Estimating informativeness of samples with Smooth Unique Information

We define a notion of information that an individual sample provides to the training of a neural network, and we specialize it to measure both how much a sample informs the final weights and how much it informs the function computed by the weights. Though related, we show that these quantities have a qualitatively different behavior. We give efficient approximations of these quantities using a linearized network and demonstrate empirically that the approximation is accurate for real-world architectures, such as pre-trained ResNets. We apply these measures to several problems, such as dataset summarization, analysis of under-sampled classes, comparison of informativeness of different data sources, and detection of adversarial and corrupted examples. Our work generalizes existing frameworks but enjoys better computational properties for heavily overparametrized models, which makes it possible to apply it to real-world networks. Training a deep neural network (DNN) entails extracting information from samples in a dataset and storing it in the weights of the network, so that it may be used in future inference or prediction. But how much information does a particular sample contribute to the trained model? The answer can be used to provide strong generalization bounds (if no information is used, the network is not memorizing the sample), privacy bounds (how much information the network can leak about a particular sample), and enable better interpretation of the training process and its outcome. To determine the information content of samples, we need to define and compute information. In the classical sense, information is a property of random variables, which may be degenerate for the deterministic process of computing the output of a trained DNN in response to a given input (inference). So, even posing the problem presents some technical challenges.