Oblivious Sketching-based Central Path Method for Linear Programming
Abstract
In this work, we propose a sketching-based central path method for solving linear programmings, whose running time matches the state of the art results [Cohen, Lee, Song STOC 19; Lee, Song, Zhang COLT 19]. Our method opens up the iterations of the central path method and deploys an "iterate and sketch" approach towards the problem by introducing a new coordinate-wise embedding technique, which may be of independent interest. Compare to previous methods, the work [Cohen, Lee, Song STOC 19] enjoys feasibility while being non-oblivious, and [Lee, Song, Zhang COLT 19] is oblivious but infeasible, and relies on $\mathit{dense}$ sketching matrices such as subsampled randomized Hadamard/Fourier transform matrices. Our method enjoys the benefits of being both oblivious and feasible, and can use $\mathit{sparse}$ sketching matrix [Nelson, Nguyen FOCS 13] to speed up the online matrix-vector multiplication. Our framework for solving LP naturally generalizes to a broader class of convex optimization problems including empirical risk minimization.
BibTeX
@InProceedings{pmlr-v139-song21e,
title = {Oblivious Sketching-based Central Path Method for Linear Programming},
author = {Song, Zhao and Yu, Zheng},
booktitle = {Proceedings of the 38th International Conference on Machine Learning},
pages = {9835--9847},
year = {2021},
editor = {Meila, Marina and Zhang, Tong},
volume = {139},
series = {Proceedings of Machine Learning Research},
month = {18--24 Jul},
publisher = {PMLR},
pdf = {http://proceedings.mlr.press/v139/song21e/song21e.pdf},
url = {https://proceedings.mlr.press/v139/song21e.html},
abstract = {In this work, we propose a sketching-based central path method for solving linear programmings, whose running time matches the state of the art results [Cohen, Lee, Song STOC 19; Lee, Song, Zhang COLT 19]. Our method opens up the iterations of the central path method and deploys an "iterate and sketch" approach towards the problem by introducing a new coordinate-wise embedding technique, which may be of independent interest. Compare to previous methods, the work [Cohen, Lee, Song STOC 19] enjoys feasibility while being non-oblivious, and [Lee, Song, Zhang COLT 19] is oblivious but infeasible, and relies on $\mathit{dense}$ sketching matrices such as subsampled randomized Hadamard/Fourier transform matrices. Our method enjoys the benefits of being both oblivious and feasible, and can use $\mathit{sparse}$ sketching matrix [Nelson, Nguyen FOCS 13] to speed up the online matrix-vector multiplication. Our framework for solving LP naturally generalizes to a broader class of convex optimization problems including empirical risk minimization.}
}