Certifiability of subexponential distributions (degree-4)

Determine whether every s-subexponential distribution on R^d is (4, O(1))-certifiably bounded; concretely, establish whether there exists a universal constant C such that for all centered s-subexponential distributions P on R^d and all vectors v in R^d, the polynomial C^4 ||v||_2^4 − E_{X∼P}[⟨v, X⟩^4] admits a sum-of-squares certificate. Here, a distribution is s-subexponential if it has (Cs·m, m)-bounded moments for all m, meaning E_{X∼P}[⟨v, X⟩^m] ≤ (Cs·m)^m ||v||_2^m for all v.

Background

The paper proves that all subgaussian distributions are certifiably subgaussian, yielding broad algorithmic consequences. A natural next step is to extend these certifiability results beyond subgaussians to heavier-tailed families such as subexponential distributions. The authors note that existing generic chaining/majorizing measures techniques used in their proof are known to fail for subexponential inputs, even when compared to canonical subexponential distributions such as Laplace, leaving open whether any nontrivial certifiable bounds can be obtained.

The open question focuses on the weakest nontrivial certifiable bound—degree-4—which underpins many algorithmic applications. Resolving it would clarify whether sum-of-squares certificates can handle subexponential tails at least at low degrees, and would guide the development of robust statistical algorithms for broader distribution classes.