On sample complexity for computational pattern recognition

In statistical setting of the 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 whether such bounds exist if we restrict our attention to computable pattern recognition 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 pattern recognition 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

Statistical Learning of Arbitrary Computable Classifiers

Statistical learning theory chiefly studies restricted hypothesis classes, particularly those with finite Vapnik-Chervonenkis (VC) dimension. The fundamental quantity of interest is the sample complexity: the number of samples required to learn to a specified level of accuracy. Here we consider learning over the set of all computable labeling functions. Since the VC-dimension is infinite and a priori (uniform) bounds on the number of samples are impossible, we let the learning algorithm decide when it has seen sufficient samples to have learned. We first show that learning in this setting is indeed possible, and develop a learning algorithm. We then show, however, that bounding sample complexity independently of the distribution is impossible. Notably, this impossibility is entirely due to the requirement that the learning algorithm be computable, and not due to the statistical nature of the problem.Comment: Expanded the section on prior work and added reference

On Computability of Pattern Recognition Problems

Abstract. In statistical setting of the 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 whether such bounds exist if we restrict our attention to computable pattern recognition 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 pattern recognition 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.