Trial and Error

A pac-learning algorithm is -space bounded, if it stores at most examples from the sample at any time. We characterize the -space learnable concept classes. For this purpose we introduce the compression parameter of a concept class and design our Trial and Error Learning Algorithm. We show : is -space learnable if and only if the compression parameter of is at most . This learning algorithm does not produce a hypothesis consistent with the whole sample as previous approaches e.g. by Floyd, who presents consistent space bounded learning algorithms, but has to restrict herself to very special concept classes. On the other hand our algorithm needs large samples; the compression parameter appears as exponent in the sample size. We present several examples of polynomial time space bounded learnable concept classes: – all intersection closed concept classes with finite VC–dimension. – convex -gons in . – halfspaces in . – unions of triangles in . We further relate the compression parameter to the VC–dimension, and discuss variants of this parameter. Received May 24, 1994 / July 4, 1995