Test Sample Accuracy Scales with Training Sample Density in Neural Networks

Generalization error bounds measure the deviation of performance on unseen test data from performance on training data. However, by providing one scalar per model, they are input-agnostic. What if one wants to predict error for a specific test sample? To answer this, we propose the novel paradigm of input-conditioned generalization error bounds. For piecewise linear neural networks, given a weighting function that relates the errors of different input activation regions together, we obtain a bound on each region’s generalization error that scales inversely with the density of training samples. That is, more densely supported regions are more reliable. As the bound is input-conditioned, it is to our knowledge the first generalization error bound applicable to the problems of detecting out-of-distribution and misclassified in-distribution samples for neural networks; we find that it performs competitively in both cases when tested on image classification tasks. When integrating the region-conditioned bound over regions, a model-level bound is obtained that implies models with fewer activation patterns, a higher degree of information loss or abstraction, generalize better.