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### Paper:

TR98-013 | 3rd March 1998 00:00

#### A New Composition Theorem for Learning Algorithms

TR98-013
Publication: 6th March 1998 12:37
Keywords:

Abstract:

We present a new approach to the composition
of learning algorithms (in various models) for
classes of constant VC-dimension into learning algorithms for
more complicated classes.
We prove that if a class $\CC$ is learnable
in time $t$ from a hypothesis class $\HH$ of constant VC-dimension
then the class $\C^\star$ of all
functions $F$ of the form $F=f(g_1,\ldots,g_m)$,
where $f$ is any function
and $g_1,\ldots,g_m\in \C$, is
learnable in time polynomial in $t$ and $m$ .
We also use a simple
argument to prove that the composition theorem cannot be extended
to classes with a nonconstant VC-dimension.

A composition theorem for the exact learning model
(with equivalence queries only)
is proven in [BBK97] {\it only} for classes $\C$
of constant VC-dimension
that have {\it constant space} learning algorithms.
Constant space algorithms are hard to find
and have large complexity.
Our algorithm is simple and has a
complexity lower than the algorithm in [BBK97].

We then show how to change a PAC-learning
algorithm of $\CC$ from $\HH$ to an
SQ-learning algorithm and to a PAC-learning
algorithm for $\CC^\star$ with malicious noise
that achieves the optimal error rate $\eta/(1-\eta)+\beta$
for any $\beta$.
This, in particular, shows that if a class of
constant VC-dimension is PAC-learnable from a
class of constant VC-dimension then
it is SQ-learnable and PAC-learnable with malicious noise.
We apply this result for SQ-learning and PAC-learning with malicious
noise a general class of geometric objects.
This class includes the set of all geometric objects
in the constant dimensional space
that are bounded by $m$ algebraic surfaces of constant
degree (for example, hyperplanes, spheres, etc.).
This result generalizes all the results known
from the literature about learning geometric objects
in the SQ-learning and PAC-learning models with malicious noise.

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