Exactly Tight Information-Theoretic Generalization Error Bound for the Quadratic Gaussian Problem

We provide a new information-theoretic generalization error bound that is exactly tight (i.e., matching even the constant) for the canonical quadratic Gaussian mean estimation problem. Despite considerable existing efforts in deriving information-theoretic generalization error bounds, applying them to this simple setting where sample average is used as the estimate of the mean value of Gaussian data has not yielded satisfying results. In fact, most existing bounds are order-wise loose in this setting, which has raised concerns about the fundamental capability of information-theoretic bounds in reasoning the generalization behavior for machine learning. The proposed new bound adopts the individual-sample-based approach proposed by Bu et al., but also has several key new ingredients. Firstly, instead of applying the change of measure inequality on the loss function, we apply it to the generalization error function itself; secondly, the bound is derived in a conditional manner; lastly, a reference distribution, which bears a certain similarity to the prior distribution in the Bayesian setting, is introduced. The combination of these components produces a general KL-divergence-based generalization error bound. We further show that although the conditional bounding and the reference distribution can make the bound exactly tight, removing them does not significantly degrade the bound, which leads to a mutual-information-based bound that is also asymptotically tight in this setting.