2023/02/12 by Steve Hanneke, Hanneke, Steve, Aryeh Kontorovich +3
Computer Science · Mathematics · #Domain Adaptation and Few-Shot Learning #FOS: Computer and information sciences #FOS: Mathematics #Machine Learning (cs.LG) #Machine Learning (stat.ML) #Statistical Methods and Inference #Statistics Theory (math.ST)
paper · pdf · doi:10.48550/arxiv.2302.06005
openalex publication_date 2023/02/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We generalize the notion of average Lipschitz smoothness proposed by Ashlagi et al. (COLT 2021) by extending it to Hölder smoothness. This measure of the "effective smoothness" of a function is sensitive to the underlying distribution and can be dramatically smaller than its classic "worst-case" Hölder constant. We consider both the realizable and the agnostic (noisy) regression settings, proving upper and lower risk bounds in terms of the average Hölder smoothness; these rates improve upon both previously known rates even in the special case of average Lipschitz smoothness. Moreover, our lower bound is tight in the realizable setting up to log factors, thus we establish the minimax rate. From an algorithmic perspective, since our notion of average smoothness is defined with respect to the unknown underlying distribution, the learner does not have an explicit representation of the function class, hence is unable to execute ERM. Nevertheless, we provide distinct learning algorithms that achieve both (nearly) optimal learning rates. Our results hold in any totally bounded metric space, and are stated in terms of its intrinsic geometry. Overall, our results show that the classic worst-case notion of Hölder smoothness can be essentially replaced by its average, yielding considerably sharper guarantees.