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Machine Learning with Discriminative Methods
Lecture 02 – PAC Learning and tail bounds intro
CS 790-134 Spring 2015
Alex Berg
Today’s lecture
• PAC Learning
• Tail bounds…
Rectangle learning
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Hypothesis H
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+
+
+
+
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Hypothesis is any axis aligned rectangle. Inside rectangle is positive.
Rectangle learning – Realizable case
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Actual boundary is also an
axis-aligned rectangle,
“The Realizable Case”
(no approximation error)
Hypothesis H
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+
+
+
+
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Rectangle learning – Realizable case
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Actual boundary is also an
axis-aligned rectangle,
“The Realizable Case”
(no approximation error)
Hypothesis H
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+
+
+
+
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A mistake for the
hypothesis H!
Measure ERROR by the probability of making a mistake.
Rectangle learning – a strategy for a learning
algorithm…
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Hypothesis H
(Output of
learning algorithm
so far…)
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+
+
+
+
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Make the smallest rectangle consistent with all the data so far.
Rectangle learning – making a mistake
Current
hypothesis
makes a mistake
on a new data
item…
Hypothesis H
(Output of
learning algorithm
so far…)
-
+
+
+
+
+
-
-
Make the smallest rectangle consistent with all the data so far.
Rectangle learning – making a mistake
Current
hypothesis
makes a mistake
on a new data
item…
+
+
Hypothesis H
(Output of
learning algorithm
so far…)
-
+
+
Use probability
+
of such a
mistake (this is
our error
measure) to find
a bound for how
likely it was we
had not yet seen
a training
example in this
region…
Make the smallest rectangle consistent with all the data so far.
Very subtle formulation…
R = Actual
decision boundary
R’ = Result of
algorithm so far (after
m sample)
From the Kearns and Vazirani Reading
From the Kearns and Vazirani Reading
PAC Learning
Flashback: Learning/fitting is a process…
Estimating the probability that a tossed coin comes up heads…
The i’th coin toss
Estimator based on n tosses
Estimate is within epsilon
Estimate is not within epsilon
Probability of being bad is inversely proportional to the number of samples…
(the underlying computation is an example of a tail bound)
From Raginsky notes
Markov’s Inequality
From Raginksy’s notes
Chebyshev’s Inequality
From Raginksy’s notes
Not quite good enough…
From Raginksy’s notes
For next class
• Read the wikipedia page for Chernoff Bound:
http://en.wikipedia.org/wiki/Chernoff_bound
• Read at least first Raginsky’s introductory notes on
tail bounds (pages 1-5)
http://maxim.ece.illinois.edu/teaching/fall14/notes/concentr
ation.pdf
Come to class with questions! It is fine to have
questions, but first spend some time trying to work
through reading/problems. Feel free to post questions
to Sakai discussion board!

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