Sample Complexity for Computational Classification Problems

In a statistical setting of the classification (pattern recognition) problem the number of examples required to approximate an unknown labelling function is linear in the VC dimension of the target learning class. In this work we consider the question of whether such bounds exist if we restrict our attention to computable classification methods, assuming that the unknown labelling function is also computable. We find that in this case the number of examples required for a computable method to approximate the labelling function not only is not linear, but grows faster (in the VC dimension of the class) than any computable function. No time or space constraints are put on the predictors or target functions; the only resource we consider is the training examples. The task of classification is considered in conjunction with another learning problem - data compression. An impossibility result for the task of data compression allows us to estimate the sample complexity for pattern recognition.