TR02-019 Authors: Nader Bshouty, Lynn Burroughs

Publication: 22nd March 2002 15:40

Downloads: 1996

Keywords:

We study the proper learnability of axis parallel concept classes

in the PAC learning model and in the exact learning model with

membership and equivalence queries. These classes include union of boxes,

DNF, decision trees and multivariate polynomials.

For the {\it constant} dimensional axis parallel concepts $C$

we show that the following problems have the same time complexity

\begin{enumerate}

\item

$C$ is ${\alpha}$-properly exactly learnable

(with hypotheses of size

at most ${\alpha}$ times the target size)

from membership and equivalence queries.

\item

$C$ is ${\alpha}$-properly PAC learnable (without membership queries)

under any product distribution.

\item

There is an ${\alpha}$-approximation algorithm

for the $\MinEqui C$ problem. (given a $g\in C$

find a minimal size $f\in C$ that is equivalent to

$g$).

\end{enumerate}

In particular, $C$ is ${\alpha}$-properly

learnable in polynomial time from membership and equivalence

queries {\bf if and only if} $C$ is ${\alpha}$-properly PAC

learnable in polynomial time under the product distribution

{\bf if and only if} $\MinEqui C$ has a polynomial

time ${\alpha}$-approximation algorithm.

Using this result we give the first proper learning algorithm

of decision trees over the constant dimensional domain and

the first negative results in proper learning from membership

and equivalence queries for many classes.

For the non-constant dimensional axis parallel concepts

we show that with the equivalence oracle

$(1)\Rightarrow (3)$. We use this to show that

(binary) decision trees are not properly learnable in polynomial time

(assuming P$\not=$NP) and DNF is not $s^\epsilon$-properly learnable

($\epsilon < 1$) in polynomial time even with an NP-oracle

(assuming $\Sigma_2^p\not=$ $P^{NP}$).