Minimax lower bounds for concept learning state, for example, that
for each sample size $n$ and learning rule $g_n$, there exists a distribution
of the observation $X$ and a concept $C$ to be learnt such that the expected
error of $g_n$ is at least a constant times $V/n$, where $V$ is the VC dimension
of the concept class. However, these bounds do not tell anything about the
rate of decrease of the error for a {\sl fixed} distribution--concept pair.\\
In this paper we investigate minimax lower bounds ...

Minimax lower bounds for concept learning state, for example, that
for each sample size $n$ and learning rule $g_n$, there exists a distribution
of the observation $X$ and a concept $C$ to be learnt such that the expected
error of $g_n$ is at least a constant times $V/n$, where $V$ is the VC dimension
of the concept class. However, these bounds do not tell anything about the
rate of decrease of the error for a {\sl fixed} distribution--concept pair.\\
In this paper we investigate minimax lower bounds in such a--stronger--sense.
We show that for several natural $k$--parameter concept classes, including
the class of linear halfspaces, the class of balls, the class of polyhedra
with a certain number of faces, and a class of neural networks, for any
{\sl sequence} of learning rules $\{g_n\}$, there exists a fixed distribution
of $X$ and a fixed concept $C$ such that the expected error is larger than
a constant times $k/n$ for {\sl infinitely many n}. We also obtain such
strong minimax lower bounds for the tail distribution of the probability
of error, which extend the corresponding minimax lower bounds.

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