Fast-Rate Loss Bounds via Conditional Information Measures with Applications to Neural Networks
This work provides improved theoretical guarantees for generalization in machine learning, particularly benefiting researchers and practitioners using neural networks, though it is incremental as it builds on existing information-theoretic methods.
The authors tackled the problem of deriving fast-rate loss bounds for randomized learning algorithms by introducing a framework based on conditional information density, achieving bounds that decay as 1/n instead of the slower 1/√n previously available. They demonstrated nonvacuous test loss estimates for neural networks on MNIST and Fashion-MNIST datasets.
We present a framework to derive bounds on the test loss of randomized learning algorithms for the case of bounded loss functions. Drawing from Steinke & Zakynthinou (2020), this framework leads to bounds that depend on the conditional information density between the the output hypothesis and the choice of the training set, given a larger set of data samples from which the training set is formed. Furthermore, the bounds pertain to the average test loss as well as to its tail probability, both for the PAC-Bayesian and the single-draw settings. If the conditional information density is bounded uniformly in the size $n$ of the training set, our bounds decay as $1/n$. This is in contrast with the tail bounds involving conditional information measures available in the literature, which have a less benign $1/\sqrt{n}$ dependence. We demonstrate the usefulness of our tail bounds by showing that they lead to nonvacuous estimates of the test loss achievable with some neural network architectures trained on MNIST and Fashion-MNIST.