Sum-of-squares hierarchies for binary polynomial optimization

We consider the sum-of-squares hierarchy of approximations for the problem of minimizing a polynomial f over the boolean hypercube 𝔹^n={0,1}^n . This hierarchy provides for each integer r ∈ℕ a lower bound f_(r) on the minimum f_min of f , given by the largest scalar λ for which the polynomial f - λ is a sum-of-squares on 𝔹^n with degree at most 2 r . We analyze the quality of these bounds by estimating the worst-case error f_min- f_(r) in terms of the least roots of the Krawtchouk polynomials. As a consequence, for fixed t ∈ [0, 1/2] , we can show that this worst-case error in the regime r ≈ t · n is of the order 1/2 - √(t(1-t)) as n tends to ∞ . Our proof combines classical Fourier analysis on 𝔹^n with the polynomial kernel technique and existing results on the extremal roots of Krawtchouk polynomials. This link to roots of orthogonal polynomials relies on a connection between the hierarchy of lower bounds f_(r) and another hierarchy of upper bounds f^(r) , for which we are also able to establish the same error analysis. Our analysis extends to the minimization of a polynomial over the q -ary cube (ℤ/ qℤ)^n . Furthermore, our results apply to the setting of matrix-valued polynomials.