A Computationally Efficient Method for Learning Exponential Family DistributionsDownload PDF

Published: 09 Nov 2021, Last Modified: 05 May 2023NeurIPS 2021 PosterReaders: Everyone
Keywords: Exponential Family, M-estimation, Maximum Likelihood Estimation, Consistency, Normality, Efficiency
TL;DR: In this work, we propose a computationally efficient proxy for the MLE for learning exponential family distributions.
Abstract: We consider the question of learning the natural parameters of a $k$ parameter \textit{minimal} exponential family from i.i.d. samples in a computationally and statistically efficient manner. We focus on the setting where the support as well as the natural parameters are appropriately bounded. While the traditional maximum likelihood estimator for this class of exponential family is consistent, asymptotically normal, and asymptotically efficient, evaluating it is computationally hard. In this work, we propose a computationally efficient estimator that is consistent as well as asymptotically normal under mild conditions. We provide finite sample guarantees to achieve an ($\ell_2$) error of $\alpha$ in the parameter estimation with sample complexity $O(\mathrm{poly}(k/\alpha))$ and computational complexity ${O}(\mathrm{poly}(k/\alpha))$. To establish these results, we show that, at the population level, our method can be viewed as the maximum likelihood estimation of a re-parameterized distribution belonging to the same class of exponential family.
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