A New Family of Generalization Bounds Using Samplewise Evaluated CMI
This work addresses the challenge of understanding generalization in machine learning, particularly for deep neural networks, by providing incremental improvements to existing theoretical bounds.
The authors tackled the problem of deriving tighter generalization bounds for machine learning models by introducing a new family of information-theoretic bounds based on samplewise evaluated conditional mutual information (CMI), which in some scenarios results in a tighter characterization of the population loss for deep neural networks than previous bounds.
We present a new family of information-theoretic generalization bounds, in which the training loss and the population loss are compared through a jointly convex function. This function is upper-bounded in terms of the disintegrated, samplewise, evaluated conditional mutual information (CMI), an information measure that depends on the losses incurred by the selected hypothesis, rather than on the hypothesis itself, as is common in probably approximately correct (PAC)-Bayesian results. We demonstrate the generality of this framework by recovering and extending previously known information-theoretic bounds. Furthermore, using the evaluated CMI, we derive a samplewise, average version of Seeger's PAC-Bayesian bound, where the convex function is the binary KL divergence. In some scenarios, this novel bound results in a tighter characterization of the population loss of deep neural networks than previous bounds. Finally, we derive high-probability versions of some of these average bounds. We demonstrate the unifying nature of the evaluated CMI bounds by using them to recover average and high-probability generalization bounds for multiclass classification with finite Natarajan dimension.