# Example of a machine Probably Approximately Correct (PAC) Learning PAC-learning

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## Example of a machine Probably Approximately Correct (PAC) Learning PAC-learning

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Example of a machine f(x,h) consists of all logical sentences about X1, X2 . Xm that contain only logical ands. Example hypotheses: X1 ^ X3 ^ X19 X3 ^ X18 X7 X1 ^ X2 ^ X2 ^ x4 ? ^ Xm Question: if there are 3 attributes, what is the complete set of hypotheses in f? (H = 8) And-Positive-Literals Machine f(x,h) consists of all logical sentences about X1, X2 . Xm that contain only logical ands. Example hypotheses: X1 ^ X3 ^ X19 X3 ^ X18 X7 X1 ^ X2 ^ X2 ^ x4 ? ^ Xm Question: if there are m attributes, how many hypotheses in f? And-Positive-Literals Machine f(x,h) consists of all logical sentences about X1, X2 . Xm that contain only logical ands. Example hypotheses: X1 ^ X3 ^ X19 X3 ^ X18 X7 X1 ^ X2 ^ X2 ^ x4 ? ^ Xm Question: if there are m attributes, how many hypotheses in f? (H = 2m)

And-Literals Machine f(x,h) consists of all logical sentences about X1, X2 . Xm or their negations that contain only logical ands. Example hypotheses: X1 ^ ~X3 ^ X19 X3 ^ ~X18 ~X7 X1 ^ X2 ^ ~X3 ^ ? ^ Xm Question: if there are 2 attributes, what is the complete set of hypotheses in f? And-Literals Machine f(x,h) consists of all logical sentences about X1, X2 . Xm or their negations that contain only logical ands. Example hypotheses: X1 ^ ~X3 ^ X19 X3 ^ ~X18 ~X7 X1 ^ X2 ^ ~X3 ^ ? ^ Xm Question: if there are 2 attributes, what is the complete set of hypotheses in f? (H = 9) And-Literals Machine f(x,h) consists of all logical sentences about X1, X2 . Xm or their negations that contain only logical ands. Example hypotheses: X1 ^ ~X3 ^ X19 X3 ^ ~X18 ~X7 X1 ^ X2 ^ ~X3 ^ ? ^ Xm Question: if there are m attributes, what is the size of the complete set of hypotheses in f?

Service Oriented Architecture Readings on Schedule Today?s Outline SAML 2.0 SAML 2.0 is widely implemented SAML 2.0 SAML 2.0 SAML 2.0 Bottom Line Important SAML 2.0 Drivers Terminology From SAML Spec SAML 2.0 Specification Defines(1) SAML 2.0 Specification Defines(2) SAML 2.0 Request Types Authentication Query Attribute Query Authorization Decision Query SAML WS Response A SAML WS Response SAML Assertions Authentication Statement Attribute Statement Authorization Decision Statement Authorization Decision Statement Web SSO Use Case Business Transaction Use Case Authorization Use Case Lower level Use Cases Lower Level Use Cases SAML Replay Attack OpenID The OpenID protocol Now, some options?

And-Literals Machine f(x,h) consists of all logical sentences about X1, X2 . Xm or their negations that contain only logical ands. Example hypotheses: X1 ^ ~X3 ^ X19 X3 ^ ~X18 ~X7 X1 ^ X2 ^ ~X3 ^ ? ^ Xm Question: if there are m attributes, what is the size of the complete set of hypotheses in f? (H = 3m) Lookup Table Machine f(x,h) consists of all truth tables mapping combinations of input attributes so that true in addition to false Example hypothesis: Question: if there are m attributes, what is the size of the complete set of hypotheses in f? Lookup Table Machine f(x,h) consists of all truth tables mapping combinations of input attributes so that true in addition to false Example hypothesis: Question: if there are m attributes, what is the size of the complete set of hypotheses in f?

A Game We specify f, the machine Nature choose hidden random hypothesis h* Nature randomly generates R datapoints How is a datapoint generated? Vector of inputs xk = (xk1,xk2, xkm) is drawn from a fixed unknown distrib: D The corresponding output yk=f(xk , h*) We learn an approximation of h* by choosing some hest in consideration of which the training set error is 0 Test Error Rate We specify f, the machine Nature choose hidden random hypothesis h* Nature randomly generates R datapoints How is a datapoint generated? Vector of inputs xk = (xk1,xk2, xkm) is drawn from a fixed unknown distrib: D The corresponding output yk=f(xk , h*) We learn an approximation of h* by choosing some hest in consideration of which the training set error is 0 For each hypothesis h , Say h is Correctly Classified (CCd) if h has zero training set error Define TESTERR(h ) = Fraction of test points that h will classify correctly = P(h classifies a random test point correctly) Say h is BAD if TESTERR(h) > e Test Error Rate We specify f, the machine Nature choose hidden random hypothesis h* Nature randomly generates R datapoints How is a datapoint generated? Vector of inputs xk = (xk1,xk2, xkm) is drawn from a fixed unknown distrib: D The corresponding output yk=f(xk , h*) We learn an approximation of h* by choosing some hest in consideration of which the training set error is 0 For each hypothesis h , Say h is Correctly Classified (CCd) if h has zero training set error Define TESTERR(h ) = Fraction of test points that i will classify correctly = P(h classifies a random test point correctly) Say h is BAD if TESTERR(h) > e

Test Error Rate We specify f, the machine Nature choose hidden random hypothesis h* Nature randomly generates R datapoints How is a datapoint generated? Vector of inputs xk = (xk1,xk2, xkm) is drawn from a fixed unknown distrib: D The corresponding output yk=f(xk , h*) We learn an approximation of h* by choosing some hest in consideration of which the training set error is 0 For each hypothesis h , Say h is Correctly Classified (CCd) if h has zero training set error Define TESTERR(h ) = Fraction of test points that i will classify correctly = P(h classifies a random test point correctly) Say h is BAD if TESTERR(h) > e PAC Learning Chose R such that alongside probability less than d we?ll select a bad hest (i.e. an hest which makes mistakes more than fraction e of the time) Probably Approximately Correct As we just saw, this can be achieved by choosing R such that i.e. R such that PAC in action

PAC in consideration of decision trees of depth k Assume m attributes Hk = Number of decision trees of depth k H0 =2 Hk+1 = (#choices of root attribute) * (# possible left subtrees) * (# possible right subtrees) = m * Hk * Hk Write Lk = log2 Hk L0 = 1 Lk+1 = log2 m + 2Lk So Lk = (2 k-1)(1+log2 m) +1 So so that PAC-learn, need What you should know Be able so that understand every step in the math that gets you so that Understand that you thus need this many records so that PAC-learn a machine alongside H hypotheses Understand examples of deducing H in consideration of various machines

## Sears, Jim Host

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