Which Algorithms Have Tight Generalization Bounds?

We study which machine learning algorithms have tight generalization bounds with respect to a given collection of population distributions. Our results build on and extend the recent work of Gastpar et al. (2023). First, we present conditions that preclude the existence of tight generalization bounds. Specifically, we show that algorithms that have certain inductive biases that cause them to be unstable do not admit tight generalization bounds. Next, we show that algorithms that are sufficiently loss-stable do have tight generalization bounds. We conclude with a simple characterization that relates the existence of tight generalization bounds to the conditional variance of the algorithm's loss.