DOCSLIB.ORG

Layered Sampling for Robust Optimization Problems

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Layered Sampling for Robust Optimization Problems Layered Sampling for Robust Optimization Problems Hu Ding 1 Zixiu Wang 1 Abstract the complexity measures like running time, space, and com- munication. In the past years, the coreset techniques have In real world, our datasets often contain outliers. been successfully applied to solve many optimization prob- Moreover, the outliers can seriously affect the lems, such as clustering (Chen, 2009; Feldman & Langberg, final machine learning result. Most existing algo- 2011; Huang et al., 2018), logistic regression (Huggins et al., rithms for handling outliers take high time com- 2016; Munteanu et al., 2018), linear regression (Dasgupta plexities (e.g. quadratic or cubic complexity). et al., 2009; Drineas et al., 2006), and Gaussian mixture Coreset is a popular approach for compressing model (Lucic et al., 2017; Karnin & Liberty, 2019). data so as to speed up the optimization algorithms. However, the current coreset methods cannot be A large part of existing coreset construction methods are easily extended to handle the case with outliers. based on the theory of sensitivity which was proposed by In this paper, we propose a new variant of coreset (Langberg & Schulman, 2010). Informally, each data point technique, layered sampling, to deal with two p P has the sensitivity φ(p) (in fact, we just need to fundamental robust optimization problems: k- compute2 an appropriate upper bound of φ(p)) to measure median/means clustering with outliers and linear its importance to the whole instance P over all possible P regression with outliers. This new coreset method solutions, and Φ(P ) = p2P φ(p) is called the total sen- is in particular suitable to speed up the iterative al- sitivity. The coreset construction is a simple sampling pro- gorithms (which often improve the solution within cedure where each point p is drawn i.i.d. from P propor- a local range) for those robust optimization prob- φ(p) tional to Φ(P ) ; each sampled point p is assigned a weight lems. Moreover, our method is easy to be imple- w(p) = Φ(P ) where m is the sample size depending on the mented in practice. We expect that our framework mφ(p) of layered sampling will be applicable to other “pseudo-dimension” of the objective function ∆ ((Feldman robust optimization problems. & Langberg, 2011; Li et al., 2001)); eventually, the set of weighted sampled points form the desired coreset S. In real world, datasets are noisy and contain outliers. More- 1. Introduction over, outliers could seriously affect the final results in data analysis (Chandola et al., 2009; Goodfellow et al., 2018). Coreset is a widely studied technique for solving many However, the sensitivity based coreset approach is not ap- optimization problems (Phillips, 2016; Bachem et al., 2017; propriate to handle robust optimization problems involving Munteanu et al., 2018; Feldman, 2020). The (informal) outliers (e.g., k-means clustering with outliers). For exam- definition is as follows. Given an optimization problem with ple, it is not easy to compute the sensitivity φ(p) because the objective function ∆, denote by ∆(P; C) the objective the point p could be inlier or outlier for different solutions; value determined by a dataset P and a solution C; a small arXiv:2002.11904v1 [cs.CG] 27 Feb 2020 moreover, it is challenging to build the relation, such as (1), set S is called a coreset if between the original instance P and the coreset S (e.g., how ∆(P; C) ∆(S; C) (1) to determine the number of outliers for the instance S?). ≈ for any feasible solution C. Roughly speaking, the coreset 1.1. Our Contributions is a small set of data approximately representing a much larger dataset, and therefore existing algorithm can run on In this paper, we consider two important robust optimization the coreset (instead of the original dataset) so as to reduce problems: k-median/means clustering with outliers and lin- ear regression with outliers. Their quality guaranteed algo- 1School of Computer Science and Technology, University rithms exist but often have high complexities that seriously of Science and Technology of China. Correspondence to: limit their applications in real scenarios (see Section 1.2 for http://staff.ustc.edu. Hu Ding <[email protected], more details). We observe that these problems can be often cn/˜huding/ >. efficiently solved by some heuristic algorithms in practice, though they only guarantee local optimums in theory. For Layered Sampling for Robust Optimization Problems algorithms (Chen, 2009; Har-Peled & Kushal, 2007; C˜ Fichtenberger et al., 2013; Feldman et al., 2013); in Local range particular, (Feldman & Langberg, 2011) proposed a LL unified coreset framework for a set of clustering problems. Solution space However, the research on using coreset to handle outliers is Figure 1. The red point represents the initial solution C~, and our still quite limited. Recently, (Huang et al., 2018) showed goal is to guarantee (2) for a local range around C~. that a uniform independent sample can serve as a coreset for clustering with outliers in Euclidean space; however, example, (Chawla & Gionis, 2013b) proposed the algorithm such uniform sampling based method often misses some k-means- - to solve the problem of k-means clustering with important points and therefore introduces an unavoidable outliers, where the main idea is an alternating minimization error on the number of outliers. (Gupta, 2018) also studied strategy. The algorithm is an iterative procedure, where it the uniform random sampling idea but under the assumption alternatively updates the outliers and the k cluster centers that each optimal cluster should be large enough. Partly in each iteration; eventually the solution converges to a lo- inspired by the method of (Mettu & Plaxton, 2004), (Chen cal optimum. The alternating minimization strategy is also et al., 2018) proposed a novel summary construction widely used for solving the problem of linear regression algorithm to reduce input data size which guarantees an with outliers, e.g., (Shen & Sanghavi, 2019). A common O(1) factor of distortion on the clustering cost. feature of these methods is that they usually start from an In theory, the algorithms with provable guarantees for k- initial solution and then locally improve the solution round median/means clustering with outliers (Chen, 2008; Kr- by round. Therefore, a natural question is ishnaswamy et al., 2018; Friggstad et al., 2018) have high can we construct a “coreset” only for a local range in the complexities and are difficult to be implemented in practice. solution space? The heuristic but practical algorithms have also been stud- ied before (Chawla & Gionis, 2013b; Ott et al., 2014). By Using such a coreset, we can substantially speed up those it- using the local search method, (Gupta et al., 2017b) pro- erative algorithms. Motivated by this question, we introduce vided a 274-approximation algorithm of k-means clustering a new variant of coreset method called layered sampling. with outliers but needing to discard more than the desired Given an initial solution C~, we partition the given data set number of outliers; to improve the running time, they also P into a consecutive sequence of “layers” surrounding C~ used k-means++ (Arthur & Vassilvitskii, 2007b) to seed the and conduct the random sampling in each layer; the union “coreset” that yields an O(1) factor approximation. Based of the samples, together with the points located in the out- on the idea of k-means++, (Bhaskara et al., 2019) proposed ermost layer, form the coreset S. Actually, our method an O(log k)-approximation algorithm. is partly inspired by the coreset construction method of k-median/means clustering (without outliers) proposed by Linear regression (with outliers). Several coreset meth- (Chen, 2009). However, we need to develop significantly ods for ordinary linear regression (without outliers) have new idea in theory to prove its correctness for the case with been proposed (Drineas et al., 2006; Dasgupta et al., 2009; outliers. The purpose of layered sampling is not to guaran- Boutsidis et al., 2013). For the case with outliers, which is tee the approximation quality (as (1)) for any solution C, also called “Least Trimmed Squares linear estimator (LTS)”, instead, it only guarantees the quality for the solutions in a uniform sampling approach was studied by (Mount et al., a local range in the solution space (the formal definition 2014; Ding & Xu, 2014). But similar to the scenario of is given in SectionL 1.3). Informally, we need to prove the clustering with outliers, such uniform sampling approach following result to replace (1): introduces an unavoidable error on the number of outliers. C ; ∆(P; C) ∆(S; C) (2) (Mount et al., 2014) also proved that it is impossible to 8 2 L ≈ achieve even an approximate solution for LTS within poly- See Figure1 for an illustration. In other words, the new nomial time under the conjecture of the hardness of affine method can help us to find a local optimum faster. Our main degeneracy (Erickson & Seidel, 1995), if the dimensionality results are shown in Theorem1 and2. The construction d is not fixed. Despite of its high complexity, several practi- algorithms are easy to implement. cal algorithms were proposed before and most of them are based on the idea of alternating minimization that improves 1.2. Related Works the solution within a local range, such as (Rousseeuw, 1984; k-median/means clustering (with outliers). k- Rousseeuw & van Driessen, 2006; Hawkins, 1994; Mount median/means clustering are two popular center-based et al., 2016; Bhatia et al., 2015; Shen & Sanghavi, 2019). clustering problems (Awasthi & Balcan, 2014). It has (Klivans et al., 2018) provided another approach based on been extensively studied for using coreset techniques to the sum-of-squares method. reduce the complexities of k-median/means clustering Layered Sampling for Robust Optimization Problems 1.3.
Load more

Recommended publications
  • Arxiv:1811.04918V6 [Cs.LG] 1 Jun 2020
    Learning and Generalization in Overparameterized Neural Networks, Going Beyond Two Layers Zeyuan Allen-Zhu Yuanzhi Li Yingyu Liang [email protected] [email protected] [email protected] Microsoft Research AI Stanford University University of Wisconsin-Madison November 12, 2018 (version 6)∗ Abstract The fundamental learning theory behind neural networks remains largely open. What classes of functions can neural networks actually learn? Why doesn't the trained network overfit when it is overparameterized? In this work, we prove that overparameterized neural networks can learn some notable con- cept classes, including two and three-layer networks with fewer parameters and smooth activa- tions. Moreover, the learning can be simply done by SGD (stochastic gradient descent) or its variants in polynomial time using polynomially many samples. The sample complexity can also be almost independent of the number of parameters in the network. On the technique side, our analysis goes beyond the so-called NTK (neural tangent kernel) linearization of neural networks in prior works. We establish a new notion of quadratic ap- proximation of the neural network (that can be viewed as a second-order variant of NTK), and connect it to the SGD theory of escaping saddle points. arXiv:1811.04918v6 [cs.LG] 1 Jun 2020 ∗V1 appears on this date, V2/V3/V4 polish writing and parameters, V5 adds experiments, and V6 reflects our conference camera ready version. Authors sorted in alphabetical order. We would like to thank Greg Yang and Sebastien Bubeck for many enlightening conversations. Y. Liang was supported in part by FA9550-18-1-0166, and would also like to acknowledge that support for this research was provided by the Office of the Vice Chancellor for Research and Graduate Education at the University of Wisconsin-Madison with funding from the Wisconsin Alumni Research Foundation.
    [Show full text]
  • Analysis of Perceptron-Based Active Learning
    JournalofMachineLearningResearch10(2009)281-299 Submitted 12/07; 12/08; Published 2/09 Analysis of Perceptron-Based Active Learning Sanjoy Dasgupta [email protected] Department of Computer Science and Engineering University of California, San Diego La Jolla, CA 92093-0404, USA Adam Tauman Kalai [email protected] Microsoft Research Office 14063 One Memorial Drive Cambridge, MA 02142, USA Claire Monteleoni [email protected] Center for Computational Learning Systems Columbia University Suite 850, 475 Riverside Drive, MC 7717 New York, NY 10115, USA Editor: Manfred Warmuth Abstract Ω 1 We start by showing that in an active learning setting, the Perceptron algorithm needs ( ε2 ) labels to learn linear separators within generalization error ε. We then present a simple active learning algorithm for this problem, which combines a modification of the Perceptron update with an adap- tive filtering rule for deciding which points to query. For data distributed uniformly over the unit 1 sphere, we show that our algorithm reaches generalization error ε after asking for just O˜(d log ε ) labels. This exponential improvement over the usual sample complexity of supervised learning had previously been demonstrated only for the computationally more complex query-by-committee al- gorithm. Keywords: active learning, perceptron, label complexity bounds, online learning 1. Introduction In many machine learning applications, unlabeled data is abundant but labeling is expensive. This distinction is not captured in standard models of supervised learning, and has motivated the field of active learning, in which the labels of data points are initially hidden, and the learner must pay for each label it wishes revealed.
    [Show full text]
  • The Sample Complexity of Agnostic Learning Under Deterministic Labels
    JMLR: Workshop and Conference Proceedings vol 35:1–16, 2014 The sample complexity of agnostic learning under deterministic labels Shai Ben-David [email protected] Cheriton School of Computer Science, University of Waterloo, Waterloo, Ontario N2L 3G1, CANADA Ruth Urner [email protected] School of Computer Science, Georgia Institute of Technology, Atlanta, Georgia 30332, USA Abstract With the emergence of Machine Learning tools that allow handling data with a huge number of features, it becomes reasonable to assume that, over the full set of features, the true labeling is (almost) fully determined. That is, the labeling function is deterministic, but not necessarily a member of some known hypothesis class. However, agnostic learning of deterministic labels has so far received little research attention. We investigate this setting and show that it displays a behavior that is quite different from that of the fundamental results of the common (PAC) learning setups. First, we show that the sample complexity of learning a binary hypothesis class (with respect to deterministic labeling functions) is not fully determined by the VC-dimension of the class. For any d, we present classes of VC-dimension d that are learnable from O~(d/)- many samples and classes that require samples of size Ω(d/2). Furthermore, we show that in this setup, there are classes for which any proper learner has suboptimal sample complexity. While the class can be learned with sample complexity O~(d/), any proper (and therefore, any ERM) algorithm requires Ω(d/2) samples. We provide combinatorial characterizations of both phenomena, and further analyze the utility of unlabeled samples in this setting.
    [Show full text]
  • Sinkhorn Autoencoders
    Sinkhorn AutoEncoders Giorgio Patrini⇤• Rianne van den Berg⇤ Patrick Forr´e Marcello Carioni UvA Bosch Delta Lab University of Amsterdam† University of Amsterdam KFU Graz Samarth Bhargav Max Welling Tim Genewein Frank Nielsen ‡ University of Amsterdam University of Amsterdam Bosch Center for Ecole´ Polytechnique CIFAR Artificial Intelligence Sony CSL Abstract 1INTRODUCTION Optimal transport o↵ers an alternative to Unsupervised learning aims at finding the underlying maximum likelihood for learning generative rules that govern a given data distribution PX . It can autoencoding models. We show that mini- be approached by learning to mimic the data genera- mizing the p-Wasserstein distance between tion process, or by finding an adequate representation the generator and the true data distribution of the data. Generative Adversarial Networks (GAN) is equivalent to the unconstrained min-min (Goodfellow et al., 2014) belong to the former class, optimization of the p-Wasserstein distance by learning to transform noise into an implicit dis- between the encoder aggregated posterior tribution that matches the given one. AutoEncoders and the prior in latent space, plus a recon- (AE) (Hinton and Salakhutdinov, 2006) are of the struction error. We also identify the role of latter type, by learning a representation that maxi- its trade-o↵hyperparameter as the capac- mizes the mutual information between the data and ity of the generator: its Lipschitz constant. its reconstruction, subject to an information bottle- Moreover, we prove that optimizing the en- neck. Variational AutoEncoders (VAE) (Kingma and coder over any class of universal approxima- Welling, 2013; Rezende et al., 2014), provide both a tors, such as deterministic neural networks, generative model — i.e.
    [Show full text]
  • Data-Dependent Sample Complexity of Deep Neural Networks Via Lipschitz Augmentation
    Data-dependent Sample Complexity of Deep Neural Networks via Lipschitz Augmentation Colin Wei Tengyu Ma Computer Science Department Computer Science Department Stanford University Stanford University [email protected] [email protected] Abstract Existing Rademacher complexity bounds for neural networks rely only on norm control of the weight matrices and depend exponentially on depth via a product of the matrix norms. Lower bounds show that this exponential dependence on depth is unavoidable when no additional properties of the training data are considered. We suspect that this conundrum comes from the fact that these bounds depend on the training data only through the margin. In practice, many data-dependent techniques such as Batchnorm improve the generalization performance. For feedforward neural nets as well as RNNs, we obtain tighter Rademacher complexity bounds by considering additional data-dependent properties of the network: the norms of the hidden layers of the network, and the norms of the Jacobians of each layer with respect to all previous layers. Our bounds scale polynomially in depth when these empirical quantities are small, as is usually the case in practice. To obtain these bounds, we develop general tools for augmenting a sequence of functions to make their composition Lipschitz and then covering the augmented functions. Inspired by our theory, we directly regularize the network’s Jacobians during training and empirically demonstrate that this improves test performance. 1 Introduction Deep networks trained in
    [Show full text]
  • Nearly Tight Sample Complexity Bounds for Learning Mixtures of Gaussians Via Sample Compression Schemes∗
    Nearly tight sample complexity bounds for learning mixtures of Gaussians via sample compression schemes∗ Hassan Ashtiani Shai Ben-David Department of Computing and Software School of Computer Science McMaster University, and University of Waterloo, Vector Institute, ON, Canada Waterloo, ON, Canada [email protected] [email protected] Nicholas J. A. Harvey Christopher Liaw Department of Computer Science Department of Computer Science University of British Columbia University of British Columbia Vancouver, BC, Canada Vancouver, BC, Canada [email protected] [email protected] Abbas Mehrabian Yaniv Plan School of Computer Science Department of Mathematics McGill University University of British Columbia Montréal, QC, Canada Vancouver, BC, Canada [email protected] [email protected] Abstract We prove that Θ(e kd2="2) samples are necessary and sufficient for learning a mixture of k Gaussians in Rd, up to error " in total variation distance. This improves both the known upper bounds and lower bounds for this problem. For mixtures of axis-aligned Gaussians, we show that Oe(kd="2) samples suffice, matching a known lower bound. The upper bound is based on a novel technique for distribution learning based on a notion of sample compression. Any class of distributions that allows such a sample compression scheme can also be learned with few samples. Moreover, if a class of distributions has such a compression scheme, then so do the classes of products and mixtures of those distributions. The core of our main result is showing that the class of Gaussians in Rd has a small-sized sample compression. 1 Introduction Estimating distributions from observed data is a fundamental task in statistics that has been studied for over a century.
    [Show full text]
  • Sample Complexity Results
    10-601 Machine Learning Maria-Florina Balcan Spring 2015 Generalization Abilities: Sample Complexity Results. The ability to generalize beyond what we have seen in the training phase is the essence of machine learning, essentially what makes machine learning, machine learning. In these notes we describe some basic concepts and the classic formalization that allows us to talk about these important concepts in a precise way. Distributional Learning The basic idea of the distributional learning setting is to assume that examples are being provided from a fixed (but perhaps unknown) distribution over the instance space. The assumption of a fixed distribution gives us hope that what we learn based on some training data will carry over to new test data we haven't seen yet. A nice feature of this assumption is that it provides us a well-defined notion of the error of a hypothesis with respect to target concept. Specifically, in the distributional learning setting (captured by the PAC model of Valiant and Sta- tistical Learning Theory framework of Vapnik) we assume that the input to the learning algorithm is a set of labeled examples S :(x1; y1);:::; (xm; ym) where xi are drawn i.i.d. from some fixed but unknown distribution D over the the instance space ∗ ∗ X and that they are labeled by some target concept c . So yi = c (xi). Here the goal is to do optimization over the given sample S in order to find a hypothesis h : X ! f0; 1g, that has small error over whole distribution D. The true error of h with respect to a target concept c∗ and the underlying distribution D is defined as err(h) = Pr (h(x) 6= c∗(x)): x∼D (Prx∼D(A) means the probability of event A given that x is selected according to distribution D.) We denote by m ∗ 1 X ∗ errS(h) = Pr (h(x) 6= c (x)) = I[h(xi) 6= c (xi)] x∼S m i=1 the empirical error of h over the sample S (that is the fraction of examples in S misclassified by h).
    [Show full text]
  • Benchmarking Model-Based Reinforcement Learning
    Benchmarking Model-Based Reinforcement Learning Tingwu Wang1;2, Xuchan Bao1;2, Ignasi Clavera3, Jerrick Hoang1;2, Yeming Wen1;2, Eric Langlois1;2, Shunshi Zhang1;2, Guodong Zhang1;2, Pieter Abbeel3& Jimmy Ba1;2 1University of Toronto 2 Vector Institute 3 UC Berkeley {tingwuwang,jhoang,ywen,edl,gdzhang}@cs.toronto.edu {xuchan.bao,matthew.zhang}@mail.utoronto.ca, [email protected], [email protected], [email protected] Abstract Model-based reinforcement learning (MBRL) is widely seen as having the potential to be significantly more sample efficient than model-free RL. However, research in model-based RL has not been very standardized. It is fairly common for authors to experiment with self-designed environments, and there are several separate lines of research, which are sometimes closed-sourced or not reproducible. Accordingly, it is an open question how these various existing MBRL algorithms perform relative to each other. To facilitate research in MBRL, in this paper we gather a wide collection of MBRL algorithms and propose over 18 benchmarking environments specially designed for MBRL. We benchmark these algorithms with unified problem settings, including noisy environments. Beyond cataloguing performance, we explore and unify the underlying algorithmic differences across MBRL algorithms. We characterize three key research challenges for future MBRL research: the dynamics bottleneck, the planning horizon dilemma, and the early-termination dilemma. Finally, to maximally facilitate future research on MBRL, we open-source our benchmark in http://www.cs.toronto.edu/~tingwuwang/mbrl.html. 1 Introduction Reinforcement learning (RL) algorithms are most commonly classified in two categories: model-free RL (MFRL), which directly learns a value function or a policy by interacting with the environment, and model-based RL (MBRL), which uses interactions with the environment to learn a model of it.
    [Show full text]
  • Theory of Deep Learning
    CONTRIBUTORS:RAMANARORA,SANJEEVARORA,JOANBRUNA,NADAVCOHEN,SIMONDU, RONGGE,SURIYAGUNASEKAR,CHIJIN,JASONLEE,TENGYUMA,BEHNAMNEYSHABUR, ZHAOSONG THEORYOFDEEP LEARNING Contents 1 Basic Setup and some math notions 13 1.1 List of useful math facts 14 1.1.1 Probability tools 14 1.1.2 Singular Value Decomposition 15 2 Basics of Optimization 17 2.1 Gradient descent 17 2.1.1 Formalizing the Taylor Expansion 18 2.1.2 Descent lemma for gradient descent 18 2.2 Stochastic gradient descent 19 2.3 Accelerated Gradient Descent 19 2.4 Local Runtime Analysis of GD 20 2.4.1 Pre-conditioners 21 3 Backpropagation and its Variants 23 3.1 Problem Setup 23 3.1.1 Multivariate Chain Rule 25 3.1.2 Naive feedforward algorithm (not efficient!) 26 3.2 Backpropagation (Linear Time) 26 3.3 Auto-differentiation 27 3.4 Notable Extensions 28 3.4.1 Hessian-vector product in linear time: Pearlmutter’s trick 29 4 4 Basics of generalization theory 31 4.0.1 Occam’s razor formalized for ML 31 4.0.2 Motivation for generalization theory 32 4.1 Some simple bounds on generalization error 32 4.2 Data dependent complexity measures 34 4.2.1 Rademacher Complexity 34 4.2.2 Alternative Interpretation: Ability to correlate with random labels 35 4.3 PAC-Bayes bounds 35 5 Advanced Optimization notions 39 6 Algorithmic Regularization 41 6.1 Linear models in regression: squared loss 42 6.1.1 Geometry induced by updates of local search algorithms 43 6.1.2 Geometry induced by parameterization of model class 46 6.2 Matrix factorization 47 6.3 Linear Models in Classification 47 6.3.1 Gradient Descent
    [Show full text]
  • A Sample Complexity Separation Between Non-Convex and Convex Meta-Learning
    A Sample Complexity Separation between Non-Convex and Convex Meta-Learning Nikunj Saunshi 1 Yi Zhang 1 Mikhail Khodak 2 Sanjeev Arora 13 Abstract complexity of an unseen but related test task. Although there is a long history of successful methods in meta-learning One popular trend in meta-learning is to learn and the related areas of multi-task and lifelong learning from many training tasks a common initialization (Evgeniou & Pontil, 2004; Ruvolo & Eaton, 2013), recent that a gradient-based method can use to solve a approaches have been developed with the diversity and scale new task with few samples. The theory of meta- of modern applications in mind. This has given rise to learning is still in its early stages, with several re- simple, model-agnostic methods that focus on learning a cent learning-theoretic analyses of methods such good initialization for some gradient-based method such as as Reptile (Nichol et al., 2018) being for convex stochastic gradient descent (SGD), to be run on samples models. This work shows that convex-case analy- from a new task (Finn et al., 2017; Nichol et al., 2018). sis might be insufficient to understand the success These methods have found widespread applications in a of meta-learning, and that even for non-convex variety of areas such as computer vision (Nichol et al., 2018), models it is important to look inside the optimiza- reinforcement learning (Finn et al., 2017), and federated tion black-box, specifically at properties of the learning (McMahan et al., 2017). optimization trajectory. We construct a simple meta-learning instance that captures the problem Inspired by their popularity, several recent learning-theoretic of one-dimensional subspace learning.
    [Show full text]
  • B.Sc Computer Science with Specialization in Artificial Intelligence & Machine Learning
    B.Sc Computer Science with Specialization in Artificial Intelligence & Machine Learning Curriculum & Syllabus (Based on Choice Based Credit System) Effective from the Academic year 2020-2021 PROGRAMME EDUCATIONAL OBJECTIVES (PEO) PEO 1 : Graduates will have solid basics in Mathematics, Programming, Machine Learning, Artificial Intelligence fundamentals and advancements to solve technical problems. PEO 2 : Graduates will have the capability to apply their knowledge and skills acquired to solve the issues in real world Artificial Intelligence and Machine learning areas and to develop feasible and reliable systems. PEO 3 : Graduates will have the potential to participate in life-long learning through the successful completion of advanced degrees, continuing education, certifications and/or other professional developments. PEO 4 : Graduates will have the ability to apply the gained knowledge to improve the society ensuring ethical and moral values. PEO 5 : Graduates will have exposure to emerging cutting edge technologies and excellent training in the field of Artificial Intelligence & Machine learning PROGRAMME OUTCOMES (PO) PO 1 : Develop knowledge in the field of AI & ML courses necessary to qualify for the degree. PO 2 : Acquire a rich basket of value added courses and soft skill courses instilling self-confidence and moral values. PO 3 : Develop problem solving, decision making and communication skills. PO 4 : Demonstrate social responsibility through Ethics and values and Environmental Studies related activities in the campus and in the society. PO 5 : Strengthen the critical thinking skills and develop professionalism with the state of art ICT facilities. PO 6 : Quality for higher education, government services, industry needs and start up units through continuous practice of preparatory examinations.
    [Show full text]
  • Computational Learning Theory – Part 2
    Computational Learning Theory – Part 2 Reading: • Mitchell chapter 7 Suggested exercises: • 7.1, 7.2, 7.5, 7.7 Machine Learning 10-701 Tom M. Mitchell Machine Learning Department Carnegie Mellon University March 22, 2010 1 What it means [Haussler, 1988]: probability that the version space is not ε-exhausted after m training examples is at most Suppose we want this probability to be at most δ 1. How many training examples suffice? 2. If then with probability at least (1-δ): Agnostic Learning Result we proved: probability, after m training examples, that H contains a hypothesis h with zero training error, but true error greater than ε is bounded Agnostic case: don’t know whether H contains a perfect hypothesis overfitting 2 General Hoeffding Bounds • When estimating the mean θ inside [a,b] from m examples • When estimating a probability θ is inside [0,1], so • And if we’re interested in only one-sided error, then 3 Sufficient condition: Holds if L requires only a polynomial number of training examples, and processing per example is polynomial Sample Complexity based on VC dimension How many randomly drawn examples suffice to ε-exhaust VSH,D with probability at least (1-δ)? ie., to guarantee that any hypothesis that perfectly fits the training data is probably (1-δ) approximately (ε) correct Compare to our earlier results based on |H|: 4 VC(H)=3 VC dimension: examples What is VC dimension of lines in a plane? • H2 = { ((w0 + w1x1 + w2x2)>0 y=1) } 5 VC dimension: examples What is VC dimension of • H2 = { ((w0 + w1x1 + w2x2)>0 y=1) } – VC(H2)=3 • For Hn = linear separating hyperplanes in n dimensions, VC(Hn)=n+1 Can you give an upper bound on VC(H) in terms of |H|, for any hypothesis space H? (hint: yes) 6 More VC Dimension Examples to Think About • Logistic regression over n continuous features – Over n boolean features? • Linear SVM over n continuous features • Decision trees defined over n boolean features F: <X1, ..
    [Show full text]