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Recent work of Klivans, Stavropoulos, and Vasilyan initiated the study of testable learning with distribution shift (TDS learning), where a learner is given labeled samples from training distribution D, unlabeled samples from test distribution D′, and the goal is to output a classifier with low error on D′ whenever the training samples pass a corresponding test. Their model deviates from all prior work in that no assumptions are made on D′. Instead, the test must accept (with high probability) when the marginals of the training and test distributions are equal. Here we focus on the fundamental case of intersections of halfspaces with respect to Gaussian training distributions and prove a variety of new upper bounds including a 2(k/ϵ)O(1)poly(d)-time algorithm for TDS learning intersections of k homogeneous halfspaces to accuracy ϵ (prior work achieved d(k/ϵ)O(1)). We work under the mild assumption that the Gaussian training distribution contains at least an ϵ fraction of both positive and negative examples (ϵ-balanced). We also prove the first set of SQ lower-bounds for any TDS learning problem and show (1) the ϵ-balanced assumption is necessary for poly(d,1/ϵ)-time TDS learning for a single halfspace and (2) a dΩ~(log1/ϵ) lower bound for the intersection of two general halfspaces, even with the ϵ-balanced assumption. Our techniques significantly expand the toolkit for TDS learning. We use dimension reduction and coverings to give efficient algorithms for computing a localized version of discrepancy distance, a key metric from the domain adaptation literature.
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This content will become publicly available on December 10, 2025
Efficient Discrepancy Testing for Learning with Distribution Shift
A fundamental notion of distance between train and test distributions from the field of domain adaptation is discrepancy distance. While in general hard to compute, here we provide the first set of provably efficient algorithms for testing localized discrepancy distance, where discrepancy is computed with respect to a fixed output classifier. These results imply a broad set of new, efficient learning algorithms in the recently introduced model of Testable Learning with Distribution Shift (TDS learning) due to Klivans et al. (2023).Our approach generalizes and improves all prior work on TDS learning: (1) we obtain universal learners that succeed simultaneously for large classes of test distributions, (2) achieve near-optimal error rates, and (3) give exponential improvements for constant depth circuits. Our methods further extend to semi-parametric settings and imply the first positive results for low-dimensional convex sets. Additionally, we separate learning and testing phases and obtain algorithms that run in fully polynomial time at test time.
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- Award ID(s):
- 2310818
- PAR ID:
- 10577203
- Publisher / Repository:
- Advances in Neural Information Processing Systems 38: Annual Conference on Neural Information Processing Systems (NeurIPS 2024)
- Date Published:
- Format(s):
- Medium: X
- Location:
- Vancouver, BC, Canada
- Sponsoring Org:
- National Science Foundation
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