Technical Report CS0855

Authors: M. Frances and A. Litman
Abstract: This paper testifies that there is no polynomial-time optimal Mistake Bound learning algorithm. This conclusion is reached via several reductions as follows. Littlestone [Lit88] has introduced a combinatorial function $K$ from classes to integers and has shown that if a subroutine computing $K$ is given, one can construct a polynomial-time optimal MB learning algorithm. We establish the reverse reduction. That is, given an optimal MB learning algorithm as a subroutine, one can compute $K$ in polynomial time. Our result combines with Littlestone's to establish that the two tasks above have the same time complexity up to a polynomial. Next, we show that the VC-dimension decision problem is polynomially reducible to the $K$ decision problem. Papadimitriou and Yannakakis [PY93] have provided a strong evidence that the VC-dimension decision problem is not in P. Therefore, it is very unlikely that there is a polynomial-time optimal Mistake Bound learning algorithm.
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