A Computationally Efficient Method for Learning Exponential Family Distributions

TL;DR

This paper introduces a new computationally efficient estimator for learning the natural parameters of bounded exponential family distributions from i.i.d. samples, achieving consistency, asymptotic normality, and finite-sample guarantees.

Contribution

It proposes a novel estimator that is computationally efficient, consistent, and asymptotically normal, with finite-sample guarantees, improving over traditional maximum likelihood methods.

Findings

Estimator achieves $ ext{O}( ext{poly}(k/eta))$ sample complexity.

Estimator has $ ext{O}( ext{poly}(k/eta))$ computational complexity.

Finite-sample error bounds are established for the estimator.

Abstract

We consider the question of learning the natural parameters of a $k$ parameter 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.