Convergence Analysis of Block Coordinate Algorithms with Determinantal Sampling
Mojmir Mutny, Michal Derezinski, Andreas Krause
Abstract
We analyze the convergence rate of the randomized Newton-like method introduced by Qu et. al. (2016) for smooth and convex objectives, which uses random coordinate blocks of a Hessian-over-approximation matrix M instead of the true Hessian. The convergence analysis of the algorithm is challenging because of its complex dependence on the structure of M. However, we show that when the coordinate blocks are sampled with probability proportional to their determinant, the convergence rate depends solely on the eigenvalue distribution of matrix M, and has an analytically tractable form. To do so, we derive a fundamental new expectation formula for determinantal point processes. We show that determinantal sampling allows us to reason about the optimal subset size of blocks in terms of the spectrum of M. Additionally, we provide a numerical evaluation of our analysis, demonstrating cases where determinantal sampling is superior or on par with uniform sampling.
BibTeX
@InProceedings{pmlr-v108-mutny20a,
title = {Convergence Analysis of Block Coordinate Algorithms with Determinantal Sampling},
author = {Mutny, Mojmir and Derezinski, Michal and Krause, Andreas},
booktitle = {Proceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics},
pages = {3110--3120},
year = {2020},
editor = {Chiappa, Silvia and Calandra, Roberto},
volume = {108},
series = {Proceedings of Machine Learning Research},
month = {26--28 Aug},
publisher = {PMLR},
pdf = {http://proceedings.mlr.press/v108/mutny20a/mutny20a.pdf},
url = {https://proceedings.mlr.press/v108/mutny20a.html},
abstract = {We analyze the convergence rate of the randomized Newton-like method introduced by Qu et. al. (2016) for smooth and convex objectives, which uses random coordinate blocks of a Hessian-over-approximation matrix M instead of the true Hessian. The convergence analysis of the algorithm is challenging because of its complex dependence on the structure of M. However, we show that when the coordinate blocks are sampled with probability proportional to their determinant, the convergence rate depends solely on the eigenvalue distribution of matrix M, and has an analytically tractable form. To do so, we derive a fundamental new expectation formula for determinantal point processes. We show that determinantal sampling allows us to reason about the optimal subset size of blocks in terms of the spectrum of M. Additionally, we provide a numerical evaluation of our analysis, demonstrating cases where determinantal sampling is superior or on par with uniform sampling.}
}