org.drip.learning.bound

Class CoveringNumberLossBound

java.lang.Object

org.drip.learning.bound.CoveringNumberLossBound

public class CoveringNumberLossBound
extends java.lang.Object

CoveringNumberLossBound provides the Upper Probability Bound that the Loss/Deviation of the Empirical from the Actual Mean of the given Learner Class exceeds 'epsilon', using the Covering Number Generalization Bounds. This is expressed as C1 (n) * N (epsilon, n) * exp (-n.epsilon^b/C2), where: - n is the Size of the Sample - 'epsilon' is the Deviation Empirical Mean from the Population Mean - C1 (n) is the sample coefficient function - C2 is an exponent scaling constant - 'b' an exponent ((i.e., the Epsilon Exponent) that depends on the setting (i.e., agnostic/classification/regression/convex etc) The References are: 1) Alon, N., S. Ben-David, N. Cesa Bianchi, and D. Haussler (1997): Scale-sensitive Dimensions, Uniform Convergence, and Learnability, Journal of Association of Computational Machinery, 44 (4) 615-631. 2) Anthony, M., and P. L. Bartlett (1999): Artificial Neural Network Learning - Theoretical Foundations, Cambridge University Press, Cambridge, UK. 3) Kearns, M. J., R. E. Schapire, and L. M. Sellie (1994): Towards Efficient Agnostic Learning, Machine Learning, 17 (2) 115-141. 4) Lee, W. S., P. L. Bartlett, and R. C. Williamson (1998): The Importance of Convexity in Learning with Squared Loss, IEEE Transactions on Information Theory, 44 1974-1980. 5) Vapnik, V. N. (1998): Statistical learning Theory, Wiley, New York.

Author:

Lakshmi Krishnamurthy

Constructor Summary

Constructors

Constructor and Description
CoveringNumberLossBound(R1ToR1 funcSampleCoefficient, double dblEpsilonExponent, double dblExponentScaler) CoveringNumberLossBound Constructor

Method Summary

Modifier and Type	Method and Description
double	deviationProbabilityUpperBound(int iSampleSize, double dblEpsilon) Compute the Upper Bound of the Probability of the Absolute Deviation between the Empirical and the Population Means
double	epsilonExponent() Retrieve the Exponential Epsilon Exponent
double	exponentScaler() Retrieve the Exponent Scaler
R1ToR1	sampleCoefficient() Retrieve the Sample Coefficient Function

Methods inherited from class java.lang.Object

equals, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Constructor Detail

CoveringNumberLossBound

public CoveringNumberLossBound(R1ToR1 funcSampleCoefficient,
                               double dblEpsilonExponent,
                               double dblExponentScaler)
                        throws java.lang.Exception

CoveringNumberLossBound Constructor

Parameters:

funcSampleCoefficient - The Sample Coefficient Function

dblEpsilonExponent - The Epsilon Exponent

dblExponentScaler - The Exponent Scaler

Throws:

java.lang.Exception - Thrown if the Inputs are Invalid

Method Detail

sampleCoefficient

public R1ToR1 sampleCoefficient()

Retrieve the Sample Coefficient Function

Returns:

The Sample Coefficient Function

epsilonExponent

public double epsilonExponent()

Retrieve the Exponential Epsilon Exponent

Returns:

The Exponential Epsilon Exponent

exponentScaler

public double exponentScaler()

Retrieve the Exponent Scaler

Returns:

The Exponent Scaler

deviationProbabilityUpperBound

public double deviationProbabilityUpperBound(int iSampleSize,
                                             double dblEpsilon)
                                      throws java.lang.Exception

Compute the Upper Bound of the Probability of the Absolute Deviation between the Empirical and the Population Means

Parameters:

iSampleSize - The Sample Size

dblEpsilon - The Deviation between Population and Empirical Means

Returns:

The Upper Bound of the Probability of the Deviation between the Empirical and the Population Means

Throws:

java.lang.Exception - Thrown if the Upper Bound of the Probability cannot be computed