Learning Sparse Combinatorial Representations via Two-stage Submodular Maximization
Eric Balkanski, Baharan Mirzasoleiman, Andreas Krause, Yaron Singer
Abstract
We consider the problem of learning sparse representations of data sets, where the goal is to reduce a data set in manner that optimizes multiple objectives. Motivated by applications of data summarization, we develop a new model which we refer to as the two-stage submodular maximization problem. This task can be viewed as a combinatorial analogue of representation learning problems such as dictionary learning and sparse regression. The two-stage problem strictly generalizes the problem of cardinality constrained submodular maximization, though the objective function is not submodular and the techniques for submodular maximization cannot be applied. We describe a continuous optimization method which achieves an approximation ratio which asymptotically approaches 1-1/e. For instances where the asymptotics do not kick in, we design a local-search algorithm whose approximation ratio is arbitrarily close to 1/2. We empirically demonstrate the effectiveness of our methods on two multi-objective data summarization tasks, where the goal is to construct summaries via sparse representative subsets w.r.t. to predefined objectives.
BibTeX
@InProceedings{pmlr-v48-balkanski16,
title = {Learning Sparse Combinatorial Representations via Two-stage Submodular Maximization},
author = {Balkanski, Eric and Mirzasoleiman, Baharan and Krause, Andreas and Singer, Yaron},
booktitle = {Proceedings of The 33rd International Conference on Machine Learning},
pages = {2207--2216},
year = {2016},
editor = {Balcan, Maria Florina and Weinberger, Kilian Q.},
volume = {48},
series = {Proceedings of Machine Learning Research},
address = {New York, New York, USA},
month = {20--22 Jun},
publisher = {PMLR},
pdf = {http://proceedings.mlr.press/v48/balkanski16.pdf},
url = {https://proceedings.mlr.press/v48/balkanski16.html},
abstract = {We consider the problem of learning sparse representations of data sets, where the goal is to reduce a data set in manner that optimizes multiple objectives. Motivated by applications of data summarization, we develop a new model which we refer to as the two-stage submodular maximization problem. This task can be viewed as a combinatorial analogue of representation learning problems such as dictionary learning and sparse regression. The two-stage problem strictly generalizes the problem of cardinality constrained submodular maximization, though the objective function is not submodular and the techniques for submodular maximization cannot be applied. We describe a continuous optimization method which achieves an approximation ratio which asymptotically approaches 1-1/e. For instances where the asymptotics do not kick in, we design a local-search algorithm whose approximation ratio is arbitrarily close to 1/2. We empirically demonstrate the effectiveness of our methods on two multi-objective data summarization tasks, where the goal is to construct summaries via sparse representative subsets w.r.t. to predefined objectives.}
}