A note on generalization bounds for losses with finite moments
This work provides incremental theoretical improvements in PAC-Bayes bounds for machine learning, addressing heavy-tailed loss functions that are common in robust statistics.
The paper tackles the problem of deriving generalization bounds for unbounded losses with heavy tails using a truncation method, resulting in bounds that interpolate between slow and fast rates depending on moment assumptions and improving dependence on confidence parameters compared to prior work.
This paper studies the truncation method from Alquier [1] to derive high-probability PAC-Bayes bounds for unbounded losses with heavy tails. Assuming that the $p$-th moment is bounded, the resulting bounds interpolate between a slow rate $1 / \sqrt{n}$ when $p=2$, and a fast rate $1 / n$ when $p \to \infty$ and the loss is essentially bounded. Moreover, the paper derives a high-probability PAC-Bayes bound for losses with a bounded variance. This bound has an exponentially better dependence on the confidence parameter and the dependency measure than previous bounds in the literature. Finally, the paper extends all results to guarantees in expectation and single-draw PAC-Bayes. In order to so, it obtains analogues of the PAC-Bayes fast rate bound for bounded losses from [2] in these settings.