DOCSLIB.ORG

Learning Nested Concept Classes with Limited Storage*

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Learning Nested Concept Classes with Limited Storage* Learning Nested Concept Classes with Limited Storage* David Heath, Simon Kasif, S. Rao Kosaraju, Steven Salzberg, and Gregory Sullivan Department of Computer Science The Johns Hopkins University Baltimore, MD 21218 Abstract ceptron methods, neural nets, and decision tree tech• Many existing learning methods use incre• niques all fit this paradigm. Most of these methods do mental algorithms that construct a general• not store the entire set of training data. When process• ization in one pass through a set of training ing is completed, the only thing they store is a general• data and modify it in subsequent passes (e.g., ized data structure such as a tree, a matrix of weights, perceptrons, neural nets, and decision trees). or a set of geometric clusters. The issue of limiting the Most of these methods do not store the en• storage of a learning algorithm is one abstraction of the tire training set, in essence employing a limited notion of a compressed representation. The question we storage requirement that abstracts the notion address is how many passes through a data set arc re• of a compressed representation. The question quired to obtain a "correct" (i.e., accurate but minimum we address is, how much additional processing in size) generalization if we are only allowed to store the time is required for methods with limited stor• generalization. This issue is equivalent to analyzing an age? Processing time for learning algorithms algorithm that has a limited storage requirement. is equated in this paper with the number of Fixed storage is an important consideration for sev• passes necessary through a data set to obtain eral reasons. First of all, some learning models al• a correct generalization. For instance, neural ways use fixed storage. Neural net learning algorithms, nets require many passes through a data set for example, have a fixed number of nodes and edges, before converging. Decision trees require fewer and only change the weights on the edges. Decision passes, but precise bounds are unknown. tree algorithms can in principle grow without bound, We consider limited storage algorithms for but in practice researchers have devised many tech• a particular concept class, nested hyperrect- niques for restricting their growth [Quinlan 1986, Ut- angles. We prove bounds that illustrate the goff 1989]. Instance-based techniques such as those of fundamental trade-off between storage require• Salzberg [1989ab] and Aha and Kibler [1989] attempt to ments and processing time required to learn an store as few examples as possible in order to minimize optimal structure. It turns out that our lower storage. Second, fixed storage is a realistic constraint bounds apply to other algorithms and concept from the perspective of cognitive modelling - human classes (e.g., decision trees) as well. Notably, learning behavior clearly must adhere to some storage imposing storage limitations on the learning limitations. Finally, there is experimental evidence that task forces one to devise a completely different restricting storage actually leads to better performance, algorithm to reduce the number of passes. We especially if the input data is noisy [Aha and Kibler 1989, also briefly discuss parallel learning algorithms. Quinlan 1986]. The intuition behind this result is that by throwing away noisy data, an algorithm can construct 1 Introduction a more accurate generalization. Many existing learning methods attempt to create con• Our results show that if a fixed-storage algorithm at• cise generalizations from a set of examples. Besides sav• tempts to create a simple concept structure, then it can• ing storage, small generalizations are easier to summa• not generalize on-line without losing accuracy By "sim• rize and communicate to others. A common learning ple" we mean a generalization that is both minimum in method will construct a generalization after one pass size and an accurate model of the data; i.e., it classifies through a set of training data, and modify it in subse• all the training examples correctly. We also show that quent passes to make it smaller or more accurate. Per- by making a number of additional passes (depending on the number of concepts) through the data set, an algo• *This research supported in part by Air Force Office of Scientific Research under Grant AFOSR-89-1151, Na• rithm can create an optimal structure. Our main goal is tional Science Foundation under Grant IRI-88-09324 and to demonstrate that there exists a fundamental trade-off NSF/DARPA under Grant CCR-8908092. between the storage available and the number of passes 1 Supported by NSF under grant CCR-8804284 and required for learning an optimal structure. There are few NSF/DARPA under grant CCR-8808092. results comparable to ours on the number of passes re- Heath, et al. 777 quired to learn a concept. Typical theory results, rather, give estimates of the size of the input data set, not of the number of presentations required. This work is an im• portant step towards formalizing the capabilities of in• cremental vs. non-incremental learning algorithms. Any algorithm that does not save all training examples is to some extent incremental, since upon presentation of new inputs the algorithm cannot re-compute a generalization using all previous examples. The learning framework considered in this paper is computing generalizations in the form of geometric con• cept classes. Many important learning algorithms fall in this category, e.g., perceptron learning [Rosenblatt 1959], instance-based learning [Aha and Kibler 1989], decision tree models [Quinlan 1986] and hyperrectangles [Salzberg 1990]. We focus on the concept class defined Figure 1: Categorizing points using rectangles by nested hyperrectangles, although many of our results are applicable to other concept classes. In fact, our im• possibility results apply to any convex partitioning of tersection closed classes, which include orthogonal rect• feature space, such as decision trees and perceptrons. angles in Rn, monomials, and other concepts. The main The learning model we consider has fixed number of assumption behind their model is that it must be possi• storage locations; i.e., it is not permitted to store and ble to classify the training examples with strictly nested process all training examples at once. The limited stor• rectangles. age requirement is applicable only during the learning (or training) phase. That is, our algorithms must operate Given that the learning community has found nested incrementally, storing a limited number of intermediate rectangles a useful concept class, and that the theoretical results during each pass through a data set and modi• community has proven some further results about this fying the partially constructed generalization in subse• same concept class, we have been led to investigate the quent passes. For both cognitive and practical reasons, ability of an algorithm to construct an optimal number much experimental learning research has focused on the of nested rectangles given only limited storage. We will development of incremental models [Utgoff 1989]. argue below that our results apply to other well-known concept classes, including the partitionings induced by 2 Nested Hyperrectangles decision trees and perceptrons. Recent experimental research on learning from exam• 3 Preliminaries ples has shown that concepts in the shape of hyperrect• angles are a useful generalization in a variety of real- An example is defined simply as a vector of real-valued world domains [Salzberg 1989ab, 1990]. In this work, numbers, plus a category label. For instance, we may be rectangular-shaped generalizations are created from the considering a problem where medical patients are rep• training examples, and are then used for classification. resented by a set of real numbers including heart rate, Rectangles may be nested inside one another to arbitrary blood pressure, etc., and our task is to categorize the depth, and new examples are classified by the innermost patients as "in-patient" or "out-patient." For our pur• rectangle containing them. Thus nested rectangles may poses, an example is just a point in Euclidean space, be thought of as exceptions to the surrounding rectan• where the dimensionality of the space is determined by gles. This learning model is called the Nested Gener• the number of attributes measured for each example. We alized Exemplar (NGE) model. Experimental results will categorize points by using axis-parallel hyperrectan• with this model thus far have shown that it compares gles, where each rectangle Ri is labeled with a category very favorably with several other models, including de• C(R{) such as "out-patient." A point is categorized by cision trees, rule-based methods, statistical techniques, the innermost rectangle containing it. Figure 1 illus• and neural nets [Salzberg 89ab]. trates how categories are assigned. In the figure, lower- Independently of the experimental work cited above, case letters indicate points belonging to categories a and Helmbold, Sloan, and Warmuth [1989] have produced b, and uppercase letter indicate the category labels of very promising theoretical results for nested rectangular the rectangles. Notice that points not contained by any concepts. In particular, they have developed a learning rectangle are assigned to category A, which corresponds algorithm for binary classification problems that creates to a default category. Only two rectangles are required strictly nested rectangles, and that makes predictions to classify all the points in Figure 1. about new examples on-line. They have proven that The general problem definition is as follows: we are their algorithm is optimal with respect to several cri• given n points in a d-dimensional space, and we are asked teria, including the probability that the algorithm pro• to construct a minimum set of strictly nested hyperrect• duces a hypothesis with small error [Valiant 1984] and angles that will correctly classify the set. We will assume the expected total number of mistakes for classification that each point belongs to one of two classes (i.e., we of the first t examples.
Load more

Recommended publications
  • Iterated Nearest Neighbors and Finding Minimal Polytopes* 1
    Iterated Nearest Neighb ors and Finding Minimal Polytop es y z David Eppstein Je Erickson Abstract Weintro duce a new metho d for nding several typ es of optimal k -p oint sets, minimizing p erimeter, diameter, circumradius, and related measures, by testing sets of the O (k ) nearest neighb ors to each p oint. We argue that this is b etter in a number of ways than previous algorithms, whichwere based on high order Voronoi diagrams. Our technique allows us for the rst time to eciently maintain minimal sets as new p oints are inserted, to generalize our algorithms to higher dimensions, to nd minimal convex k -vertex p olygons and p olytop es, and to improvemany previous results. Weachievemany of our results via a new algorithm for nding rectilinear nearest neighb ors in the plane in time O (n log n + kn). We also demonstrate a related technique for nding minimum area k -p oint sets in the plane, based on testing sets of nearest vertical neighbors to each line segment determined by a pair of p oints. A generalization of this technique also allows us to nd minimum volume and b oundary measure sets in arbitrary dimensions. 1 Intro duction Anumb er of recent pap ers have discussed problems of selecting, from a set of n p oints, the k points optimizing some particular criterion [2, 14, 20]. Criteria that have b een studied include diameter [2], variance [2], area of the convex hull [20], convex hull p erimeter [14,20], and rectilinear diameter and p erimeter [2].
    [Show full text]
  • Scenechecker: Boosting Scenario Verification Using Symmetry
    SceneChecker: Boosting Scenario Verification using Symmetry Abstractions? Hussein Sibai , Yangge Li , and Sayan Mitra University of Illinois at Urbana-Champaign Coordinated Science Laboratory {sibai2,li213,mitras}@illinois.edu Abstract. We present SceneChecker, a tool for verifying scenarios involving vehicles executing complex plans in large cluttered workspaces. SceneChecker converts the scenario verification problem to a standard hybrid system verification problem, and solves it effectively by exploiting structural properties in the plan and the vehicle dynamics. SceneChecker uses symmetry abstractions, a novel refine- ment algorithm, and importantly, is built to boost the performance of any existing reachability analysis tool as a plug-in subroutine. We evaluated SceneChecker on several scenarios involving ground and aerial vehicles with nonlinear dynamics and neural network controllers, employing different kinds of symmetries, using different reachability subroutines, and following plans with hundreds of way- points in complex workspaces. Compared to two leading tools, DryVR and Flow*, SceneChecker shows 14× average speedup in verification time, even while using those very tools as reachability subroutines1. Keywords: Hybrid systems · Safety verification · Symmetry. 1 Introduction Remarkable progress has been made in safety verification of hybrid and cyber-physical systems in the last decade [2,3,4,5,6,7,8,9]. The methods and tools developed have been applied to check safety of aerospace, medical, and autonomous vehicle control systems [4,5,10,11]. The next barrier in making these techniques usable for more com- plex applications is to deal with what is colloquially called the scenario verification problem. A key part of the scenario verification problem is to check that a vehicle or an agent can execute a plan through a complex environment.
    [Show full text]
  • Optimal Payments in Dominant-Strategy Mechanisms for Single-Parameter Domains
    Optimal Payments in Dominant-Strategy Mechanisms for Single-Parameter Domains Victor Naroditskiy Maria Polukarov Nicholas R. Jennings School of Electronics and Computer Science, University of Southampton, United Kingdom fvn,mp3,[email protected] Abstract We study dominant-strategy mechanisms in allocation domains where agents have one-dimensional types and quasi-linear utilities. Taking an allocation function as an input, we present an algorithmic technique for finding opti- mal payments in a class of mechanism design problems, including utilitarian and egalitarian allocation of homogeneous items with nondecreasing marginal costs. Our results link optimality of payment functions to a geometric condi- tion involving triangulations of polytopes. When this condition is satisfied, we constructively show the existence of an optimal payment function that is piecewise linear in agent types. 1. Introduction A celebrated positive result in dominant-strategy implementation is the class of Groves mechanism [10] for quasi-linear environments. For unrestricted type spaces, the Groves class describes all deterministic dominant-strategy mechanisms [21]. When types are given by single numbers|in single-parameter domains|it characterizes all efficient mechanisms. Mechanisms within the Groves class differ from one another in the amounts of payments they col- lect from the agents. Until recently, most of the attention in the literature was given to a single Groves mechanism called VCG.1 Our work develops a 1Also known as the Vickrey-Clarke-Groves, the pivotal, or the Clarke mechanism. technique for finding the best mechanism from the Groves class (and a more general class that includes non-efficient mechanisms) for a given objective. To see the need to optimize over mechanisms in the Groves class, consider allocating free items among a group of participants each having a private value for receiving an item (or, more generally, scenarios with no residual claimant absorbing the surplus or covering the deficit).
    [Show full text]
  • A Concentration Result of Estimating Phi-Divergence Using Data
    A Concentration Result of Estimating Phi-Divergence using Data Dependent Partition Fengqiao Luo1 and Sanjay Mehrotra2 1,2Department of Industrial Engineering and Management Science, Northwestern University September 18, 2018 Abstract Estimation of the φ-divergence between two unknown probability distributions using empirical data is a fundamental problem in information theory and statistical learning. We consider a multi-variate generalization of the data dependent partitioning method for estimating divergence between the two unknown distributions. Under the assumption that the distribution satisfies a power law of decay, we provide a convergence rate result for this method on the number of samples and hyper-rectangles required to ensure the estimation error is bounded by a given level with a given probability. 1 Introduction d Let P and Q be two probability distributions on defined on (R , Rd ), where Rd is the Borel measure on B B Rd. The φ-divergence of Q from P is defined as: dP Dφ(P Q)= φ dQ. (1) || Rd dQ Z The φ-divergence family includes the Kullback-Leibler (KL) divergence (Kullback and Leibler, 1951), Hellinger distance (Nikulin, 2001), total variation distance, χ2-divergence, α-divergence among others. Many other information-theoretic quantities such as entropy and mutual information can be formulated as special cases of φ-divergence. When the distributions P and Q are unknown, the estimate of D (P Q) based on the i.i.d. φ || samples from P and Q is a fundamental problem in information theory and statistics. The estimation of arXiv:1801.00852v1 [math.PR] 2 Jan 2018 φ-divergence as well as entropy and mutual information has many important applications.
    [Show full text]
  • Dynamic Approximate Maximum Independent Set of Intervals, Hypercubes and Hyperrectangles
    Dynamic Approximate Maximum Independent Set of Intervals, Hypercubes and Hyperrectangles Monika Henzinger Faculty of Computer Science, University of Vienna, Vienna, Austria [email protected] Stefan Neumann Faculty of Computer Science, University of Vienna, Vienna, Austria [email protected] Andreas Wiese Department of Industrial Engineering, Universidad de Chile, Santiago, Chile [email protected] Abstract Independent set is a fundamental problem in combinatorial optimization. While in general graphs the problem is essentially inapproximable, for many important graph classes there are approximation algorithms known in the offline setting. These graph classes include interval graphs and geometric intersection graphs, where vertices correspond to intervals/geometric objects and an edge indicates that the two corresponding objects intersect. We present dynamic approximation algorithms for independent set of intervals, hypercubes and hyperrectangles in d dimensions. They work in the fully dynamic model where each update inserts or deletes a geometric object. All our algorithms are deterministic and have worst-case update times that are polylogarithmic for constant d and ε > 0, assuming that the coordinates of all input objects are in [0,N]d and each of their edges has length at least 1. We obtain the following results: For weighted intervals, we maintain a (1 + ε)-approximate solution. For d-dimensional hypercubes we maintain a (1 + ε)2d-approximate solution in the unweighted case and a O(2d)-approximate solution in the weighted case. Also, we show that for maintaining an unweighted (1 + ε)-approximate solution one needs polynomial update time for d ≥ 2 if the ETH holds. For weighted d-dimensional hyperrectangles we present a dynamic algorithm with approximation ratio (1 + ε) logd−1 N.
    [Show full text]
  • Approximate Polytope Membership Queries and Applications
    UNIVERSITÉ CÔTE D’AZUR Approximate Polytope Membership Queries and Applications Mémoire d’habilitation à diriger des recherches Spécialité Informatique Guilherme Dias da Fonseca Université Clermont Auvergne, LIMOS, and INRIA Sophia Antipolis [email protected] Jean Cardinal Université Libre de Bruxelles Rapporteur Timothy Chan University of Illinois at UC Rapporteur Nabil Mustafa ESIEE Paris Rapporteur Jean-Daniel Boissonnat INRIA, Sophia Antipolis Garant Victor Chepoi Aix-Marseille Université Examinateur Bruno Martin Université de Nice Sophia Antipolis Examinateur David M. Mount University of Maryland Examinateur June 8, 2018 Abstract In the polytope membership problem, we are given a convex polytope P Rd for constant d 2, and the objective is to preprocess P into a data structure so that,⊂ given any query ≥ point q Rd, it is possible to determine efficiently whether q P . We consider this problem2 in an approximate setting. Given an approximation parameter2 " > 0, an "- approximate polytope membership query returns a positive result if q P , a negative result if the distance from q to P is greater than " diam(P ), and it may2 return either result otherwise. · We presented the first data structures especially tailored to approximate polytope (d 1)=4 membership. Initially, we showed how to answer queries in O(1=" − ) time using (d 1)=2 optimal O(1=" − ) storage. Later, we improved the analysis of the same data structure (d 1)=8 to O(log(1=")=" − ) query time for the same optimal storage. Switching to a different approach, we finally obtained an optimal data structure with O(log(1=")) query time (d 1)=2 and O(1=" − ) storage.
    [Show full text]
  • Limiting Polytope Geometry for Rigid Rods, Disks, and Spheres Frank H
    Journal of Statistical Physics, VoL 1, No. 1, 1969 Limiting Polytope geometry for Rigid Rods, Disks, and Spheres Frank H. Stillinger, Jr., 1 and Zevi W. Salsburg ~ Received February 13, 1969 The available configuration space for finite systems of rigid particles separates into equivalent disconnected regions if those systems are highly compressed. This paper presents a study of the geometric properties of the limiting high-compression regions (polytopes) for rods, disks, and spheres. The molecular distribution functions represent cross sections through the convex polytopes, and for that reason they are obliged to exhibit single-peak behavior by the Brtinn-Minkowski inequality. We demonstrate that increasing system dimensionality implies tendency toward nearest-neighbor particle-pair localization away from contact. The relation between the generalized Euler theorem for the limiting polytopes and cooperative "jamming" of groups of particles is explored. A connection is obtained between the moments of inertia of the polytopes (regarded as solid homogeneous bodies) and crystal elastic properties. Finally, we provide a list of unsolved problems in this geometrical many-body theory. KEY WORDS: Rigid spheres; Rigid disks; Rigid rods; Elasticity; High pressure; Polytopes; Convexity; Crystal anharmonicity; Pair correlation functions; Multidi- mensional geometry; Crystalline order; Crystal defects. 1. INTRODUCTION The hard-sphere system has been one of the most important and convenient models available to the theoretician during the development of modern statistical mechanics. Its attractiveness is largely due to easy visualization of energetically allowed particle configurations. Also, the relevant mathematical analysis, while generally far from trivial, is still relatively simpler than corresponding analysis for other molecular models which specify "softer" collisions.
    [Show full text]
  • On Special Orthogonal Group and Variable Selection
    SJÄLVSTÄNDIGA ARBETEN I MATEMATIK MATEMATISKA INSTITUTIONEN, STOCKHOLMS UNIVERSITET On special orthogonal group and variable selection av Iris van Rooijen 2016 - No 9 MATEMATISKA INSTITUTIONEN, STOCKHOLMS UNIVERSITET, 106 91 STOCKHOLM On special orthogonal group and variable selection Iris van Rooijen Självständigt arbete i matematik 30 högskolepoäng, avancerad nivå Handledare: Yishao Zhou 2016 Abstract We will look at Euler angle representation of rotations, and the even subalgebras in Clifford algebra which form a double cover of the special orthogonal group. Some of their properties and metrics will be compared and some suggestions made on how they could be applied in Nordlings variable selection system for high dimensions. This will hopefully serve as an introduction to create interest in the subject and shed some light the difficulties that occur at different stages. Contents 1 Introduction 1 2 About variable selection and linear regression 2 2.1 Defenitions of variable selection . 2 2.1.1 Optimisation versus robustness . 2 2.1.2 Some different techniques . 2 2.1.3 Validation of predictability or robustness . 3 2.2 Introduction to variable selection using linear regression . 4 2.3 Linear regression and uncertainty . 5 2.3.1 Properties of a rotation matrix . 5 2.3.2 Short on Nordling’s system . 5 3 Rotation 14 3.1 Representation of the rotation matrix . 14 3.1.1 Euler angles . 15 n 1 3.1.2 Generalized Euler theorem of rotations, SO(n) and S − 19 3.1.3 Quaternions H, H H and groups SO(3) and SO(4) . 24 × 3.1.4 Clifford algebra, an extension of quaternions to higher dimensions .
    [Show full text]
  • Computing a Minimum-Width Cubic and Hypercubic Shell∗
    Computing a Minimum-Width Cubic and Hypercubic Shell∗ Sang Won Baey 16th April, 2019 01:21 Abstract In this paper, we study the problem of computing a minimum-width axis-aligned cubic shell that encloses a given set of n points in a three-dimensional space. A cubic shell is a closed volume between two concentric and face-parallel cubes. Prior to this work, there was no known algorithm for this problem in the literature. We present the first nontrivial algo- rithm whose running time is O(n log2 n). Our approach easily extends to higher dimension, resulting in an O(nbd=2c logd−1 n)-time algorithm for the hypercubic shell problem in d ≥ 3 dimension. Keywords: facility location, geometric optimization, exact algorithm, cubic shell, hypercu- bic shell, minimum width 1 Introduction The minimum-width circular annulus problem asks to find an annulus of the minimum width, determined by two concentric circles, that encloses a given set P of n points in the plane. It has an application to the points-to-circle matching problem, the minimum-regret facility location, and the roundness problem. After early results on the circular annulus problem [12, 13], the currently best algorithm that computes a minimum-width circular annulus that encloses n input 3 + points takes O(n 2 ) time [2, 3]. Along with these applications and with natural theoretical interests, the minimum-width an- nulus problem and its variants have recently been attained a lot of attention by many researchers, resulting in various efficient algorithms. Abellanas et al. [1] considered minimum-width rect- angular annuli that are axis-parallel, and presented two algorithms taking O(n) or O(n log n) time: one minimizes the width over rectangular annuli with arbitrary aspect ratio and the other does over rectangular annuli with a prescribed aspect ratio, respectively.
    [Show full text]
  • Maximal and Convex Layers of Random Point Sets
    MAXIMAL AND CONVEX LAYERS OF RANDOM POINT SETS by Cuong Phu Nguyen Submitted in partial fulfillment of the requirements for the degree of Master of Computer Science at Dalhousie University Halifax, Nova Scotia October 2017 c Copyright by Cuong Phu Nguyen, 2017 This thesis is dedicated to my wife, who has been patiently waiting until it is finished. ii Table of Contents List of Figures ...................................... v Abstract .......................................... vi List of Abbreviations and Symbols Used ...................... vii Acknowledgements ................................... viii Chapter 1 Introduction .............................. 1 1.1 Motivation . .2 1.2 Our contribution . .4 1.3 Outline . .5 Chapter 2 Previous work ............................. 6 2.1 Algorithms computing maximal and convex layers . .6 2.2 Constant factors in the number of comparisons . .8 2.3 Complexity of geometric structure . .9 Chapter 3 Algorithms for maximal and convex layers ........... 10 3.1 Algorithm Overview . 10 3.1.1 The original algorithm to compute one layer . 10 3.1.2 An overview of our algorithm to compute multiple layers . 12 3.2 Maximal Layers . 12 3.3 Convex Layers in Two and Three Dimensions . 19 3.4 Convex Layers in Four or Higher Dimensions . 24 Chapter 4 Expected Size of the First k Layers ............... 28 4.1 Constant Maximal Layers in 2D . 28 4.2 The First k Maximal Layers . 31 4.3 The First k Orthant Layers . 33 Chapter 5 Conclusions ............................... 36 iii Bibliography ....................................... 38 Appendix A LazySelect for Elements Far From the Median ......... 41 iv List of Figures 1.1 Maximal layers and Convex layers in two dimensional space. .2 3.1 The inner and outer regions used in Bentley et al.'s and our algorithm illustrated for the 2D case.
    [Show full text]
  • Arxiv:1506.00034V3 [Cs.IT]
    Bracketing numbers of convex and m-monotone functions on polytopes Charles R. Doss1 School of Statistics University of Minnesota 224 Church St SE #313 Minneapolis, MN 55455 Abstract We study bracketing covering numbers for spaces of bounded convex functions in the Lp norms. Bracketing numbers are crucial quantities for understanding asymptotic behavior for many statistical nonparametric estimators. Bracketing number upper bounds in the supre- mum distance are known for bounded classes that also have a fixed Lipschitz constraint. However, in most settings of interest, the classes that arise do not include Lipschitz con- straints, and so standard techniques based on known bracketing numbers cannot be used. In this paper, we find upper bounds for bracketing numbers of classes of convex functions without Lipschitz constraints on arbitrary polytopes. Our results are of particular interest in many multidimensional estimation problems based on convexity shape constraints. Additionally, we show other applications of our proof methods; in particular we define a new class of multivariate functions, the so-called m-monotone functions. Such functions have been considered mathematically and statistically in the univariate case but never in the multivariate case. We show how our proof for convex bracketing upper bounds also arXiv:1506.00034v3 [cs.IT] 14 Apr 2020 applies to the m-monotone case. Keywords: bracketing entropy, Kolmogorov metric entropy, convex functions, convex polytope, covering numbers, nonparametric estimation, convergence rates 2010 MSC: Primary: 52A41, 41A46; Secondary: 52A27, 52B11, 52C17, 62G20 1. Introduction and Motivation To quantify the size of an infinite dimensional set, the pioneering work of [35] studied the so-called metric entropy of the set, which is the logarithm of the metric covering number 1Supported by NSF grant DMS-1712664 Preprint submitted to Journal of Approximation Theory April 16, 2020 of the set.
    [Show full text]
  • A Nearest Hyperrectangle Learning Method
    Machine Learning, 6, 251-276 (1991) © 1991 Kluwer Academic Publishers, Boston. Manufactured in The Netherlands. A Nearest Hyperrectangle Learning Method STEVEN SALZBERG ([email protected]) Department of Computer Science, Johns Hopkins University, Baltimore, MD 21218 Editor: Tom Dietterich Abstract. This paper presents a theory of learning called nested generalized exemplar (NGE) theory, in which learning is accomplished by storing objects in Euclidean n-space, E n, as hyperrectangles. The hyperrectangles may be nested inside one another to arbitrary depth. In contrast to generalizationprocesses that replace symbolic formulae by more general formulae, the NGE algorithm modifies hyperrectanglesby growing and reshaping them in a well-definedfashion. The axes of these hyperrectanglesare defined by the variables measured for each exam- ple. Each variable can have any range on the real line; thus the theory is not restricted to symbolicor binary values. This paper describes some advantages and disadvantages of NGE theory, positions it as a form of exemplar- based learning, and compares it to other inductive learning theories. An implementationhas been tested in three different domains, for which results are presented below: prediction of breast cancer, classificationof iris flowers, and prediction of survival times for heart attack patients. The results in these domains support the claim that NGE theory can be used to create compact representations with excellent predictive accuracy. Keywords. Exemplar, induction, generalization, prediction, incremental learning, exceptions 1. Introduction This paper presents a theory of learning from examples called Nested Generalized Exemplar (NGE) theory. NGE theory is derived from a learning model called exemplar-based learn- ing that was proposed originally as a model of human learning by Medin and Schaffer (1978).
    [Show full text]