Bayesian experimental design using regularized determinantal point processes
Michal Derezinski, Feynman Liang, Michael Mahoney
Abstract
We establish a fundamental connection between Bayesian experimental design and determinantal point processes (DPPs). Experimental design is a classical task in combinatorial optimization, where we wish to select a small subset of $d$-dimensional vectors to minimize a statistical optimality criterion. We show that a new regularized variant of DPPs can be used to design efficient algorithms for finding $(1+\epsilon)$-approximate solutions to experimental design under four commonly used optimality criteria: A-, C-, D- and V-optimality. A key novelty is that we offer improved guarantees under the Bayesian framework, where prior knowledge is incorporated into the criteria. Our algorithm returns a $(1+\epsilon)$-approximate solution when the subset size $k$ is $\Omega\left(\frac{d_A}{\epsilon} + \frac{\log(11/epsilon)}{\epsilon^2}\right)$, where $d_A << d$ is an effective dimension determined by prior knowledge (via a precision matrix $\mathbf{A}$).This is the first approximation guarantee where the dependence on $d$ is replaced by an effective dimension. Moreover, the time complexity of our algorithm significantly improves on existing approaches with comparable guarantees.
BibTeX
@InProceedings{pmlr-v108-derezinski20a,
title = {Bayesian experimental design using regularized determinantal point processes},
author = {Derezinski, Michal and Liang, Feynman and Mahoney, Michael},
booktitle = {Proceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics},
pages = {3197--3207},
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/derezinski20a/derezinski20a.pdf},
url = {https://proceedings.mlr.press/v108/derezinski20a.html},
abstract = {We establish a fundamental connection between Bayesian experimental design and determinantal point processes (DPPs). Experimental design is a classical task in combinatorial optimization, where we wish to select a small subset of $d$-dimensional vectors to minimize a statistical optimality criterion. We show that a new regularized variant of DPPs can be used to design efficient algorithms for finding $(1+\epsilon)$-approximate solutions to experimental design under four commonly used optimality criteria: A-, C-, D- and V-optimality. A key novelty is that we offer improved guarantees under the Bayesian framework, where prior knowledge is incorporated into the criteria. Our algorithm returns a $(1+\epsilon)$-approximate solution when the subset size $k$ is $\Omega\left(\frac{d_A}{\epsilon} + \frac{\log(11/epsilon)}{\epsilon^2}\right)$, where $d_A << d$ is an effective dimension determined by prior knowledge (via a precision matrix $\mathbf{A}$).This is the first approximation guarantee where the dependence on $d$ is replaced by an effective dimension. Moreover, the time complexity of our algorithm significantly improves on existing approaches with comparable guarantees.}
}