# Weak Learning DNF under uniform d by cuiliqing

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```									Weak Learning DNF under uniform
distribution
• A parity function weakly approximates f
– Find this function

• KM algorithm
– Form a tree, pruning a node if there are no
large coefficients starting with that substring
Strong learning: Boosting
• Recall Freund’s algo
– Construct weak hypotheses h1,h2,h3…
– At step i:
Distribution Di: more weight to x on which hi,…,hi-1 were wrong
Form hypo hi on Di
– Combine hypotheses using some rule

• Does parity approximate f on Di?
– Yes.. Answer on next slide
• Needs a distribution-independent weak learner
– We don’t have one for DNF
Strong learning DNF
• Let f be a DNF having s-terms
• Lemma: f has a fourier coefficient of value
at least 1/(2s+1)
Can we tweak KM?
• Cool fact: KM works for real-valued
functions as well

• Idea: Construct function g such that

• Depends on L(g) in running time
Converting (f,D) to (g,U)
• Notice that            on dist D is same as
on uniform

• We can learn           on U!!

• Need MQ oracle for g => MQ oracle for D
• L(g) should not be too large
“D close to uniform”
An appropriate Boosting algorithm
• Final hypothesis – majority of his
• Di = D(x)i(x)  has to be normalized
i(x)  prob that hypotheses are almost equally
divided over x
• Stop if Dis become too small

• Notice: easily computable
• Close to uniform – within a small factor

```
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