Scalable MCMC Sampling for Nonsymmetric Determinantal Point Processes

Insu Han, Mike Gartrell, Elvis Dohmatob, Amin Karbasi
Proceedings of the 39th International Conference on Machine Learning, PMLR 162:8213-8229, 2022.

Abstract

A determinantal point process (DPP) is an elegant model that assigns a probability to every subset of a collection of $n$ items. While conventionally a DPP is parameterized by a symmetric kernel matrix, removing this symmetry constraint, resulting in nonsymmetric DPPs (NDPPs), leads to significant improvements in modeling power and predictive performance. Recent work has studied an approximate Markov chain Monte Carlo (MCMC) sampling algorithm for NDPPs restricted to size-$k$ subsets (called $k$-NDPPs). However, the runtime of this approach is quadratic in $n$, making it infeasible for large-scale settings. In this work, we develop a scalable MCMC sampling algorithm for $k$-NDPPs with low-rank kernels, thus enabling runtime that is sublinear in $n$. Our method is based on a state-of-the-art NDPP rejection sampling algorithm, which we enhance with a novel approach for efficiently constructing the proposal distribution. Furthermore, we extend our scalable $k$-NDPP sampling algorithm to NDPPs without size constraints. Our resulting sampling method has polynomial time complexity in the rank of the kernel, while the existing approach has runtime that is exponential in the rank. With both a theoretical analysis and experiments on real-world datasets, we verify that our scalable approximate sampling algorithms are orders of magnitude faster than existing sampling approaches for $k$-NDPPs and NDPPs.

Cite this Paper

BibTeX
@InProceedings{pmlr-v162-han22b, title = {Scalable {MCMC} Sampling for Nonsymmetric Determinantal Point Processes}, author = {Han, Insu and Gartrell, Mike and Dohmatob, Elvis and Karbasi, Amin}, booktitle = {Proceedings of the 39th International Conference on Machine Learning}, pages = {8213--8229}, year = {2022}, editor = {Chaudhuri, Kamalika and Jegelka, Stefanie and Song, Le and Szepesvari, Csaba and Niu, Gang and Sabato, Sivan}, volume = {162}, series = {Proceedings of Machine Learning Research}, month = {17--23 Jul}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v162/han22b/han22b.pdf}, url = {https://proceedings.mlr.press/v162/han22b.html}, abstract = {A determinantal point process (DPP) is an elegant model that assigns a probability to every subset of a collection of $n$ items. While conventionally a DPP is parameterized by a symmetric kernel matrix, removing this symmetry constraint, resulting in nonsymmetric DPPs (NDPPs), leads to significant improvements in modeling power and predictive performance. Recent work has studied an approximate Markov chain Monte Carlo (MCMC) sampling algorithm for NDPPs restricted to size-$k$ subsets (called $k$-NDPPs). However, the runtime of this approach is quadratic in $n$, making it infeasible for large-scale settings. In this work, we develop a scalable MCMC sampling algorithm for $k$-NDPPs with low-rank kernels, thus enabling runtime that is sublinear in $n$. Our method is based on a state-of-the-art NDPP rejection sampling algorithm, which we enhance with a novel approach for efficiently constructing the proposal distribution. Furthermore, we extend our scalable $k$-NDPP sampling algorithm to NDPPs without size constraints. Our resulting sampling method has polynomial time complexity in the rank of the kernel, while the existing approach has runtime that is exponential in the rank. With both a theoretical analysis and experiments on real-world datasets, we verify that our scalable approximate sampling algorithms are orders of magnitude faster than existing sampling approaches for $k$-NDPPs and NDPPs.} }
Endnote
%0 Conference Paper %T Scalable MCMC Sampling for Nonsymmetric Determinantal Point Processes %A Insu Han %A Mike Gartrell %A Elvis Dohmatob %A Amin Karbasi %B Proceedings of the 39th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2022 %E Kamalika Chaudhuri %E Stefanie Jegelka %E Le Song %E Csaba Szepesvari %E Gang Niu %E Sivan Sabato %F pmlr-v162-han22b %I PMLR %P 8213--8229 %U https://proceedings.mlr.press/v162/han22b.html %V 162 %X A determinantal point process (DPP) is an elegant model that assigns a probability to every subset of a collection of $n$ items. While conventionally a DPP is parameterized by a symmetric kernel matrix, removing this symmetry constraint, resulting in nonsymmetric DPPs (NDPPs), leads to significant improvements in modeling power and predictive performance. Recent work has studied an approximate Markov chain Monte Carlo (MCMC) sampling algorithm for NDPPs restricted to size-$k$ subsets (called $k$-NDPPs). However, the runtime of this approach is quadratic in $n$, making it infeasible for large-scale settings. In this work, we develop a scalable MCMC sampling algorithm for $k$-NDPPs with low-rank kernels, thus enabling runtime that is sublinear in $n$. Our method is based on a state-of-the-art NDPP rejection sampling algorithm, which we enhance with a novel approach for efficiently constructing the proposal distribution. Furthermore, we extend our scalable $k$-NDPP sampling algorithm to NDPPs without size constraints. Our resulting sampling method has polynomial time complexity in the rank of the kernel, while the existing approach has runtime that is exponential in the rank. With both a theoretical analysis and experiments on real-world datasets, we verify that our scalable approximate sampling algorithms are orders of magnitude faster than existing sampling approaches for $k$-NDPPs and NDPPs.
APA
Han, I., Gartrell, M., Dohmatob, E. & Karbasi, A.. (2022). Scalable MCMC Sampling for Nonsymmetric Determinantal Point Processes. Proceedings of the 39th International Conference on Machine Learning, in Proceedings of Machine Learning Research 162:8213-8229 Available from https://proceedings.mlr.press/v162/han22b.html.

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