A Kolmogorov Complexity Approach to Generalization in Deep Learning

Deep artificial neural networks can achieve an extremely small difference between training and test accuracies on identically distributed training and test sets, which is a standard measure of generalization. However, the training and test sets may not be sufficiently representative of the empirical sample set, which consists of real-world input samples. When samples are drawn from an underrepresented or unrepresented subset during inference, the gap between the training and inference accuracies can be significant. To address this problem, we first reformulate a classification algorithm as a procedure for searching for a source code that maps input features to classes. We then derive a necessary and sufficient condition for generalization using a universal cognitive similarity metric, namely information distance, based on Kolmogorov complexity. Using this condition, we formulate an optimization problem to learn a more general classification function. To achieve this end, we extend the input features by concatenating encodings of them, and then train the classifier on the extended features. As an illustration of this idea, we focus on image classification, where we use channel codes on the input features as a systematic way to improve the degree to which the training and test sets are representative of the empirical sample set. To showcase our theoretical findings, considering that corrupted or perturbed input features belong to the empirical sample set, but typically not to the training and test sets, we demonstrate through extensive systematic experiments that, as a result of learning a more general classification function, a model trained on encoded input features is significantly more robust to common corruptions, e.g., Gaussian and shot noise, as well as adversarial perturbations, e.g., those found via projected gradient descent, than the model trained on uncoded input features.