.31.1. Passive vs Active Machine LearningThis type of learning is called passive learning because the machinelearner is ignorant of the informativeness [51] of the examples. Settles etal. [50] point out that we can find redundancies within each class in mas-sive data sets, and due to the presence of noisy data, all examples are notrepresentative of a particular class. It is important to sample only the mostrepresentative and informative examples from each class.1.1.2 Active LearningAL [8, 10, 20, 46, 50, 57] proposes a machine learning technique where thelearner intelligently enquires the usefulness of the examples for learning. Inactive learning, the learner asks intelligent questions to sample the mostuseful examples from the complete data set. By removing noisy examplesfrom the training process and including only the informative examples themachine learner can learn as good as in passive learning but with less train-ing data. AL is advantageous in scenarios where labeling the data is costly,either due to financial restrictions or lack of access to the experts or re-sources.The setup of active learning is based upon the ability of a machine learnerto query the labels of examples it decides to include in the training set. Theassumption is that the unlabeled examples are either cost free or very easyto acquire. The labeling system of the selected samples take the form of anannotator (either human or computer) also known as an Oracle, providinglabels for the unlabeled data.Different settings of active learning can be formed to allow the learner toask intelligent questions. The two most common settings are stream-basedselective sampling and pool-based active learning. In stream-based selectivesampling one unlabeled example is sampled at each iteration and the ma-chine learner decides either to query its label or to reject it. The decisionis made based on query strategies like setting a threshold on the informa-tiveness measure. In pool-based active learning, all the unlabeled instancesare collected at the beginning of the task. The pool is assumed to be staticthat is new unlabeled data are not introduced to the pool after the activelearning begins. Although this is not a necessary restriction. In pool-basedsampling, the learner ranks all the unlabeled instances according to someinformativeness measure and samples the ones with the best scores. In thisthesis, all the experiments are formed in a pool-based active learning setup.The crux of active learning is the query strategy or sampling method thathelps to train a machine learner with very few examples. Various samplingtechniques have been proposed for active learning. We discuss them in de-41.2. Scalability in Unsupervised Learningtail in Chapter 2. In this thesis, we use use a novel representative samplingtechnique called Sigma point sampling for extracting the informative exam-ples from the unlabeled pool. we later use the extracted subset for trainingthe active learner.1.2 Scalability in Unsupervised LearningUnsupervised learning such as clustering, Gaussian Mixture Model (GMM)are notoriously unscalable for high dimensional massive data sets. The clus-tering algorithms are mostly O(N2) on the size of the data set. GMM has atime complexity of O(NKD3) [40], where N is the size of the data set, K isthe number of components and D is the dimension of the data. The cubiccomplexity on the dimension comes from the co-variance matrix inversionin GMM. Due to this polynomial dependency on either the dimension ofthe data or size of the data set, these algorithms are very inefficient. Oneof the proposed methods to solve this problem is extracting a meaningfulsubset of the complete data set that summarizes it [3, 5–7, 16, 17, 35]. Therepresentative subset is called a coreset if it approximates the loss on thecomplete data set. Coresets have been proven to be very effective in scalingunsupervised learning algorithms [7].In this thesis we solve this problem as follows:• We propose a novel technique for representative subset constructionthat captures the underlying distribution of the data set from a latentspace and can be scaled on massive high-dimensional data sets.• The proposed sampling technique is very easy to implement and thesampled subset is a very good representative of the complete data setwhich can be used for further machine learning.1.3 Thesis ProposalIn this thesis, we dive deeper into the existing research and propose a poten-tially better approach for representative subsetting. We also discuss how therepresentative subset constructed using our approach improves active learn-ing. The objective of active learning is to train a machine learning modelusing a subset of the most informative labeled data points from a data setthat comprises of a very high number of examples. It addresses the issuethat a budget constraint might limit acquiring labels for the data. Also, evenif collecting unlabeled data is easy, the unlabeled data set might consist of51.3. Thesis Proposalexamples that are either redundant or noisy. Active learning model shouldbe able to avoid those examples by using an effective query algorithm.(a) Unlabeled data (b) Grouped unlabeled data(c) Selected samplesFigure 1.1: Sub-figure (a) shows unlabeled data. (b) shows a rough estimateof the groups present in the data achieved through a very efficient algorithme.g. initialization step of k-means++. (c) shows the selected samples fromeach groupThe goal of representative subset is to represent a complete data setusing a smaller set (in some cases a subset) of points that summarizes thecomplete data set. A na¨ıve approach to constructing a representative sub-set is randomly selecting the members. Noisy training data in the randomlysampled subset is a bottleneck for achieving optimum performance for amachine learner. For random selection, the minimum coreset size requiredto achieve satisfactory result can be calculated as equation (1.1). Figure 1.1shows a better approach to this problem. In this sub-sampling technique thecomplete data set is partitioned into clusters and some of the examples are61.3. Thesis Proposalsampled from each cluster based on pre-defined sampling strategy. Pourka-mali et al. [41] use a similar approach to prove that the subset summarizesthe complete data set well and can be used for active learning. Bachem et al.[6] on the other hand propose another approach of representative samplingcalled coreset constriction that summarizes massive unlabeled data sets forscaling unsupervised learning which sometimes can be computationally ex-pensive (Section 1.2).Even though the existing representative sampling techniques are usefulfor effective representative subset generation, none of them aim to capturethe underlying distribution of the data set in the representative subset. [41]uses K-means to distribute the representative subset candidates through-out the data set proportionally. [5],[6] constructs the coreset (representativesubset) in such a way that it approximates the cost of optimization of thecomplete unlabeled data set in the subset. These representative subset con-struction techniques query the best candidates and add them to a subset.They do not take into consideration that adding data directly from the un-labeled pool might make the representative subset susceptible to noise.This thesis proposes a very effective sub-sampling technique that approx-imates the underlying data distribution of the complete data set. Sigmapoints selection is a very popular tool used in Unscented Kalman Filters(UKF) [63]. It has shown to be very effective in capturing the statisticsof the distribution that goes through non-linear transformations in estima-tion theory in different applications, such as object tracking and navigation[61, 65]. [33, 60, 66] use sigma points in UKF for the estimation of theunknown state-space in deep learning. UKF, which is a derivative-free ap-proach, is very easy to compute and it is more effective than other Kalmanfilters.In this thesis, we combined sigma point sampling with GMM to estimatethe priors of the data set and divide the data into soft clusters. We use thesoft-clusters to estimate the density and distribution of the data and use thedistribution parameters to compute the sigma points. In our research, thecollection of sigma points act as the representative subset of the data set.Finally, we use the coreset in a pool-based active learning scenario. Thebenefits of using sigma point sampling over other representative samplingapproaches are as follows:• Firstly, in our approach, the representative subset approximates thedistribution of the complete data set with very few data points. Weempirically show that approximating the distribution is a better ap-proach than the existing techniques and leads to better accuracy.71.4. Thesis Outline• Secondly, we use the sigma points instead of the actual data to trainour active learner, which reduces the risk of including the inherentnoise from the data points themselves.• Thirdly, the sigma point sampling approach is parameterized whichadds more control to the user while constructing the representativesubset.In the upcoming chapters, this thesis discusses in detail the followinghypotheses:• Sigma point sampling is a novel framework for representative subsetconstruction that estimates the underlying distribution of the completedata set.• Sigma point representative subset outperforms the other representa-tive sampling approaches by measuring and comparing their perfor-mance in active learning.• Sigma point representative subset can be used along with a dimen-sionality reduction tool to implement it on massive high dimensionaldata sets.• Sigma point sampling can be used in both classification and regression,and it overcomes some of the challenges of the existing active learningframeworks.1.4 Thesis OutlineThis thesis is organized as follows: Chapter 2 describes the background ofsigma point sampling and active learning. The first section of Chapter 3discusses the application of sigma point sampling in classification of hand-written digits. The second section of Chapter 3 shows an application inregression. Chapter 4 delves into the details of using sigma point samplingwith transfer learning. Chapter 5 discusses the experiments to show thevalidity of the hypotheses. Finally, Chapter 6 summarizes the findings ofmy research and concludes the thesis.8Chapter 2BackgroundIn this chapter, we first discuss the different sampling techniques in activelearning. Then we discuss the working of unscented transformation. Wecompare the sigma point sampling technique with the sampling strategiesdiscussed in this chapter. Chapter 5 discusses the comparison of differentsampling techniques with our sigma point sampling graphically. Becauseour representative sampling strategy is not coreset based sampling, we com-pared it with other representative sampling strategies as well as uncertaintysampling and query-by-committee which are good choices of sampling inactive learning.2.1 Problem Statement in Pool-based ActiveLearningSettles et al. [51] summarize the pool-based active learning setup, wherecollecting unlabeled data is easy, but labeling them is costly. Pool-basedactive learning starts with collecting all the unlabeled input data to createthe unlabeled pool U = {xi}ui=1 and a labeled pool L = {xi, yi}li=1 where(u + l) is the total number of input instances, and yi is the label for xi.The learner outputs a hypothesis h0 : X→ y by training on the latest L atthe first iteration. In the next step, the learner asks the label for the mostinformative instance x∗ ∈ U and adds the (x∗, y∗) pair to L. The learnertrains on the new labeled space L + (x∗, y∗) and outputs a new hypothesish1 : X → y in the second iteration. This iterative process continues untilthe learner exhausts the labeling budget B.2.2 Uncertainty Sampling[11, 30, 31] use uncertainty sampling to choose points closest to the decisionboundary of a binary classifier carrying the lowest confidence from the unla-beled pool. In this pool-based query strategy the learner ranks the examplesbased on the class confidence and the examples with confidence closest to92.3. Query-By-Committee0.5 are queried. Equation (2.1) shows the sampling strategy.x∗ = argmaxx(1− Pθ(yˆ|x))(2.1)where, yˆ = argmaxyPθ(y|x) and x∗ is the query strategy.Scheffer et al. [48] propose a multi-class variant of least-confident sam-pling called margin sampling, which selects points with minimum probabilitydifference between the most and second most probable classes(yˆ1 and yˆ2 re-spectively) to train the classifier. Equation (2.2) shows the mathematicalformat of margin sampling.x∗ = argminx(Pθ(yˆ1|x)− Pθ(yˆ2|x))(2.2)Entropy sampling [52, 53] is a generalization of uncertainty samplingstrategies. In this sampling approach, entropy is calculated for each examplein the unlabeled pool in each iteration for all possible labels, and the examplewith the highest entropy is added to the labeled pool. Equation (2.3) showsthe entropy sampling.x∗ = argmaxx−∑iPθ(yi|x) logPθ(yi|x) (2.3)2.3 Query-By-CommitteeSeung et al. [54] propose QBC which works by creating a committee ofmodels C = {θ(1), θ(2), ·, θ(k)} and chooses samples from U on which thecommittee members disagree the most. Different strategies can be used toquantify the degree of disagreement, e.g., Dagan et al. [12] propose voteentropy sampling to train a Hidden Markov Model classifier for parts-of-speech tagging. Mccallumzy et al. [36] use Kullback Leibler (KL) divergenceto measure member disagreements for text classification. For a samples x,all possible labels yi = {y1, y2, ·, yN} and a total of k committee membersthe query strategy can be expressed as equation (2.4).V E = argmaxx−∑iV (yi)klogV (yi)k(2.4)where, V (yi) is the total votes for label yi.102.4. Representative Sampling2.4 Representative SamplingIn representative sampling, a subset of the complete data set is used for ac-tive learning. The subset approximates the complete data set based on somecriteria. Usually, the general goal of representative sampling is to includeas much versatile examples as possible from the complete data set withoutbeing redundant. Several representative sampling exists in the literatureand the next subsections discuss two of them.2.4.1 CoresetA coreset is a computational geometric approach [1] that was introducedas an approximation technique to extract a small most useful (core) point-set C, from a collection of points D such that the result from an expensivealgorithm applied on C can be translated to D. Most of the applicationsof coreset sampling in machine learning are either for scaling clustering al-gorithms on large data sets or parallelizing the computation of clusteringalgorithms [3–5, 7, 16, 18, 34]. In that sense, coreset is a representativesubset or a summary of a data set that can be further used for inference.However, the interpretation of a coreset and the techniques used for coresetsampling vary from one task to another. Bachem et al.[6] formulate thecreation of a coreset in machine learning as an optimization problem. Theyassumed that the input data x ∈ X where X ∈ Rn and their correspondingweights are given. For all possible solutions θ ∈ Θ, the cost of optimizationon the complete data set is,cost(X, θ) =∑x∈Xw(x)l(x, θ) (2.5)where, l(x, θ) is a additive loss function used in any machine learning ob-jective such as logistic regression, support vector machine, linear regression,k-means clustering etc.A weighted subset C of the original data set is called a -coreset of X ifthe following holds:|cost(X, θ)− cost(C, θ)| ≤ εcost(X, θ) (2.6)Since, achieving the optimal solution is difficult, any solution θ ∈ Θis queried whose cost of optimization cost(C, θ) falls within the (1 ± ε) ×cost(X, θ) range is used as a coreset. The simplest approach for coreset112.4. Representative Samplingconstruction is assigning equal weights to all the examples in the data set.For a data set of size M the weights of all examples are wM (x) =1M . Thecost of optimization on complete data set for a query θ ∈ Θ is,cost(X, θ) =∑x∈X1Ml(x, θ) = Ex[l(x, θ)] (2.7)where, x ∈ X is chosen uniformly at random. If a weighted subset C, ofsize m is chosen uniformly at random from X and the weights of the pointsin the subset is equal to wC(x) = 1m , then the cost function cost(C,Θ) is anunbiased estimator of cost(X,Θ) for any θinΘ if the following holds,EC [cost(C,Θ)] =∑x∈C1mEx[l(x, θ)] = cost(X,Θ) (2.8)It can be shown that the variance of cost(C,Θ) is finite, which meansthat when m→∞ it converges to cost(X,Θ). But this convergence makesm impractically large and not any better than the size of training data set(m) in PAC learning (1.1). In order to make the sampling process betterthe authors propose Importance sampling which assigns more weights tothe more important examples. This assignment of non-uniform weights toexamples causes a subset C to estimate the cost of complete data set with lessexamples. This algorithm of coreset construction takes into considerationonly the multiplicative error (ε × cost(X, θ)) as shown in equation (2.6).Bachem et al. [7] improves this approach of coreset construction to includeboth multiplicative and additive error.2.4.2 K-means Based CoresetAnother method to form a representative subset is to add different variationsof samples from the data distribution without being redundant. Pourkamaliet al. [41] shows an example of this method. The authors propose a k-meansbased sub-sampling technique on the latent space of an image data set foractive learning. The authors show that features extracted with VAE can beused for active learning. Their geometric approach is easier to implementin the latent space and scalable on large data sets. The authors map thereal data with dimension D into a lower dimensional space d for all n pointsin the data set Z = {z1, z2, · · ·, zn}∀zi ∈ Rd. Then they partition thelatent variables into K clusters. Given that the sampling budget for activelearning is set to B, m = BK points are sampled uniformly at random from122.5. Unscented Transformeach cluster. This process of coreset construction, minimizes the maximumEuclidean distance between the coreset zj ∈ C and the rest of the exampleszi ∈ Z\C. The authors achieve better test accuracy with the latent variablescompared to the observed variables in active learning.2.5 Unscented TransformUnscented Transformation (UT) [24] is a popular method for estimating theparameters of a system that goes through a non-linear transformation, suchas in integrated navigation and robotics. Sigma points in UT capture theGaussian approximation of the current state and help estimate the distribu-tion parameters of the new state after non-linear transformation. Contraryto traditional probabilistic sampling algorithms such as the Monte Carlosampling method [14], UT estimates an unknown distribution with only(2×N + 1) sigma points for an N -dimensional data set (Figure 2.1). Wanet al. [65] use the unscented kalman fiter UKF, a use case of UT, to estimatethe prior distribution (mean and covariance) of the current state, and afternon-linear transformation, approximate the posterior mean and covarianceaccurately for both Gaussian and non-Gaussian input.Merwe et al. [63] generalize derivativeless and deterministic sampling-based Gaussian approximation filters under Sigma Point Kalman Filters(SPKF). The proposed sigma point calculation in SPKF is a modification ofscaled unscented transform [23] to remove the scaling errors over increasingdimensions. Consider a random variable x of dimension N with mean µx andcovariance Σx, which propagates through a non-linear function y = f(x). Asigma point matrix X is formed with dimension (2N + 1 ×N) to estimatethe statistics of y. Equation (2.9) shows the calculation of sigma points.X0 = µXi = µ+(√(N + λ)Σ)ii = 1, · · ·, NXi = µ−(√(N + λ)Σ)i−N i = (N + 1), · · ·, 2NW(m)0 =λ(N + λ)W(c)0 =λ(N + λ)+ (1− α2 + β)W(m)i =W(c)0 =12(N + λ)i = 1, · · ·, 2N(2.9)132.5. Unscented TransformOne of the sigma points is the mean µ itself, and the rest of the 2Nsigma points spread out around the mean symmetrically using the squareroot of scaled covariance term√(N + λ)Σ, where λ = α2(N + κ)−N . Thecalculation of sigma points is parameterized with α, β, and κ. α is theprimary scaling parameter, and κ is the secondary scaling parameter thatcontrols the spread of the sigma points. In our experiments we vary α forscaling the sigma points. β is responsible for capturing prior distributioninformation. For Gaussian prior β = 2 is recommended by [63].A higher value of α weights the mean µ more than the other sigma pointsand spreads the points further from µ than in the case of smaller α.Figure 2.1: Efficient prediction of unknown distribution of a non-linear func-tion y = f(x) using Unscented Transform. For 2-dimensional data it uses(2× 2 + 1) = 5 sigma points in x [63].This chapter outlines the concepts of active learning and the samplingstrategies used in active learning. We delved into the details of representativesampling and its usage in active learning. We also briefly discussed thebackground of sigma point sampling. In the next chapter, we will discussusage of sigma point sampling in active learning and build a pipeline toimplement it on high dimensional massive data sets.14Chapter 3Sigma Point Sampling, aStrategy for Improved ActiveLearningIn this chapter, we explain the integration of the novel representative sam-pling approach in active learning. Pourkamali et al. [41] show that geometricapproach for sampling works better on the latent space than the observedvariable space. To that end, in my experiments (Chapter 5), we use sigmapoint sampling on the latent space. To generate the latent space, on theMNIST data set [29] we use Variational Autoencoder (VAE) which we ex-plain briefly in the first section of this chapter and on the other data set(Section 4.1.2) we use multi layer perceptron as an encoder.3.1 Variational Autoencoder (VAE)We use VAE to create latent space of image data set because of its ability tolearn non-linear representations of the data. Kingma et al. [25] introducedVAE as a directed probabilistic graphical model that replaces the determin-istic nature of Auto-encoders with stochastic data generation, which is morerobust as a generative model. They solve the intractability problem of themarginal likelihood of data p(x) while calculating the posterior distributionpθ(z|x) by approximating it with a neural network architecture (inferencemodel) with Probability Density Function (PDF) qφ(z|x). Reconstructeddata is generated from z with PDF pθ(x|z) from another neural net archi-tecture (generator model). We can treat z as a hidden state representa-tion of the original data, making the whole architecture autoencoder-likewith qφ(z|x) being stochastic encoder and pθ(x|z) being stochastic decoder.Kingma et al. [26] describes VAE with figure 3.1. Evidence Lower Bound(ELBO) on the marginal likelihood of data is the objective function of VAEand is given by (3.1).153.1. Variational Autoencoder (VAE)Figure 3.1: The data set distribution D lies in a high dimensional x-spacewhile the latent space is relatively simpler and low dimensional (figure showsa spherical latent space to emphasize the simplicity of latent space and thecomplexity of the observed variable space). The generator model estimatesthe joint distribution pθ(x, z), which for a isotropic Gaussian distributioncan be factorized into pθ(x, z) = pθ(z)pθ(x|z). pθ(z) is the prior distribu-tion which we assume to be pθ(z) ∼ N(z; 0, I) and pθ(x|z) is the stochasticdecoder. The stochastic encoder qφ(z|x) approximates the true posteriordistribution pθ(z|x). [26] 163.2. Sigma Point Sampling on Latent SpaceL(θ, φ; x(i)) =−DKL(qφ(z|x(i))||pθ(z))+ Eqφ(z|x(i))[log pθ(x(i)|z)] (3.1)The first term on the right-hand side of equation (3.1) is a regularizerterm that checks whether the approximate posterior distribution of latentvariable z is the same as its prior distribution, i.e., pθ(z). The commonpractice is to draw prior from isotropic Gaussian, pθ(z) ∼ N(z; 0, I) and theoutput of the encoder qφ(z|x(i)) is µ(i) and σ(i). The second expression inequation (3.1) is the reconstruction term and expressed as the log-likelihoodof pθ(x(i)|z) over the posterior distribution of latent variable qφ(z|x(i)). Byoptimizing L(θ, φ; x(i)) with respect to θ and φ both the generator and theinference model are optimized.Because of the effectiveness to learn any complex function of neural net-works, it makes the probabilistic encoder-decoder architecture an excellenttool to learn non-linear representations of the hidden variables.3.2 Sigma Point Sampling on Latent SpaceWe use the sigma point sampling to construct a representative subset andtrain a learner in a pool-based active learning scenario. Sigma points cancapture the mean and covariance of a completely unknown distribution withdecent accuracy with much fewer samples than Monte Carlo-type samplingtechniques used in particle filters [27]. We show that sigma point samplingalong with GMM can summarize a large data set with unknown manifoldseffectively. Figure 3.2 shows our coreset construction approach using sigmapoints.The objective of the representative subset is to find a collection of pointsthat capture the underlying distribution of the complete data. To tackle thehigh dimensionality of our data, we use a VAE. VAE can stochastically fitto any random distribution and approximate the hidden variables in a lowdimensional latent space better than traditional dimensionality reductionalgorithms [13]. Even after the feature space is formed using VAE, thelatent manifolds have complex properties that are difficult to capture witha single isotropic Gaussian assumption of VAE. To resolve that, we applythe GMM [47] on the latent space to approximate the latent manifolds witha mixture of known distributions and create soft clusters. GMM representsdata as a weighted sum of Gaussian distributions. Equation (3.2) shows the173.2. Sigma Point Sampling on Latent SpaceFigure 3.2: Representative querying using sigma point sampling. Sigmapoints X are highlighted in red. xj ∈ U are unlabeled data in the encodedlatent space. A Sigma point takes label from their nearest neighbor.The(Xj , yj) pair is added to the labeled pool for training active learner.representation of data x as a mixture of Gaussian.p(x|w, µ,Σ) =K∑k=1wk(2pi)n/2|Σk| 12exp{− 12(x− µk)TΣ−1k (x− µk)}(3.2)where K is the total number of Gaussian components used to representan n-dimensional data set x and∑Kk=1wk = 1. Given the mean vectorsand covariance matrices of the soft clusters, (µk,Σk), k = {1, 2, · · ·,K}parameters for soft cluster k, we use equation (2.9) to calculate the sigmapoints for that cluster. We sample 2n+ 1 sigma points from each cluster forthe n-dimensional latent space acquired from the VAE. Given that we useK Gaussian mixtures to represent the latent space, we get∑Kk=1(2n+ 1) =K(2n + 1) sigma points in total. Equation (3.3) shows the calculation ofsigma points from k = {1, 2, · · ·,K} clusters of the mixture of Gaussians.183.2. Sigma Point Sampling on Latent SpaceX k0 = µk, k = {1, 2, · · ·,K}X ki = µk +(√(n+ λ)Σk)i, i = 1, · · ·, n and k = {1, 2, · · ·,K}X ki = µk −(√(n+ λ)Σk)i−n, i = (n+ 1), · · ·, 2n and k = {1, 2, · · ·,K}(3.3)After we acquire the sigma points, we find their nearest neighbors toestimate their labels from the real data set. Because we run this operationon the latent space, the time complexity is O(n×K(2n+ 1)×M) ≈ O(M).The final coreset comprises of (Xj , yj) pairs where j = 1, 2, · · ·,K × (2n+ 1).We use support vector machine (SVM) with grid search cross validation totrain on the coreset.Algorithm 1: Active learning with sigma point sampling1: Inputs: Input data x ∈ X(M×N), Query budget B2: VAE on x to create a latent space z, with dimension (M × n), wheren N3: Unlabeled pool U = {zi}Mi=1, where zi ∈ Rn4: Apply GMM on U with K components to create soft clusters5: Obtain (µk,Σk), k ∈ 1, · · ·,K for each cluster6: Sample sigma points X ki from each cluster using the (µk,Σk), wherei = {1, · · ·, (2n+ 1)}, k = {1, · · ·,K}7: For each X ki find nearest neighbor zBj=1 ∈ U and query labels yBj8: if B ≤ K × (2n+ 1) then9: L = {Xl, yl}K×(2n+1)l=110: Train learner on L11: output hypothesis h : z→ y12: else13: α = α - 0.114: Repeat step 6 to 11 with updated α15: end ifAlgorithm 1 shows our proposed approach. We start with a high dimen-sional data set x ∈ RN and a query budget B. We use an encoder (in ourcase, convolution Variational Autoencoder) to create a latent space z ∈ Rn,which we treat as the unlabeled pool. We apply GMM on the unlabeled poolto retrieve the soft clusters, which we later use to sample sigma points X ki .We find the nearest neighbors of the sigma points in the latent space andattach their labels to form the (Xi, yi) pairs. We use the pairs as labeleddata set in our active learning. Finding the nearest neighbors and using193.2. Sigma Point Sampling on Latent Spacethe labels is the query step in our representative sampling algorithm. Sincewe only query one nearest for each sigma point and we have K × (2n + 1)sigma points in total, we can check for the query budget with the conditionB ≤ K × (2n + 1). If it fails, we reduce α and rerun the steps from sigmapoint generation to retrain the model.We choose to train the active learner on sigma points instead of thedata for the following reasons. Firstly, a representative subset constructedwith sigma points is a good summarization because it captures the truedistribution of the data. Secondly, the sigma points are less affected bynoise in the data because we can control the spread of sigma points bytuning α. Usually, making α too large causes generation of sigma pointsthat represent noisy data. Thirdly, to query the labels for the sigma points,we use the labels of their nearest neighbour in the latent space. Becausein this experiment, the query budget (B) for active learning is set on thelatent data space xj ∈ U , we can query label yj for at most B examples(considering a unit cost for each query). We can adjust α to sample pointswith a fixed budget of B to satisfy B ≤ K × (2n + 1). We also find in ourexperiments that sigma points in latent space perform better than latentspace data.This chapter detailed the procedure to implement the proposed sigmapoint sampling to generate representative subset of a data set and use thesubset for active learning. We saw the application of the pipeline on thedata available to us.To summarize,• The proposed pipeline is applied on the latent space of the data in orderto save computation by using the lower dimensional latent space.• The latent space also captures rich features about the data extractedwith an encoder which makes machine learning more effective.• The active learning setup in the pipeline also saves computation andreduces labeling costs by applying machine learning only on a repre-sentative subset of the data.The next chapter explains the integration of sigma point sampling withtransfer learning and active learning. It also details the application of thenovel pipeline on a new data set.20Chapter 4Applications of Sigma PointSampling in Industry 4.0In this chapter, we show two applications of the sigma point sampling onindustrial data sets for aerospace composite parts manufacturing. With theadvent of advanced computing platforms and intelligent systems, manufac-turing has become automated. An integral part of this automated manufac-turing is data driven predictive models at different stages of the process e.g.supply, data monitoring and product quality. In this study, we focus onlyon designing a predictive model to maintain the quality of manufacturing onaerospace material data. In aerospace industry, the manufacturing processestake very long to be able to generate sufficient amount of product qualitydata, which means that such data sets are usually sparse.Even though simulated data can be generated by using novel softwares,they are costly and time consuming to generate. Another problem in thismanufacturing process is that every raw material acts differently in themanufacturing processes and models trained on one data set usually performpoorly on another data set that revolves around using a different material.This brings a possibility to design a new predictive system that is agnosticto the varying data distribution caused by different materials in the samemanufacturing process. The naive approach of training a new model onevery data set is not optimal to the dynamic and fast evolving needs ofsmart manufacturing (Industry 4.0) [28].Manufacturing composite parts requires high levels of precision and de-pendency on the manufacturing tools. Constant monitoring of each stepof the manufacturing process is very crucial for producing cured parts thatshow expected material properties. In this experiment, we show that transferlearning with sigma point sampling can help to automate the manufactur-ing process of cured parts. Figure 4.1 shows the curing process that we aretrying to automate.In aerospace manufacturing, raw materials are put inside an autoclaveand through a curing cycle the composite parts are produced. A manufac-turer usually decides the presets of the curing process, it is called Manu-214.1. Transfer Learning on Composites Manufacturing DataFigure 4.1: Autoclave curing process in aerospace manufacturing. [43]facturer’s Recommended Curing Cycle (MRCC). Thermal lag and exothermare the thermal history specifications during the curing process that are theprimary quality metrics to evaluate the mechanical, chemical and physicalproperties of the composite parts. Thermal lag is the deviation betweenthe part temperature and the autoclave temperature during the cure cycle.Exotherm denotes the maximum temperature achieved by the part materialduring the cure cycle. In this study, we setup thresholds for these two met-rics. Parts that exceed these thresholds are considered failed products. Inthe experiment, we use a binary classifier to detect whether a curing pro-cess configuration will lead to failed product or acceptable composite parts.Then we transfer the knowledge of this classifier from one resin material toanother to show that our transfer learning process with sigma points sam-pling can detect successful or failed configuration on the new raw materialwith high accuracy using very few samples.4.1 Transfer Learning on CompositesManufacturing DataTL [67] focuses on training on a data set and transferring the knowledge toa similar data set with a different distribution. TL has proven to be very ef-224.1. Transfer Learning on Composites Manufacturing Datafective in manufacturing where acquiring new data distribution is common.Xu et al. [70] use deep transfer learning to design a fault detection system inmanufacturing process by training a model on simulated data and transfer-ring the knowledge on physical system data. Sun et al. [56] transfer learntfeatures and weights from historical failure data to new objects. The goal ofthis chapter is to present a novel variant of inductive transfer learning [73]that combines sigma point sampling to increase the effectiveness of trans-ferred knowledge from one data set to another in aerospace manufacturing.The results show that we achieve significant accuracy on the target data setwith very few training examples by using sigma point sampling.Transfer learning (TL) is usually defined with domains and tasks. Theseterms are defined below.Definition 1 A domain D consists of a feature space X and a marginalprobability distribution P (X), where X = {x1, x2, · · ·, xn} ∈ X . Therefore,D = {X , P (X)}.Definition 2 A task T consists of a label space Y and an objective functionf : X → Y. Therefore, given a specific domain D = {X , P (X)} the task isdefined as T = {Y, f}. f is learnt from sample data.Usually the source domain DS for a particular task TS is observed withinstance-label pairs DS = {(x, y)|xi ∈ XS , yi ∈ YS , i = 1, 2, · · ·, nS}. Theobjective function fS is learnt from the instance-label pairs.Definition 3 Given observations corresponding to a source domain andsource task (DS , TS) and some observations corresponding to target domainand task (DT , TT ), where DS 6= DT or TS 6= TT , the goal of transfer learningis to improve the performance of target predictive function fT on DT usingthe knowledge from DS and TS.In this definition of transfer learning we use only a single source for learningand a single target task for transferring the knowledge. A common scenariofor transfer learning is where source domain consists of abundant labeledinstances but labeled target domain instances are limited. Since both sourceand target domains have labels, the transfer learning is called inductivetransfer learning.In our data set on manufacturing parameters, the source domain andtarget domain are different (DS 6= DT or TS 6= TT )), which makes thislearning heterogeneous transfer learning. In smart composites manufactur-ing, changing domains can arise from changing the raw materials in the234.1. Transfer Learning on Composites Manufacturing Datamanufacturing process which changes the target task (TT ). We use transferlearning to reduce the effort and computational cost for training new modelson new target data sets. TL has proven to be very effective in getting highaccuracy on different but similar domains by transferring knowledge fromthe source domain [56, 70, 71].4.1.1 Transfer Learning with Sigma Point SamplingWe propose a modified approach to transfer learning so that we can applythe transferred model trained on the source domain to a small subset ofthe target domain. Figure 4.2 shows our proposed transfer learning pipelinewith sigma point sampling. At the initial step we consider that the observedinstances from the target domain are unlabeled. We use feature-based trans-fer learning [38] on the first step of target domain. We learn latent features(ZS) of the source data (Xsource) with a fully-connected encoder (g) andapply the same encoder on the unlabeled target domain to generate similarlatent space (ZT ).Figure 4.2: Transfer Learning with Sigma Point Sampling.We apply sigma point sampling on the latent space (ZT ) instead of theobserved feature space for the following reasons. Firstly, [41] show thatactive learning with representative sampling performs better in the latentspace because an encoder captures rich features from the observed variablespace. In chapter 3 we apply sigma point sampling on latent space. Inorder to keep our experiments comparable we follow the same pipeline inthis chapter. Secondly, the latent space representations in our experiments244.1. Transfer Learning on Composites Manufacturing Datahave reduced dimension which helps implementing the sigma point samplingalgorithm with reduced computation.In the second step we use parameter-based transfer learning [38] wherewe transfer the source classifier model (hS) to the target domain withoutfreezing. Since the target task (TT ) is different from source task (TS) we finetune the classifier (hT ) here. We trained the final output layer for targetdomain from scratch without transferring weights from source.We implement our TL with sigma point sampling pipeline in two stages.First, we use an encoder g : X → Z to generate a target latent space thatis similar to the source latent space. Then we use sigma point samplingon the latent space of the target domain to generate sigma points to trainthe classifier (hT ). In the final step, we use a classifier h : Z → Y. Wefine-tune the weights from classifier (hS) to fit the classification task of thetarget data set. The sampling strategy is similar to the process describedin Section 3.2. The results of our novel technique show that active learningwith representative sampling can be successfully integrated with transferlearning. We achieve very high accuracy and F1 score for detecting thedefective curing process for different raw composite materials. Section 5.2describe the experimental setup on the composites manufacturing and theresults of the experiments.4.1.2 Composite Material data setIn this experiment, we use a data set consisting of curing process configu-rations for aerospace composite material manufacturing. Figure 4.3 showsthe two-hold autoclave curing process that include the observed variablesfor the curing configurations. First step is a heating process followed by anisothermal hold. This step is repeated twice until the autoclave achieves atemperature of 180◦. The next step is cool-down process. The temperatureof the autoclave is brought down from 180◦ to 20◦ at a rate of 3.5◦/minute.As shown in Figure 4.1, apart from the current temperature and the heatingrate inside the autoclave, gas pressure also affects the quality of the com-posite material produced in the process. This is mostly because the physicalproperties of the tool material along with the gas pressure and gas velocityinside the autoclave determine the thermal history around the part material.Hence, we also keep track of the Heat Transfer Co-efficient (HTC) above andbelow the part material.For our application in this experiment, we choose two aerospace-gradecomposite material data sets namely, AS4/8552 as the source data set andAS4/8551 with a different resin system as the target data set. The observed254.1. Transfer Learning on Composites Manufacturing Datavariables for both the data sets are described in the Table 4.1.Figure 4.3: Two-step autoclave cure cycle.[45]Table 4.1: Observed variables used in predictive modeling of composite ma-terials data set.Variable Min value Max valueTool thickness (mm) 2.5 20Part thickness (mm) 2.5 20Heating rate-ramp 1 (◦C/min) 1 5Isothermal hold 1 temp (◦C) 100 130Isothermal hold 1 duration (seconds) 30 90Heating rate-ramp 2 (◦C/min) 1 5Isothermal hold 2 temp (◦C) 175 185Isothermal hold 2 duration (seconds) 30 90Top-side HTC (W/m2K) 10 200Bottom-side HTC (W/m2K) 5 150264.2. Regression with Sigma Point Sampling on Composites Manufacturing DataThe data for the source and target data set were generated using RAVENcomposites processing simulation software [45]. The output from RAVENof the observed variables mentioned in Table 4.1 are two quality metrics forthe composite parts namely, thermal lag and exotherm. The definitionsof these two terms has been described in Chapter 4. Even though thesemetrics are quantitative, we frame the problem of fault detection as a binaryclassification problem. As per the MRCC we choose Exotherm ≤ 5◦Cand Thermal lag ≤ 20◦C as successful cases and any value above thesethresholds are considered as failed cases. In all our experiments, we keepthe tool material fixed to Aluminium.4.2 Regression with Sigma Point Sampling onComposites Manufacturing DataIn this section of the thesis, we describe how easily our sigma point samplingcan be adopted for linear regression instead of classification described in theprevious sections. To demonstrate the usability, we use the source data setfrom the transfer learning study (AS4/8552). The data set consists of 1000examples and we used 80% data for training and active learning and 20%data for testing. The explanatory variables are described in table 4.1. Asshown in figure 4.1 the dependent variables are thermal lag and exotherm.To measure the performance of the linear regression model we used threedifferent metrics namely, R2, MAE and MSE. Whereas MAE and MSE helpto understand the scatter of the residuals points around the linear regressionline, R2 helps to understand the quality of fit of the model. We achieved anupper bound R2 value of 0.85 which shows a good fit of the multi-outputlinear regression model. Because the linear regression model outputs twodependent variables we used variance weighted R2 value which takes intoaccount all output variances individually. To further prove that the linearmodel fit is good, we applied threshold on both the real and predicted valuesof thermal lag and exotherm (dependent variables) in the test data set.The process of thresholding is described in Section 4.1 (Exotherm ≤ 5◦Cand Thermal lag ≤ 20◦C for success). Thresholding generates binaryvalues for outputs where 0s are successful cases and 1s are failed cases.27Chapter 5ExperimentsIn this chapter, we perform experiments on two data sets to empirically provethat SPS is a valid representative sampling strategy for active learning. Weshow that the sigma points themselves can be used for further machinelearning and the subset created with sigma points are representative of thecomplete data set. We also perform a sensitivity analysis on the effect ofthe value of α on the active learning accuracy. The sensitivity analysisexperiment shows the justification for choosing α.5.1 Sigma Point Sampling on MNIST data setIn this section, we first describe the setup of our experiment. We discuss thedata sets used in our experiments and the neural network architecture usedin variational autoencoder for dimensionality reduction. Later we discussthe experimental results on multi-class classification tasks.5.1.1 Experimental SetupWe compare my proposed sampling method with others using the MNIST[29] data set on three, four, five, and ten digits (labels). Increasing the num-ber of labels in our experiment increases the size of the data set, changesthe distribution of the data, and changes the manifolds of the latent space.We choose four different classification tasks on the same data set by increas-ing the number of labels in the experiment to demonstrate the robustnessand scalability of our model. We compare sigma point sampling approachwith coreset based representative sampling techniques [3, 41], least confidentuncertainty sampling [30, 31], entropy sampling [48], and margin sampling,query by committee [12, 36, 54]. We report the accuracy comparisons on thetest set and show that our method works either better or equally well whencompared with the existing approaches. We use Support Vector Machine(SVM) [64] with grid search cross validation as the active learning classifier.Since we are using SVM for multiclass classification we use One Versus Rest(OVR) settings. We use five fold cross validation with linear and Radial285.1. Sigma Point Sampling on MNIST data setBasis Function (RBF) kernel in SVM to remove the constraint of a fixedhypothesis on the active learner.Table 5.1 shows the labels of MNIST used in our study and the size of thetrain and test set for each classification task. The MNIST data set consistsTable 5.1: Labels used for classification problems with their correspondingdata set sizes used for training and testing.Number of labels Labels Training set Test set3 0,3,9 18003 29994 0,3,7,9 24745 41345 0,2,3,7,9 30226 505910 0-9 60000 10000of handwritten digits of dimension (28 × 28) [29]. We use VAE for dimen-sionality reduction and feature learning. We use the Keras framework toimplement the VAE. The convolution VAE used in our experiment has fourconvolution layers (except the final layer) in the encoder and the decoderwith 32, 64, 64 and 64 channels with Rectified Linear Unit (ReLU) acti-vation. Maxpooling function is used in the encoder along with 25 percentdropout. Similarly, in the decoder upsampling function is used along with25 percent dropout. The last layer of the encoder is a dense layer with 128output channels with ReLU activation. The bottleneck layer has two consec-utive dense layer without any activation1to estimate the parameters (meanand variance) of the latent space with a isotropic Gaussian assumption andsample the latent variable z from them. The final layer of the decoder isa convolution layer with a single output channel with sigmoid activation.This is to bound the output pixel values of the reconstructed images byVAE within zero and one. We run 100 epochs for three, four, five, and tenclasses with early stopping (10 epoch patience). We use Adam optimizer tooptimize our loss function. We use Binary Cross-entropy to calculate thereconstruction loss between real and reconstructed images. For three, four,five, and ten class classification we choose a latent space of dimension 3,5,5,and 7 respectively.1https://www.tensorflow.org/tutorials/generative/cvae295.1. Sigma Point Sampling on MNIST data set20 40 60 80 100120140Number of labelled points0.840.860.880.900.920.940.960.98AccuracyUniform coreset samplingRandom coreset samplingSensitivity coreset samplingSigma point coreset sampling(a) Three-class classification40 80 120160200240280320360400440Number of labelled points0.900.910.920.930.940.950.960.970.98AccuracyUniform coreset samplingRandom coreset samplingSensitivity coreset samplingSigma point coreset sampling(b) Four-class classification40 80 120160200240280320360400440Number of labelled points0.820.840.860.880.900.920.940.96AccuracyUniform coreset samplingRandom coreset samplingSensitivity coreset samplingSigma point coreset sampling(c) Five-class classification50 100150200250300350400450500Number of labelled points0.700.750.800.850.90AccuracyUniform coreset samplingRandom coreset samplingSensitivity coreset samplingSigma point coreset sampling(d) Ten-class classificationFigure 5.1: Accuracy comparison of sigma point sampling vs. other repre-sentative sampling methodsWe represent the latent space with a mixture of Gaussian distribu-tions with K components. For three, four, and five-class classification, weuse 80 components in GMM, and for ten-class classification, we use 100components. We collect the (µk, Σk) parameters from each soft cluster,k = {1, 2, ···,K}. We treat these parameter pairs as input to the sigma pointsampling algorithm. We know from equation (3.3) that for an n-dimensionallatent space, the sigma point sampling algorithm generates (2n + 1) sigmapoints from each soft cluster. We collect K(2n + 1) sigma points from all305.1. Sigma Point Sampling on MNIST data setclusters. We run KNeighborsClassifier (with argument n neighbors=1) fromthe scikit-learn package in python to find the nearest neighbor of each sigmapoint and predict their labels. We generate (Xi, yi) pairs for all K(2n + 1)sigma points and construct the final representative subset C = (Xi, yi), fori = {1, 2, · · ·,K(2n+ 1)}.20 40 60 80 100120140Number of labelled points0.880.900.920.940.960.98AccuracyEntropy SamplingMargin SamplingUncertainty SamplingSigma point sampling ( = 0.1)(a) Three-class classification40 80 120160200240280320360400440Number of labelled points0.920.930.940.950.960.970.98AccuracyEntropy SamplingMargin SamplingUncertainty SamplingSigma point sampling ( = 0.5)(b) Four-class classification40 80 120160200240280320360400440Number of labelled points0.860.880.900.920.940.96AccuracyEntropy SamplingMargin SamplingUncertainty SamplingSigma point sampling ( = 0.3)(c) Five-class classification50 100150200250300350400450500Number of labelled points0.600.650.700.750.800.850.90AccuracyEntropy SamplingMargin SamplingUncertainty SamplingSigma point sampling ( = 0.7)(d) Ten-class classificationFigure 5.2: Accuracy comparison of sigma point sampling vs. maximumuncertainty sampling, margin sampling and entropy samplingWe train the active learner on this representative subset. To train theSVM we iteratively add 20, 40, 40, and 50 pairs of (Xi, yi) from C to L forthree, four, five, and ten-class classification, respectively. For representative315.1. Sigma Point Sampling on MNIST data setsampling we run each trial 500 times and report the average test accu-racy with standard deviation for each classification task. For uncertaintysampling methods and QBC we run each of the experiments once becauserunning them 500 times is time-consuming and we did not observe too muchvariation in each run. To form the committee for QBC we use three differentclassifiers namely, support vector machine with linear kernel, random forestclassifier and K-nearest neighbour classifier.20 40 60 80 100120140Number of labelled points0.880.900.920.940.960.98AccuracyMax disagreement SamplingVote Entropy SamplingConsensus Entropy SamplingSigma point sampling ( = 0.1)(a) Three-class classification40 80 120160200240280320360400440Number of labelled points0.880.900.920.940.960.98AccuracyMax disagreement SamplingVote Entropy SamplingConsensus Entropy SamplingSigma point sampling ( = 0.5)(b) Four-class classification40 80 120160200240280320360400440Number of labelled points0.750.800.850.900.95AccuracyMax disagreement SamplingVote Entropy SamplingConsensus Entropy SamplingSigma point sampling ( = 0.3)(c) Five-class classification50 100150200250300350400450500Number of labelled points0.600.650.700.750.800.850.90AccuracyMax disagreement SamplingVote Entropy SamplingConsensus Entropy SamplingSigma point sampling ( = 0.7)(d) Ten-class classificationFigure 5.3: Accuracy comparison of sigma point sampling vs. consensusentropy sampling, maximum disagreement sampling and vote entropy sam-pling of Query by committee.325.1. Sigma Point Sampling on MNIST data setThe codes and results for our experiments can be found here. 25.1.2 Experimental ResultsIn this section we show that sigma points can be treated as members of arepresentative subset, and we can use them for training learners in a pool-based active learning setup. We first compare the test accuracy of sigmapoint sampling with uniform sampling, k-means based random coreset sam-pling [41], and sensitivity coreset sampling [6] (Figure 5.1). Secondly, wecompare our approach with maximum uncertainty sampling, entropy sam-pling, margin sampling (Figure 5.2). Then we compare the test accuracy ofour approach with consensus entropy sampling [36], maximum disagreementsampling, and vote entropy sampling [12] of QBC (Figure 5.3).5.1.3 Comparison of different sampling techniquesFrom Figure 5.1, we report that the test accuracy for the sigma point sam-pling outperforms all other representative sampling methods in all four clas-sification tasks. We can observe from Figure 5.2 that margin sampling per-forms the best in most cases but sigma point sampling is really effectivewhen we have very less training points. The test accuracy for 20 trainingexamples in three-class classification, 40 training examples in four-class clas-sification and five-class classification, and 50 training examples in ten-classclassification are highest for sigma point sampling. We observe similar phe-nomena in Figure 5.3 as well. Compared to sampling techniques used inQBC, sigma point sampling either performs better or equally well on verysmall amount of training data.We can summarize the drawbacks of other sampling techniques and ad-vantages of sigma point sampling in three folds. Firstly, all sampling tech-niques other than representative sampling techniques query one exampleat a time from the unlabeled pool and calculate informativeness for everyexample in the unlabeled pool. Hence, they are computationally very expen-sive. Secondly, the committee based querying strategies are very inconsistentacross our experiments. For example, in three, four, and five-class classi-fication maximum disagreement sampling performs best (after sigma pointsampling) but in ten-class classification vote entropy sampling outperformsthe other committee based querying. Vote entropy sampling performs poorlyin the other experiments which shows the inconsistency whereas sigma pointsampling consistently either outperforms or works equally well as compared2https://github.com/anonymous-moniker/repo1335.1. Sigma Point Sampling on MNIST data setto all other sampling techniques. Thirdly, [68] describes the necessary qual-ities of an ideal active learning setup which include removing redundancy ofdata in the labeled pool, including all variations of data distribution in thelabeled pool. Even though random coreset sampling [41] attempts to includedifferent data distributions in the labeled pool it fails to reduce redundancyand some of the examples in the labeled pool might be outliers. Sensitivitysampling however focuses on sampling examples that best optimizes the costfunction.Sigma Point Sampling on the other hand, captures all variations of datadistribution through GMM and we include at least one member from eachGaussian component. Because we use sigma points for training our modelwe reduce redundancy of data distribution as a higher value α spreads thesigma points within the data distribution. When compared to the otherrepresentative sampling strategies, sigma point sampling outperforms them.5.1.4 Sensitivity AnalysisIn Section 3.2, we argue that the representative subset can be constructedwith a maximum of B queried labels (considering unit cost for each query)from real data space to accommodate the query budget B for active learning.Algorithm 1 shows that, we tune the value of α to accommodate the querybudget. In this section, we discuss the effect of the primary scaling param-eter α of sigma point sampling on the test accuracy of the active learner.We show that the test accuracy is optimum for a certain range of α, for aspecific classification task. Figure 5.4 shows the variation of the spread ofsigma points across two Gaussian distributions for a two-dimensional toydata with increasing α. The black dots represent the sigma points. It isevident from the figure that as the value of α increases the sigma pointsspread more.In our experiments, we vary α between 0.1 and 0.9, as recommended inthe original paper [63]. Higher value of α produces sigma points with higherscaled covariance and the sigma points spread further, which is validated inequation 3.3. Figures, 5.5, 5.6 and 5.7 show the spread of sigma point onthe 2 dimensional latent space of MNIST data set (with digits 0,3,9) andhow the sigma points represent the latent space for varying values of α. 5.5shows the overlap of the sigma points (shown using black dot markers) onlatent space data (shown with coloured square markers). We can see thatthe density of the data points in the latent space is not homogeneous. Weuse 80 Gaussian components to estimate the density of this latent spaceand the Gaussian components in the sparse regions have higher covariance345.1. Sigma Point Sampling on MNIST data setthan those in the dense regions. Figure 5.6 shows the spread of sigma pointsinside each Gaussian component. We can use the primary scaling parameterα (which uses scaled covariance) to further control the scaling of the sigmapoints inside the Gaussian components.2.0 1.5 1.0 0.5 0.0 0.5 1.0 1.51.51.00.50.00.51.01.52.0(a) α = 0.12.0 1.5 1.0 0.5 0.0 0.5 1.0 1.51.51.00.50.00.51.01.52.0(b) α = 0.32.0 1.5 1.0 0.5 0.0 0.5 1.0 1.51.51.00.50.00.51.01.52.0(c) α = 0.7Figure 5.4: Effect of α on the spread of sigma points. As the value of αincreases the sigma points spread more across a Gaussian distribution. Thespread of the sigma points affects their representative quality.355.1. Sigma Point Sampling on MNIST data set−4 −2 0 2 4 6z1−4−2024z 2Latent space dataSigma points(a) α = 0.1−4 −2 0 2 4 6z1−4−2024z 2Latent space dataSigma points(b) α = 0.3−4 −2 0 2 4 6z1−4−2024z 2Latent space dataSigma points(c) α = 0.5−4 −2 0 2 4 6z1−4−2024z 2Latent space dataSigma points(d) α = 0.7Figure 5.5: Spread of the sigma points in the 2 dimensional latent space ofMNIST data set (with digits 0,3, and 9) for different α values.Finally, the figure 5.7 shows the collection of 80×(2×2+1) = 400 points(where K = 80 and N = 2) as a representative subset of the latent spacedata set with 18003 data points. Varying α generates different representativesubsets.365.1. Sigma Point Sampling on MNIST data set−4 −2 0 2 4 6z1−4−2024z 2Sigma points(a) α = 0.1−4 −2 0 2 4 6z1−4−2024z 2Sigma points(b) α = 0.3−4 −2 0 2 4 6z1−4−2024z 2Sigma points(c) α = 0.5−4 −2 0 2 4 6z1−4−2024z 2Sigma points(d) α = 0.7Figure 5.6: Spread of the sigma points inside each Gaussian mixture com-ponent and throughout the 2 dimensional latent space of MNIST data set(with digits 0,3, and 9) for different α values. In the regions where the latentspace data is sparse the Gaussian components have higher covariance.375.1. Sigma Point Sampling on MNIST data set−4 −2 0 2 4 6z1−4−2024z 2Sigma points(a) α = 0.1−4 −2 0 2 4 6z1−4−2024z 2Sigma points(b) α = 0.3−4 −2 0 2 4 6z1−4−2024z 2Sigma points(c) α = 0.5−4 −2 0 2 4 6z1−4−2024z 2Sigma points(d) α = 0.7Figure 5.7: Representation of the 2 dimensional latent space of MNIST dataset (with digits 0,3, and 9) with the sigma points for different α values. Inthe regions where the latent space data is sparse, the higher covariance ofthe Gaussian components causes the sigma points to be more spread outthan the dense regions. By changing the value of α, which is a scaled co-variance parameter for sigma points, we can control the spread of the sigmapoints inside the Gaussian components even further to get a satisfactoryrepresentation.Figure 5.8 shows the variation of test accuracy with varying α for three,385.1. Sigma Point Sampling on MNIST data setfour, five, and ten-class classification. In these experiments, we keep thenumber of Gaussian components (K) at 80, 80, 80 and 100, respectively, forthree, four, five, and ten-class classification.Our experiment shows that, for a certain value of K, the best test accu-racy depends on α. As expected, for a small data set, a small α gives thebest test accuracy, and for a bigger data set, a high value of α spreads thesigma points enough to represent the data set best. Figure 5.8 shows thatfor three, four, five, and ten-class classification, the best test accuracy areachieved at α equal to 0.1, 0.3, 0.5, and 0.7, respectively.20 40 60 80 100120140Number of labelled points0.9400.9450.9500.9550.9600.9650.9700.975Accuracy =0.1=0.3=0.5=0.7=0.9(a) Three-class classification40 80 120160200240280320360400440Number of labelled points0.96000.96250.96500.96750.97000.97250.97500.97750.9800Accuracy =0.1=0.3=0.5=0.7=0.9(b) Four-class classification40 80 120160200240280320360400440Number of labelled points0.920.930.940.950.96Accuracy =0.1=0.3=0.5=0.7=0.9(c) Five-class classification50 100150200250300350400450500Number of labelled points0.840.860.880.900.92Accuracy =0.1=0.3=0.5=0.7=0.9(d) Ten-class classificationFigure 5.8: Effect of α on multi-class classification accuracy395.1. Sigma Point Sampling on MNIST data setThis is expected because for three-class classification the unlabeled dataset size is small which means a small value of α should be sufficient togenerate the sigma points that are representative of the complete data set.For four and five-class classification the value of K is the same as that ofthe three-class but the unlabeled data set size increases with the number ofclasses (Table 5.1). So it is expected that higher values of α are needed asthe number of classes in the experiment increases. Our experiment showsthat, for a certain value of K, the best test accuracy depends on α. Asexpected, for a small data set, a small α gives the best test accuracy, andfor a bigger data set, a high value of α spreads the sigma points enoughto represent the data set best. Figure 5.8 shows that for three, four, five,and ten-class classification, the best test accuracy are achieved at α equal to0.1, 0.3, 0.5, and 0.7, respectively. This is expected because for three-classclassification the unlabeled data set size is small which means a small valueof α should be sufficient to generate the sigma points that are representativeof the complete data set. For four and five-class classification the value of Kis the same as that of the three-class but the unlabeled data set size increaseswith the number of classes (Table 5.1). So it is expected that higher valuesof α are needed as the number of classes in the experiment increases.It is visible in Figure 5.1b and Figure 5.1c that α = 0.1 under-representsthe data sets and hence produces the worst accuracy in both cases. It is alsovisible that sigma points with α = 0.7 performs very good when the traininglabeled data set size is over 400 examples in four and five-class classification.This is because, for a higher number of sampled points we need versatileexamples in the labeled set (L) representative of the complete data set.Higher values of α generate more versatile sigma points because when theyspread more across a Gaussian, the distance between them increases. It issafe to assume that in a latent space, with Euclidean assumption, sigmapoints that have higher intra-cluster distance are more versatile. For ten-class classification, α = 0.7 performs better after 150 training samples andperforms better than the other α values, which is why we declare it the bestα value this task. This is expected given that the unlabeled pool in thistask has 60000 examples. It is also visible that α = 0.9 spreads the sigmapoints far from the most representative data in all cases.405.2. Transfer Learning with Sigma Point Sampling on Composite Manufacturing data set5.2 Transfer Learning with Sigma Point Samplingon Composite Manufacturing data setIn this section, we discuss the experimental setup and results on the com-posite manufacturing process data set.For the experiment on transfer learning with composite materials dataset, we assumed the source data set (AS4/8552) has 1000 examples andwe used 80% data to train the predictive modeling and 20% data was setaside for validation. In our experiment, the target data set (AS4/8551) has1000 examples. For the target data set (AS4/8551), we set aside 200 pointsfor testing and evaluation of our transfer learning model and apply sigmapoint sampling (and random sampling for comparison) on the remaining800 examples. We vary the training set for target data from 20 examples to200 examples. This setup is realistic for both transfer learning and activelearning.To develop the transfer learning portion of the experiment, we designthe neural network (NN) architecture on the Keras platform of Python.Usually, the initial layers of a NNs act as feature extractors that is useful tolearn the low-level features of data. On the other hand, the last few layers ofthe NNs are more specific to the tasks [72]. Hence, in our experiment the NNarchitecture has two stages namely an encoder and a classifier. The classifierfrom source data is less suitable for transfer hence we unfreeze the weightsand retrain it on the target data. However, we use the same encoder fromthe source data set to extract features from the target data. The encoderhas one input layer with ten channels and three hidden layers. The firsttwo hidden layers have ten output channels, whereas the final layer of theencoder has five output channels. The reason for less number of channels inthe last layer is to use the encoder as a dimensionality reduction tool as wellas a feature extractor. The classifier has two hidden layers with five andten output channels respectively while the last output layer of the classifierhas one layer (for binary classification) with sigmoid activation. All hiddenlayers have Scaled Exponential Linear Unit (SELU) activation because itperformed relatively better than other activation functions. We kept thelearning rate fixed at 0.01 throughout the training with Adam optimizer.To prevent overfitting on the training data we used early stopping on 100epochs.In the next section, we show that the knowledge learnt from the sourcedata set can be used in the target data set for fault detection with very fewtraining examples.415.2. Transfer Learning with Sigma Point Sampling on Composite Manufacturing data set5.2.1 Experimental resultsIn this section, we share results of the transfer learning study with sigmapoint sampling. To demonstrate the effectiveness of our sigma point sam-pling in an ATL scenario we compare it with various other sampling tech-niques namely entropy sampling, margin sampling, uncertainty sampling,random sampling and Gaussian processes [44]. Before jumping into ATL,we show in figure 5.9 that even for simple active learning (without transferlearning) on the target data set (AS4/8552) sigma point sampling (SPS)performs better than all other sampling strategies. In this figure, we startwith 20 training points and we compare the test accuracy and test F1-score(on the test data set with 200 examples) for up to 200 training samples. Forall the experiments, we keep a test set of 200 examples from the target dataset.20 40 60 80 100120140160180200Number of labelled points0.20.30.40.50.60.70.80.9AccuracyEntropy SamplingMargin SamplingUncertainty SamplingSigma point samplingRandomGaussian Process(a) Accuracy comparison of SPS vs.random20 40 60 80 100120140160180200Number of labelled points0.00.20.40.60.8F1 ScoreEntropy SamplingMargin SamplingUncertainty SamplingSigma point samplingRandomGaussian Process(b) F1-score comparison of SPS vs. ran-domFigure 5.9: Comparison of performance metrics with increasing number oftraining examples in AS4/8552 for SPS vs. random samplingFor this experiment, we use the pre-trained encoder from the sourcedata set to generate the latent space and apply the sampling techniqueson the latent space to choose training examples. We train the classifierwith initialized weights on the training examples. For a particular numberof training points (say 20 training examples), we run 100 trials (samplingand training the classifier). Figure 5.9 shows standard deviation (SD) bandfor SPS and random sampling because each trial we get slightly different425.2. Transfer Learning with Sigma Point Sampling on Composite Manufacturing data setsamples. For all other sampling techniques we do not have SD band becauseall the 100 trials in those sampling strategies led to almost the same accuracy(and F1 score).For a fair comparison, we should also consider the upper limit of the ac-curacy on the target data set. We achieved a test accuracy of 92.59% whentrained on the complete data set. The figure 5.9a shows that with 20 ran-domly selected training examples 83% test accuracy was achieved whereastraining with 20 sigma points achieved a little over 86% test accuracy. Thetest accuracy of the other sampling strategies are very close to that of ran-dom sampling. We see the same phenomena while comparing the F1 score(Figure 5.9a). As expected, the test accuracy increases with increasing num-ber of training examples. With 200 randomly chosen training examples weachieved a test accuracy of 88% whereas training with 200 sigma pointsachieved a 89.5% test accuracy. Other sampling techniques perform inferiorto SPS well. The same can be said for the F1-score on the test data set(Figure 5.9a). Apart from lacking in accuracy entropy, margin and uncer-tainty sampling are very unreliable on this data set. We can conclude thatfrom the sudden drops in the accuracy and f1-score in figures 5.9a and 5.9b.For the experiments of active transfer learning (ATL), we compare SPSwith random sampling because among all other sampling strategies (exceptSPS) it performed the best (Figure 5.9).435.2. Transfer Learning with Sigma Point Sampling on Composite Manufacturing data set0 5 10 15 20 25 30Epochs0.00.10.20.30.40.50.60.70.80.91.0AccuracyAccuracy with epochsFresh Random TrainFresh Random Test(a) Random sampling (RS)0 5 10 15 20 25 30Epochs0.00.10.20.30.40.50.60.70.80.91.0AccuracyAccuracy with epochsFreeze Random TrainFreeze Random Test(b) Random sampling with TL (TL)0 5 10 15 20 25 30Epochs0.00.10.20.30.40.50.60.70.80.91.0AccuracyAccuracy with epochsFresh ( = 0.7) TrainFresh ( = 0.7) Test(c) SPS (AL)0 5 10 15 20 25 30Epochs0.00.10.20.30.40.50.60.70.80.91.0AccuracyAccuracy with epochsUnfreeze ( = 0.7) TrainUnfreeze ( = 0.7) Test(d) SPS with TL (ATL)Figure 5.10: Comparison of accuracy in Random vs TL vs AL vs ATL on20 pointsFigures 5.10,5.11,5.12 show the train and test curves for random sam-pling with fresh NN (RS), random sampling with TL, sigma point samplingwith fresh NN (AL), sigma point sampling with transfer learning (ATL) forup to 30 epochs. In order to make the number of passes over data for NNequal for all trials in every experiment we chose 30 epochs without earlystopping. In my experiment, we found that running more than 30 epochson 20 training samples without early stopping results in overfitting in mostcases.445.2. Transfer Learning with Sigma Point Sampling on Composite Manufacturing data set0 5 10 15 20 25 30Epochs0.00.10.20.30.40.50.60.70.80.91.0AccuracyAccuracy with epochsFresh Random TrainFresh Random Test(a) Random sampling (RS)0 5 10 15 20 25 30Epochs0.00.10.20.30.40.50.60.70.80.91.0AccuracyAccuracy with epochsUnfreeze Random TrainUnfreeze Random Test(b) Random sampling with TL (TL)0 5 10 15 20 25 30Epochs0.00.10.20.30.40.50.60.70.80.91.0AccuracyAccuracy with epochsFresh (K=120, = 0.5) TrainFresh (K=120, = 0.5) Test(c) SPS (AL)0 5 10 15 20 25 30Epochs0.00.10.20.30.40.50.60.70.80.91.0AccuracyAccuracy with epochsUnfreeze (K=120, = 0.5) TrainUnfreeze (K=120, = 0.5) Test(d) SPS with TL (ATL)Figure 5.11: Comparison of accuracy in Random vs TL vs AL vs ATL on100 pointsFigure 5.10 shows the train and test curves of the classifier NN with20 training examples and 200 test examples that we set aside at the verybeginning of the transfer learning study from the target data set. In figure5.10 ATL outperforms RS, TL, AL in train and test accuracy. The confi-dence intervals (standard deviation fringes) of the train and test curves arethe narrowest in ATL which shows that adding transferred knowledge fromsource data set not only increases the test accuracy of the model but alsoincreases the confidence of performance of the model.455.2. Transfer Learning with Sigma Point Sampling on Composite Manufacturing data set0 5 10 15 20 25 30Epochs0.00.10.20.30.40.50.60.70.80.91.0AccuracyAccuracy with epochsFresh Random TrainFresh Random Test(a) Random sampling (RS)0 5 10 15 20 25 30Epochs0.00.10.20.30.40.50.60.70.80.91.0AccuracyAccuracy with epochsUnfreeze Random TrainUnfreeze Random Test(b) Random sampling with TL (TL)0 5 10 15 20 25 30Epochs0.00.10.20.30.40.50.60.70.80.91.0AccuracyAccuracy with epochsFresh (K=80, = 0.7) TrainFresh (K=80, = 0.7) Test(c) SPS (AL)0 5 10 15 20 25 30Epochs0.00.10.20.30.40.50.60.70.80.91.0AccuracyAccuracy with epochsUnfreeze (K=80, = 0.7) TrainUnfreeze (K=80, = 0.7) Test(d) SPS with TL (ATL)Figure 5.12: Comparison of accuracy in Random vs TL vs AL vs ATL on200 pointsFigure 5.11 shows the train and test curves with 100 training examplesand 200 test examples (test set with 200 examples is same for all the experi-ments across the transfer learning study). ATL outperforms RS, TL and ALin train and test accuracy on every epoch proving that adding transferredknowledge improves the model training. The confidence intervals (CI) in fig-ure 5.11 are narrower when compared to the CI in figure 5.10 which showsthat adding more training samples improves the confidence of the model.Finally, figure 5.12 shows train and test curves of the classifier with 200465.3. Regression Learning with Sigma Point Sampling on Composite Manufacturing data setpoints. Even though we do not get a significant improvement in the finaltrain and test accuracy, ATL slightly outperforms RS, TL and AL. ATLalso has the narrowest CI of all the experiments, solidifying the claim thatadding more sigma points for training with transferred weights from sourcedata set improves test accuracy and model confidence.5.3 Regression Learning with Sigma PointSampling on Composite Manufacturing datasetIn this section, we discuss my findings on the linear regression study oncomposite manufacturing data set. For linear regression study, we used theAS4/8552 data set with 1000 examples. The data set has ten independent(observed) variables listed in table 4.1 and two dependent variables namelythermal lag and exotherm from the autoclave. We set aside 200 examplesfor testing and evaluating the performance of the linear regression model.We kept 800 examples for training and active learning. To set the upperlimit performance of the linear model we used all 800 examples for trainingand measured the R2 score, MAE and MSE on the test data set. MAE andMSE are good performance metrics for a linear regression model and theygive a good idea about how far the residuals are from the linear regressionline. Whereas R2 score measures the amount of variance explained by thelinear regression model and hence it measures the quality of fit of the linearregression model.Table 5.2: Labels used for classification problems with their correspondingdata set sizes used for training and testing.Performance metric ValueR2score 0.85MAE 3.44MSE 26.56For linear regression, we use sklearn module in Python, with 5-foldcross validation. The submodule LinearRegression from sklearn supportsmultiple outputs for a linear regression model which is essential for our dataset. To measure the performance metrics (R2, MAE, MSE), we used the475.3. Regression Learning with Sigma Point Sampling on Composite Manufacturing data setvariance weighted argument for multiple outputs. Table 5.2 shows the upperlimit performance metrics.20 40 60 80 100120140160180200Number of examples45678MAERandomSPS (a) Mean Absolute Error (MAE)20 40 60 80 100120140160180200Number of examples050100150200MSERandomSPS (b) Mean Squared Error MSE20 40 60 80 100120140160180200Number of examples0.20.00.20.40.60.81.0R2RandomSPS(c) R2 ScoreFigure 5.13: Performance metrics MAE, MSE and R2 Score on test data setfor Regression with SPS5.3.1 Experimental resultsFigures 5.13 show the comparison of random sampling with sigma pointsampling on the performance metrics MAE, MSE and R2 score. Sigmapoint sampling performs slightly better than random sampling in all themetrics. It also has a better confidence interval especially when the trainingdata set is very small. By using 200 examples for training, we achievedalmost similar performance as the upper limit (Table 5.2).48Chapter 6Conclusion and Future WorkIn this thesis, we discussed representative subset construction and its use inactive learning. We explored the theory and application of a new represen-tative sampling approach. To our knowledge, it is the only representativesampling technique that approximates the distribution of the complete dataset with a much smaller subset of points. Our novel representative subsetconstruction approach uses sigma points (used in UKF) along with GMMto estimate the density and distribution of the data set. We use the rep-resentative subset constructed with sigma points to train the active learnercompared to all other existing approaches that use real data points to doso.In the first experiment, we saw how we can use the latent space of adata set to apply sigma point sampling. The latent space of the data set hasa smaller dimension which significantly reduces the cost of implementingthe soft clusters (GMM) on it compared to the observed variable space.Our experiments showed that the active learner trained with sigma pointsfrom the latent space outperforms all the representative sampling techniquesused for active learning. We also showed that our approach almost alwaysperforms better than any other query strategy used for pool-based activelearning.On the second set of experiments, we showed an application of the sigmapoint sampling approach on the composite material manufacturing processin Industry 4.0. In this experiment, we proposed a new approach to activetransfer learning ATL [43]. We showed that SPS helps in integrating ac-tive learning (AL) with TL, resulting in Sigma Point Sampling based ActiveTransfer Learning (SPSATL). We show that SPS helps in training on a newdata set with transferred weights much more efficiently without compromis-ing the accuracy. We also show that the proposed sampling strategy can beintegrated with both classification and regression learning. To improve uponthe existing ATL approach for optimizing the manufacturing process, we ap-plied our SPS with TL (ATL) technique on the feature space of the data set.We found that SPS outperformed all other query strategies on this data set.We also performed a series of experiments to check the effect of our sampling49Chapter 6. Conclusion and Future Workapproach on the neural networks used for transfer learning. We found thatSPS with TL (ATL) results in faster convergence of the neural networks.The final mean accuracy in the train and test phase is slightly higher withsmaller variance in SPS with TL (ATL) when compared to simple randomsampling (RS), random sampling with TL, and simple SPS without trans-ferring. We replicated this experiment in three different settings by varyingthe size of the training set. The three experimental settings used 20, 100and 200 points respectively for training the neural network. We found thatincreasing the training data reduces the variance of accuracy in both train-ing and testing. We also found that increasing the training set brings thetrain and test accuracy over each epoch closer to each other, which indicatesan absence of over-fitting of the model.Labeling cost associated with data sets can be a hindrance while applyingtraditional machine learning methodologies in cases where labeling cost ishigh. Traditional PAC ML techniques are data hungry and active learningsolves that problem by smartly querying the useful data. In this thesis, wesaw an application of active learning where labeling the data might be costlybecause it needs to through a proprietary software. Our batch-mode activelearning pipeline includes a novel query strategy (sigma point representativesubset) which captures the distribution of the complete data set with veryfew data points. Our active learning pipeline works with dimensionalityreduction tools such as VAE and principal component analysis (PCA).One limitation of this research is that our pipeline requires us to applyGMM on the latent space first and then use sigma point sampling for repre-sentative subset construction. We will create an end-to-end active learningon latent space pipeline with sigma points in the future. Our pipeline as-sumes that the latent space follows the Euclidean geometry which is notalways correct. Even though ideally we should ask the labels of the sigmapoints to an oracle we use k-nearest neighbor for querying the label of thenearest data point because it is readily available to us. 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Thesis/Dissertation
eng
Computer Science
Vancouver : University of British Columbia Library
Attribution-NonCommercial-NoDerivatives 4.0 International
Graduate
Narayan, Apurva
Improved sampling strategy for representative set construction
Text