We consider the parameter estimation problem of a probabilistic generative model prescribed using a natural exponential family of distributions. For this problem, the typical maximum likelihood estimator usually overfits under limited training sample size, is sensitive to noise and may perform poorly on downstream predictive tasks. To mitigate these issues, we propose a distributionally robust maximum likelihood estimator that minimizes the worst-case expected log-loss uniformly over a parametric Kullback-Leibler ball around a parametric nominal distribution. Leveraging the analytical expression of the Kullback-Leibler divergence between two distributions in the same natural exponential family, we show that the min-max estimation problem is tractable in a broad setting, including the robust training of generalized linear models. Our novel robust estimator also enjoys statistical consistency and delivers promising empirical results in both regression and classification tasks.

Since everything is parametric, I'd expect explicit rates of convergence involvind all probalem complexity parameters (n, m, p, etc.) To make the rest of my points clear, let me recall the following notations are used in the paper: - n: the dimensionality of the covariate (i.e feature vector) X. Thus X is random vector in R n. BTW, in the context of ML or stats, I'd use another notation here, as n conventionally stands for "sample size".

This paper proposes a method for distributionally robust optimization under KL ambiguity sets for exponential families. Although KL ambiguity sets have their drawbacks, in particular not covering any changes in the inputs x, the present work produces a standard conic problem for a wide problem class via a novel analysis, provides good theoretical analysis, and yields good numerical results for a variety of small-scale classification problems. With the various clarifications that came up in the reviews, this paper makes a solid contribution to the DRO literature and will be quite welcome to the NeurIPS audience.

We consider the parameter estimation problem of a probabilistic generative model prescribed using a natural exponential family of distributions. For this problem, the typical maximum likelihood estimator usually overfits under limited training sample size, is sensitive to noise and may perform poorly on downstream predictive tasks. To mitigate these issues, we propose a distributionally robust maximum likelihood estimator that minimizes the worst-case expected log-loss uniformly over a parametric Kullback-Leibler ball around a parametric nominal distribution. Leveraging the analytical expression of the Kullback-Leibler divergence between two distributions in the same natural exponential family, we show that the min-max estimation problem is tractable in a broad setting, including the robust training of generalized linear models. Our novel robust estimator also enjoys statistical consistency and delivers promising empirical results in both regression and classification tasks.