# An Efficient Tester-Learner for Halfspaces

## TL;DR

This paper introduces the first efficient algorithm for testable learning of halfspaces under Gaussian and strongly log-concave distributions, achieving near-optimal error bounds even with adversarial noise, by combining new tests with moment-matching techniques.

## Contribution

It presents the first efficient testable learning algorithm for halfspaces in the testable learning model, handling both Massart and adversarial noise with optimal or near-optimal error guarantees.

## Key findings

- Polynomial-time algorithm for Massart noise with error opt + ε
- Quasipolynomial-time algorithm for adversarial noise with error O(opt) + ε
- New tests leveraging labels to improve moment-matching in testable learning

## Abstract

We give the first efficient algorithm for learning halfspaces in the testable learning model recently defined by Rubinfeld and Vasilyan (2023). In this model, a learner certifies that the accuracy of its output hypothesis is near optimal whenever the training set passes an associated test, and training sets drawn from some target distribution -- e.g., the Gaussian -- must pass the test. This model is more challenging than distribution-specific agnostic or Massart noise models where the learner is allowed to fail arbitrarily if the distributional assumption does not hold. We consider the setting where the target distribution is Gaussian (or more generally any strongly log-concave distribution) in $d$ dimensions and the noise model is either Massart or adversarial (agnostic). For Massart noise, our tester-learner runs in polynomial time and outputs a hypothesis with (information-theoretically optimal) error $\mathsf{opt} + \epsilon$ for any strongly log-concave target distribution. For adversarial noise, our tester-learner obtains error $O(\mathsf{opt}) + \epsilon$ in polynomial time when the target distribution is Gaussian; for strongly log-concave distributions, we obtain $\tilde{O}(\mathsf{opt}) + \epsilon$ in quasipolynomial time. Prior work on testable learning ignores the labels in the training set and checks that the empirical moments of the covariates are close to the moments of the base distribution. Here we develop new tests of independent interest that make critical use of the labels and combine them with the moment-matching approach of Gollakota et al. (2023). This enables us to simulate a variant of the algorithm of Diakonikolas et al. (2020) for learning noisy halfspaces using nonconvex SGD but in the testable learning setting.