Quantify the approximation gap of the quadratic-feature SDP relaxation to the generalized problem of moments

Ascertain quantitative bounds on the approximation gap between the quadratic-feature semidefinite programming relaxation described in Proposition 13 for Hoeffding’s inequality (two independent and identically distributed random variables on [0,1] with mean μ, using ϕ(x) = (1, x, x^2)) and the original generalized problem of moments for independent variables (Problem (1)). Specifically, derive explicit error bounds or convergence guarantees that measure how close the relaxed saddle-point formulation is to the exact optimal value of the generalized problem of moments under the stated assumptions.

Background

The paper proposes a feature-based variational framework and, for two variables with quadratic features, provides an SDP relaxation that substitutes the exact moment tensor by a constrained positive semidefinite matrix. While this makes computation feasible, the authors emphasize that the quality of the relaxation relative to the exact generalized problem of moments is not yet characterized.

They explicitly leave for future work the task of quantifying the distance between the relaxed formulation and the exact problem, i.e., establishing principled bounds on the relaxation gap.