[Paper Review] On the sum-of-squares degree of symmetric quadratic functions

Troy Lee, Anupam Prakash+2 more|arXiv (Cornell University)|Jan 11, 2016

Advanced Optimization Algorithms Research23 references5 citations

TL;DR

This paper establishes tight bounds on the sum-of-squares (SOS) degree required to approximate symmetric quadratic functions of the form $f_k(x) = (|x| - k)(|x| - k - 1)$ on the boolean hypercube, both in $\ell_\infty$ and $\ell_1$-norms. It proves that the SOS degree for $\ell_1$-approximation of $f_k$ is $\Theta(\sqrt{k(n-k)})$, and uses this to resolve open problems in proof complexity and quantum query complexity.

ABSTRACT

We study how well functions over the boolean hypercube of the form $f_k(x)=(|x|-k)(|x|-k-1)$ can be approximated by sums of squares of low-degree polynomials, obtaining good bounds for the case of approximation in $\ell_{\infty}$-norm as well as in $\ell_1$-norm. We describe three complexity-theoretic applications: (1) a proof that the recent breakthrough lower bound of Lee, Raghavendra, and Steurer on the positive semidefinite extension complexity of the correlation and TSP polytopes cannot be improved further by showing better sum-of-squares degree lower bounds on $\ell_1$-approximation of $f_k$; (2) a proof that Grigoriev's lower bound on the degree of Positivstellensatz refutations for the knapsack problem is optimal, answering an open question from his work; (3) bounds on the query complexity of quantum algorithms whose expected output approximates such functions.

Motivation & Objective

• To determine the minimal degree of sum-of-squares polynomials that approximate symmetric quadratic functions $f_k(x) = (|x| - k)(|x| - k - 1)$ on the boolean hypercube.
• To establish tight bounds on the $\ell_\infty$ and $\ell_1$-approximate SOS degrees for these functions.
• To resolve open questions in complexity theory, including the optimality of Grigoriev's Positivstellensatz lower bound for knapsack and limitations on SOS-based extensions of polytopes.
• To provide new bounds on the query complexity of quantum algorithms that approximate such symmetric functions.

Proposed method

• Analyzes the SOS approximation of symmetric functions $f_k(x) = (|x| - k)(|x| - k - 1)$ using univariate polynomial techniques and symmetric function identities.
• Applies Blekherman’s theorem on non-negative polynomials to construct low-degree SOS representations for $A_k(\mathbf{x}) = |\mathbf{x}|(|\mathbf{x}|-1)\cdots(|\mathbf{x}|-k+1)$.
• Uses the identity $A_{k+2}(\mathbf{x}) + b = g(\mathbf{x})(|\mathbf{x}| - r)$ with $b = -r(r-1)\cdots(r-k-1)$ to construct a degree-$2k+4$ Positivstellensatz refutation for the knapsack system.
• Reduces the SOS approximation problem to semidefinite programming feasibility, leveraging the moment matrix and monomial basis $m_x$ of degree $\leq d$.
• Employs the $\ell_1$-norm approximation via $\sum_x |f(x) - h(x)| \leq \varepsilon$, with $\varepsilon = \delta 2^n$ corresponding to average error $\delta$.
• Uses the fact that $e_k(x_1^2, \dots, x_n^2)$ is a degree-$2k$ SOS, and relates $A_k$ to elementary symmetric polynomials and constraints $x_i^2 - x_i = 0$.

Experimental results

Research questions

• RQ1 What is the minimal SOS degree required to $\varepsilon$-approximate $f_k(x) = (|x| - k)(|x| - k - 1)$ in $\ell_\infty$-norm?
• RQ2 What is the minimal SOS degree required to $\varepsilon$-approximate $f_k$ in $\ell_1$-norm, and how does it scale with $k$ and $n$?
• RQ3 Is Grigoriev’s $2k+4$ degree lower bound for Positivstellensatz refutations of the knapsack system tight?
• RQ4 Can the SOS degree of $f_k$ be used to prove limitations on semidefinite programming hierarchies for polytopes like the correlation and TSP polytopes?
• RQ5 What are the implications of these SOS bounds for the query complexity of quantum algorithms approximating symmetric functions?

Key findings

• The $\ell_1$-approximate SOS degree of $f_k$ is $\Theta(\sqrt{k(n-k)})$, matching the exact degree of the function and showing tightness of the bound.
• The $\ell_\infty$-approximate SOS degree of $f_k$ is $\Theta(\sqrt{k(n-k)} + \sqrt{n\log(1/\varepsilon)})$, analogous to the approximate degree of symmetric boolean functions.
• Grigoriev’s $2k+4$ degree lower bound for Positivstellensatz refutations of the knapsack system is optimal, as a degree-$2k+4$ refutation exists.
• The SOS degree of $f_k$ cannot be used to improve the $\ell_1$-approximation lower bounds for the correlation and TSP polytopes beyond what was already shown by Lee, Raghavendra, and Steurer.
• The results imply that quantum algorithms approximating $f_k$ non-adaptively require $\Omega(\sqrt{k(n-k)})$ queries, matching the known lower bound for the OR function.
• The construction of a degree-$2k+4$ Positivstellensatz refutation for $k < r < k+1$ confirms the tightness of Grigoriev’s bound and resolves an open question from [Gri01].