## Abstract

We study distribution-free nonparametric regression following a notion of average smoothness initiated by Ashlagi et al. (2021), which measures the “effective” smoothness of a function with respect to an arbitrary unknown underlying distribution. While the recent work of Hanneke et al. (2023) established tight uniform convergence bounds for average-smooth functions in the realizable case and provided a computationally efficient realizable learning algorithm, both of these results currently lack analogs in the general agnostic (i.e. noisy) case. In this work, we fully close these gaps. First, we provide a distribution-free uniform convergence bound for average-smoothness classes in the agnostic setting. Second, we match the derived sample complexity with a computationally efficient agnostic learning algorithm. Our results, which are stated in terms of the intrinsic geometry of the data and hold over any totally bounded metric space, show that the guarantees recently obtained for realizable learning of average-smooth functions transfer to the agnostic setting. At the heart of our proof, we establish the uniform convergence rate of a function class in terms of its bracketing entropy, which may be of independent interest.

Original language | English |
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Pages (from-to) | 719-731 |

Number of pages | 13 |

Journal | Proceedings of Machine Learning Research |

Volume | 237 |

State | Published - 1 Jan 2024 |

Event | 35th International Conference on Algorithmic Learning Theory, ALT 2024 - La Jolla, United States Duration: 25 Feb 2024 → 28 Feb 2024 |

## Keywords

- agnostic learning
- average smoothness
- bracketing numbers
- generalization
- metric space

## All Science Journal Classification (ASJC) codes

- Artificial Intelligence
- Software
- Control and Systems Engineering
- Statistics and Probability