Sparse high-dimensional regression: Exact scalable algorithms and phase transitions
Name
1709.10029.pdf
Description
Submitted version
Size
597.29 KB
Format
Adobe PDF
Checksum (MD5)
39c6baee4e8028fd43ce979bc7fac40c
Author(s)
Bertsimas, Dimitris
Van Parys, Bart
Date Issued
2020
Journal
The Annals of Statistics
Publisher
Institute of Mathematical Statistics
Version
Original manuscript
Abstract
We present a novel binary convex reformulation of the sparse regression problem that constitutes a new duality perspective. We devise a new cutting plane method and provide evidence that it can solve to provable optimality the sparse regression problem for sample sizes n and number of regressors p in the 100,000s, that is, two orders of magnitude better than the current state of the art, in seconds. The ability to solve the problem for very high dimensions allows us to observe new phase transition phenomena. Contrary to traditional complexity theory which suggests that the difficulty of a problem increases with problem size, the sparse regression problem has the property that as the number of samples n increases the problem becomes easier in that the solution recovers 100% of the true signal, and our approach solves the problem extremely fast (in fact faster than Lasso), while for small number of samples n, our approach takes a larger amount of time to solve the problem, but importantly the optimal solution provides a statistically more relevant regressor. We argue that our exact sparse regression approach presents a superior alternative over heuristic methods available at present.
MIT Department
Massachusetts Institute of Technology. Operations Research Center
Terms of Use
Creative Commons Attribution-Noncommercial-Share Alike
Persistent DSpace Link
DOI of Published Version
https://doi.org/10.1214/18-AOS1804
