Chained Generalisation Bounds
This work provides incremental theoretical advancements for researchers in machine learning theory by offering tighter generalization bounds, but it is limited to toy examples and lacks broad empirical validation.
The paper tackles the problem of deriving tighter upper bounds for the expected generalization error in supervised learning by introducing a chaining technique that lifts regularity assumptions from the loss function to its gradient, resulting in novel bounds based on metrics like Wasserstein distance and showing significant improvements in toy examples, particularly when hypothesis distributions are concentrated.
This work discusses how to derive upper bounds for the expected generalisation error of supervised learning algorithms by means of the chaining technique. By developing a general theoretical framework, we establish a duality between generalisation bounds based on the regularity of the loss function, and their chained counterparts, which can be obtained by lifting the regularity assumption from the loss onto its gradient. This allows us to re-derive the chaining mutual information bound from the literature, and to obtain novel chained information-theoretic generalisation bounds, based on the Wasserstein distance and other probability metrics. We show on some toy examples that the chained generalisation bound can be significantly tighter than its standard counterpart, particularly when the distribution of the hypotheses selected by the algorithm is very concentrated. Keywords: Generalisation bounds; Chaining; Information-theoretic bounds; Mutual information; Wasserstein distance; PAC-Bayes.