Bregman Deviations of Generic Exponential Families
This work provides a theoretical framework for time-uniform concentration bounds in exponential families, which is incremental but useful for practitioners in statistics and machine learning.
The paper tackles the problem of concentration in generic exponential families by establishing a time-uniform bound on the Bregman divergence between the true parameter and its finite sample estimate, introducing the Bregman information gain. It shows competitive results compared to state-of-the-art alternatives in numerical comparisons.
We revisit the method of mixture technique, also known as the Laplace method, to study the concentration phenomenon in generic exponential families. Combining the properties of Bregman divergence associated with log-partition function of the family with the method of mixtures for super-martingales, we establish a generic bound controlling the Bregman divergence between the parameter of the family and a finite sample estimate of the parameter. Our bound is time-uniform and makes appear a quantity extending the classical information gain to exponential families, which we call the Bregman information gain. For the practitioner, we instantiate this novel bound to several classical families, e.g., Gaussian, Bernoulli, Exponential, Weibull, Pareto, Poisson and Chi-square yielding explicit forms of the confidence sets and the Bregman information gain. We further numerically compare the resulting confidence bounds to state-of-the-art alternatives for time-uniform concentration and show that this novel method yields competitive results. Finally, we highlight the benefit of our concentration bounds on some illustrative applications.