Technical Report MSC-2006-30

Title: On Exact Learning Halfspaces with Random Consistent Hypothesis Oracle
Authors: Ehab Wattad
Supervisors: Nader Bshouty
Abstract: We investigate several learning strategies for exact learning halfspaces over the domain $\{0,1,\ldots,n-1\}^d$ and study the query complexity and the time complexity of exact learning using those strategies. Our strategies are based on the $\RCH$-oracle that returns a random consistent hypothesis with the counterexamples received from the equivalence query oracle.

We first give a new polynomial time learning algorithm that uses the RCH-oracle for learning halfspaces from majority of halfspaces. We show that the query complexity of this algorithm is less (by some constant factor) than the best known algorithm that <<HS.pdf>> learns halfspaces from halfspaces.

We then study the query complexity of exact learning when limited number of calls to the $\RCH$-oracle is allowed in each trial, i.e., before each equivalence query. We first show that an $\tilde O(d)$ calls to the RCH-oracle in each trial is sufficient for learning in polynomial number of queries. We then show that any ``reasonable'' strategy must use the $\RCH$-oracle at least $\Omega(\sqrt{d})$ times in each trial.

Then we show that if only one call to $\RCH$-oracle is allowed in each trial then the query complexity of the learning algorithm is $2^{\Theta(d)}\log n$. We then give a tight lower bound $2^{\Omega(d)}+\Omega(d^2\log n)$. This proves that this learning algorithm does not run in polynomial time for $d=\omega(\log\log n)$.

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