Abstract
Data-driven algorithms can adapt their internal structure or parameters to inputs from unknown application-specific distributions, by learning from a training sample of inputs. Several recent works have applied this approach to problems in numerical linear algebra, obtaining significant empirical gains in performance. However, no theoretical explanation for their success was known. In this work we prove generalization bounds for those algorithms, within the PAC-learning framework for data-driven algorithm selection proposed by Gupta and Roughgarden (SICOMP 2017). Our main results are closely matching upper and lower bounds on the fat shattering dimension of the learning-based low rank approximation algorithm of Indyk et al. (NeurIPS 2019). Our techniques are general, and provide generalization bounds for many other recently proposed data-driven algorithms in numerical linear algebra, covering both sketching-based and multigrid-based methods. This considerably broadens the class of data-driven algorithms for which a PAC-learning analysis is available.
Original language | English |
---|---|
Pages (from-to) | 2013-2024 |
Number of pages | 12 |
Journal | Proceedings of Machine Learning Research |
Volume | 178 |
State | Published - 2022 |
Externally published | Yes |
Event | 35th Conference on Learning Theory, COLT 2022 - London, United Kingdom Duration: 2 Jul 2022 → 5 Jul 2022 https://proceedings.mlr.press/v178 |
Keywords
- PAC-learning
- data-driven algorithms
- fat shattering dimension
- low rank approximation
- multigrid
- numerical linear algebra
- pseudo-dimension
- sketching
All Science Journal Classification (ASJC) codes
- Artificial Intelligence
- Software
- Control and Systems Engineering
- Statistics and Probability