Comment on ‘A multiple process solution …’ (B. Macwhinney)

2004 ◽  
Vol 31 (4) ◽  
pp. 941-943
Author(s):  
PARTHA NIYOGI
Keyword(s):  
Learning Algorithm ◽  
Computational Learning Theory ◽  
Vc Dimension ◽  
Empirical Process Theory ◽  
Generalization Problem ◽  
The Family ◽  
Finite Set ◽  
Multiple Process ◽  
Probably Approximately Correct ◽  
Class Of Functions

The central question posed by the so called ‘logical problem of language acquisition’ is how it comes to be that children are able to GENERALIZE from a finite set of linguistic data to acquire (learn, develop, grow) a computational system (grammar) that applies to novel examples not encountered before. The difficulty of this generalization problem was first posed cogently by Gold and while Macwhinney discusses the Gold framework and the linguistic literature on this matter, it is worth noting that the Gold framework is not the only one. There are at least two important new sources of insight from computational learning theory in the decades following Gold that need to be kept in mind. First, there is the development of empirical process theory that forms the basis of any analysis of statistical learning (see summary in Vapnik, 1998). Applying this approach to language (see Niyogi, 1998 for a treatment), one concludes that the family of learnable grammars must have a finite Vapnik Chervonenkis (VC) dimension. The VC dimension is a combinatorial measure of the complexity of a class of functions. Grammars may be viewed as functions mapping sentences to their grammaticality value. In this more sophisticated sense of the VC dimension, the class of grammars must be constrained. Second, there is the development of the theory of computational complexity suggesting that while a learning algorithm might exist, it may not be efficient, i.e. run in polynomial time. These two developments come together in the influential Probably Approximately Correct (PAC) model (Valiant, 1984).

scholarly journals Extremal problems for a class of functions of positive real part and applications

1986 ◽  
Vol 41 (2) ◽  
pp. 152-164
Author(s):  
V. V. Anh ◽  
P. D. Tuan
Keyword(s):  
Lower Bound ◽  
Extremal Problems ◽  
Positive Real Part ◽  
Positive Real ◽  
Sharp Bounds ◽  
The Family ◽  
Radius Of Convexity ◽  
Class Of Functions

AbstractIn this paper we determine the lower bound on |z| = r < 1 for the functional Re{αp(z) + β zp′(z)/p(z)}, α ≧0, β ≧ 0, over the class Pk (A, B). By means of this result, sharp bounds for |F(z)|, |F',(z)| in the family and the radius of convexity for are obtained. Furthermore, we establish the radius of starlikness of order β, 0 ≦ β < 1, for the functions F(z) = λf(Z) + (1-λ) zf′ (Z), |z| < 1, where ∞ < λ <1, and .

The VC Dimension and Pseudodimension of Two-Layer Neural Networks with Discrete Inputs

1996 ◽  
Vol 8 (3) ◽  
pp. 625-628 ◽  
Author(s):  
Peter L. Bartlett ◽  
Robert C. Williamson
Keyword(s):  
Neural Networks ◽  
Basis Function ◽  
Upper Bounds ◽  
Pac Learning ◽  
Learning Performance ◽  
Sigmoid Function ◽  
Vc Dimension ◽  
Learning Framework ◽  
Probably Approximately Correct ◽  
Training Examples

We give upper bounds on the Vapnik-Chervonenkis dimension and pseudodimension of two-layer neural networks that use the standard sigmoid function or radial basis function and have inputs from {−D, …,D}n. In Valiant's probably approximately correct (pac) learning framework for pattern classification, and in Haussler's generalization of this framework to nonlinear regression, the results imply that the number of training examples necessary for satisfactory learning performance grows no more rapidly than W log (WD), where W is the number of weights. The previous best bound for these networks was O(W4).

Lower Bounds on the VC Dimension of Smoothly Parameterized Function Classes

1995 ◽  
Vol 7 (5) ◽  
pp. 1040-1053 ◽  
Author(s):  
Wee Sun Lee ◽  
Peter L. Bartlett ◽  
Robert C. Williamson
Keyword(s):  
Lower Bounds ◽  
Differentiable Manifold ◽  
Function Class ◽  
Basis Functions ◽  
Vc Dimension ◽  
Probably Approximately Correct Learning ◽  
Learning Framework ◽  
Function Classes ◽  
Probably Approximately Correct ◽  
The Relationship

We examine the relationship between the VC dimension and the number of parameters of a threshold smoothly parameterized function class. We show that the VC dimension of such a function class is at least k if there exists a k-dimensional differentiable manifold in the parameter space such that each member of the manifold corresponds to a different decision boundary. Using this result, we are able to obtain lower bounds on the VC dimension proportional to the number of parameters for several thresholded function classes including two-layer neural networks with certain smooth activation functions and radial basis functions with a gaussian basis. These lower bounds hold even if the magnitudes of the parameters are restricted to be arbitrarily small. In Valiant's probably approximately correct learning framework, this implies that the number of examples necessary for learning these function classes is at least linear in the number of parameters.

scholarly journals FAIR FACILITY ALLOCATION IN EMERGENCY SERVICE SYSTEM

2020 ◽  
Vol 21 (4) ◽  
pp. 1058-1071
Author(s):  
Jaroslav Janáček ◽  
Lýdia Gábrišová ◽  
Miroslav Plevný
Keyword(s):  
Emergency Service ◽  
Service System ◽  
Allocation Problem ◽  
Dynamic Programming Principle ◽  
Objective Functions ◽  
The Family ◽  
Finite Set ◽  
Problem Instances ◽  
Original Approach ◽  
Average User

The request of equal accessibility must be respected to some extent when dealing with problems of designing or rebuilding of emergency service systems. Not only the disutility of the average user but also the disutility of the worst situated user must be taken into consideration. Respecting this principle is called fairness of system design. Unfairness can be mitigated to a certain extent by an appropriate fair allocation of additional facilities among the centres. In this article, two criteria of collective fairness are defined in the connection with the facility allocation problem. To solve the problems, we suggest a series of fast algorithms for solving of the allocation problem. This article extends the family of the original solving techniques based on branch-and-bound principle by newly suggested techniques, which exploit either dynamic programming principle or convexity and monotony of decreasing nonlinearities in objective functions. The resulting algorithms were tested and compared performing numerical experiments with real-sized problem instances. The new proposed algorithms outperform the original approach. The suggested methods are able to solve general min-sum and min-max problems, in which a limited number of facilities should be assigned to individual members from a finite set of providers.

scholarly journals Learning Union of Integer Hypercubes with Queries

2021 ◽  
pp. 243-265
Author(s):  
Oliver Markgraf ◽  
Daniel Stan ◽  
Anthony W. Lin
Keyword(s):  
Open Problem ◽  
Learning Algorithm ◽  
Computational Learning Theory ◽  
Natural Generalization ◽  
Finite Union ◽  
Computational Learning ◽  
Linear Arithmetic ◽  
Classic Problem ◽  
Fixed Dimension ◽  
Boolean Formulas

AbstractWe study the problem of learning a finite union of integer (axis-aligned) hypercubes over the d-dimensional integer lattice, i.e., whose edges are parallel to the coordinate axes. This is a natural generalization of the classic problem in the computational learning theory of learning rectangles. We provide a learning algorithm with access to a minimally adequate teacher (i.e. membership and equivalence oracles) that solves this problem in polynomial-time, for any fixed dimension d. Over a non-fixed dimension, the problem subsumes the problem of learning DNF boolean formulas, a central open problem in the field. We have also provided extensions to handle infinite hypercubes in the union, as well as showing how subset queries could improve the performance of the learning algorithm in practice. Our problem has a natural application to the problem of monadic decomposition of quantifier-free integer linear arithmetic formulas, which has been actively studied in recent years. In particular, a finite union of integer hypercubes correspond to a finite disjunction of monadic predicates over integer linear arithmetic (without modulo constraints). Our experiments suggest that our learning algorithms substantially outperform the existing algorithms.

Extrema of a Class of Functions on a Finite Set

1974 ◽  
Vol 26 (4) ◽  
pp. 806-819
Author(s):  
Kenneth W. Lebensold
Keyword(s):  
Positive Integer ◽  
Traveling Salesman Problem ◽  
Traveling Salesman ◽  
Polygonal Path ◽  
Finite Set ◽  
Vertex Set ◽  
The Traveling Salesman Problem ◽  
The Cost ◽  
Special Case ◽  
Class Of Functions

In this paper, we are concerned with the following problem: Let S be a finite set and Sm* ⊂ 2S a collection of subsets of S each of whose members has m elements (m a positive integer). Let f be a real-valued function on S and, for p ∊ Sm*, define f(P) as Σs∊pf (s). We seek the minimum (or maximum) of the function f on the set Sm*.The Traveling Salesman Problem is to find the cheapest polygonal path through a given set of vertices, given the cost of getting from any vertex to any other. It is easily seen that the Traveling Salesman Problem is a special case of this system, where S is the set of all edges joining pairs of points in the vertex set, Sm* is the set of polygons, each polygon has m elements (m = no. of points in the vertex set = no. of edges per polygon), and f is the cost function.

A new type of coding problem

2001 ◽  
Vol 38 (1-4) ◽  
pp. 139-147 ◽  
Author(s):  
G. Brightwell ◽  
Gyula Katona
Keyword(s):  
The Family ◽  
Finite Set ◽  
New Type

Let X be an n-element finite set, and 0 Let X be an n-element finite set, and are pairs of disjoint k-element subsets of X (that is, {A1  =  B1} = {A2  =  B2} = k, A1 \ B1 = A2 \ B2 = Define the distance between these pairs by d(f A1;B1 g; f A2; B2 g)=min fj A1 - A2 j +  B1 - B2 j; j A1 - B2 j + j B1 - A2 jg . Itisknown ([2]) that the family of all k-element subsets of X can be paired (with one exception if their number is odd) in such a way that the distance between any two pairs is at least k. Here we answer questions arising for distances larger than k.

On cardinality questions concerning automaton mappings

2003 ◽  
Vol 40 (1-2) ◽  
pp. 71-82
Author(s):  
A. Ádám ◽  
M. Laczkovich
Keyword(s):  
The Family ◽  
Finite Set ◽  
Positive Length ◽  
Automaton Mapping

Let F+(X) be the set of words of positive length over a finite set X. By an automaton mapping (over (X,Y)) we understand a mapping of F+(X) into a finite set Y where |Y|?1). The family of all mappings over (X,Y) may be considered as an infinite automaton U having 2? states. U has at most 2^{2?} subautomata and at most 2? countable subautomata. We show that these bounds are actually attained.

scholarly journals The uniform limit of Lipschitz functions on a Banach space

1973 ◽  
Vol 15 (3) ◽  
pp. 325-331
Author(s):  
J. V. Herod
Keyword(s):  
Banach Space ◽  
Lipschitz Functions ◽  
Bounded Subset ◽  
Real Numbers ◽  
Uniform Limit ◽  
The Family ◽  
Class Of Functions

With R the set of real numbers and S a Banach space, let ℒ be the class of functions A from R × S to S which have following properties: (1) if B is a bounded subset of S then the family {A(·, P): P is in B} is equicontinuous; i.e., if t is a number and ε > 0 then there is a positive number δ such that if |s – t | < δ and P is in B then | A(s, P) – A(t, P)| < ε.

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