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

Bloomsburg University of Pennsylvania, US has reference to this Academic Journal, PAC-learning Andrew W. Moore Associate Professor School of Computer Science Carnegie Mellon University cs.cmu /~awm awm@cs.cmu 412-268-7599 Note so that other teachers in addition to users of these slides. Andrew would be delighted if you found this source material useful in giving your own lectures. Feel free so that use these slides verbatim, or so that modify them so that fit your own needs. PowerPoint originals are available. If you make use of a significant portion of these slides in your own lecture, please include this message, or the following link so that the source repository of Andrew?s tutorials: cs.cmu /~awm/tutorials . Comments in addition to corrections gratefully received. Probably Approximately Correct (PAC) Learning Imagine we?re doing classification alongside categorical inputs. All inputs in addition to outputs are binary. Data is noiseless. There?s a machine f(x,h) which has H possible settings (a.k.a. hypotheses), called h1, h2 . hH. 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?

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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?

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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

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