Efficiently solve the saddle-point optimization in the quadratic-feature SDP relaxation for Hoeffding’s inequality (two i.i.d. variables)

Develop an efficient solution method for the saddle-point optimization that arises in the quadratic-feature semidefinite programming relaxation introduced in Proposition 13 for Hoeffding’s inequality with two independent and identically distributed random variables X1 and X2 supported on [0,1] with mean μ, where the feature vector is ϕ(x) = (1, x, x^2) and the relaxation replaces sup_{σ∈K(μ)} Tr(H σσ^⊤) by sup_{M ≽ 0} Tr(H M) under moment-consistency constraints. The objective is to produce algorithms that reliably and efficiently solve the resulting min–max (saddle-point) problem defined by minimizing over H subject to feature-based upper-bound constraints while maximizing over feasible moment matrices M.

Background

Section 3.4.2 introduces a feature-based variational formulation for concentration bounds, specializing to quadratic features for two i.i.d. variables in Hoeffding’s setting. Proposition 13 provides an SDP relaxation by replacing the moment tensor with a positive semidefinite matrix constrained by relationships between first and second moments.

The authors note that, even with this relaxation, solving the overall optimization requires addressing a saddle-point structure (minimization over the upper-bounding function parameters H coupled with maximization over feasible moment matrices M). They explicitly defer the efficient solution of this saddle-point problem to future work.