Almost Free: Self-concordance in Natural Exponential Families and an Application to Bandits

TL;DR

This paper establishes self-concordance properties for natural exponential families with subexponential tails and applies these results to derive improved regret bounds for generalized linear bandit algorithms, including cases with Poisson, exponential, and gamma distributions.

Contribution

It proves self-concordance for certain exponential families and uses this to derive novel regret bounds for generalized linear bandits with subexponential tails.

Findings

Self-concordance established for subexponential tail families.

Regret bounds for generalized linear bandits are scale-sensitive and exponential-free.

First regret bounds for bandits with Poisson, exponential, and gamma distributions.

Abstract

We prove that single-parameter natural exponential families with subexponential tails are self-concordant with polynomial-sized parameters. For subgaussian natural exponential families we establish an exact characterization of the growth rate of the self-concordance parameter. Applying these findings to bandits allows us to fill gaps in the literature: We show that optimistic algorithms for generalized linear bandits enjoy regret bounds that are both second-order (scale with the variance of the optimal arm's reward distribution) and free of an exponential dependence on the bound of the problem parameter in the leading term. To the best of our knowledge, ours is the first regret bound for generalized linear bandits with subexponential tails, broadening the class of problems to include Poisson, exponential and gamma bandits.