Learning faster than promised by the Vapnik-Chervonenkis dimension

We investigate the sample size needed to infera separating line between IWO I’CIII~C~ planar region> usiny Valiant’s model of the complesity of learning t‘rom I-andom ewmples [4]. A rheorem proved III II] using the Vapnlk~C‘hervonenki~ dimension give, an O((I /c)ln( 1 IF)) upper hound on the sample size sufficient to infer a separating line with error lehs than E between two COTIVSX planar regions. This theorem requires that wth high probability uti_v separating hne CCIIIsilent wiO1 such a sample have hmall CI-rm. The preseni paper gives a lower hound showmg that under this requil-ement the sample size cannot bc irnpro\cd. II i$ further +hrwn that if this requirement is weakened to require only that a particular line which ii Larigenl to Ihe cun\;~x hull+ of the hampIe points m the two regions have Tmall error then the In(l,‘c) trim can be eliminated flon~ the upper borlnd.