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arXiv:1312.6652 · Electronic colloquium on computational complexity

Rounding Sum-of-Squares Relaxations

Boaz Barak, Jonathan A. Kelner, David Steurer

Published Dec 23, 2013• 0 citations
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Abstract

We present a general approach to rounding semidefinite programming relaxations obtained by the Sum-of-Squares method (Lasserre hierarchy). Our approach is based on using the connection between these relaxations and the Sum-of-Squares proof system to transform a *combining algorithm* -- an algorithm that maps a distribution over solutions into a (possibly weaker) solution -- into a *rounding algorithm* that maps a solution of the relaxation to a solution of the original problem. Using this approach, we obtain algorithms that yield improved results for natural variants of three well-known problems: 1) We give a quasipolynomial-time algorithm that approximates the maximum of a low degree multivariate polynomial with non-negative coefficients over the Euclidean unit sphere. Beyond being of interest in its own right, this is related to an open question in quantum information theory, and our techniques have already led to improved results in this area (Brandão and Harrow, STOC '13). 2) We give a polynomial-time algorithm that, given a d dimensional subspace of R^n that (almost) contains the characteristic function of a set of size n/k, finds a vector $v$ in the subspace satisfying $|v|_4^4 > c(k/d^{1/3}) |v|_2^2$, where $|v|_p = (E_i v_i^p)^{1/p}$. Aside from being a natural relaxation, this is also motivated by a connection to the Small Set Expansion problem shown by Barak et al. (STOC 2012) and our results yield a certain improvement for that problem. 3) We use this notion of L_4 vs. L_2 sparsity to obtain a polynomial-time algorithm with substantially improved guarantees for recovering a planted $μ$-sparse vector v in a random d-dimensional subspace of R^n. If v has mu n nonzero coordinates, we can recover it with high probability whenever $μ< O(\min(1,n/d^2))$, improving for $d < n^{2/3}$ prior methods which intrinsically required $μ< O(1/\sqrt(d))$.

DOI
10.48550/arxiv.1312.6652

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We present a general approach to rounding semidefinite programming relaxations obtained by the Sum-of-Squares method (Lasserre hierarchy). Our approach is based on using the connection between these relaxations and the Sum-of-Squares proof system to transform a combining algorithm -- an algorithm that maps a distribution over solutions into a (possibly weaker) solution -- into a rounding algorithm that maps a solution of the relaxation to a solution of the original problem. Using this approach, we obtain algorithms that yield improved results for natural variants of three well-known problems: 1) We give a quasipolynomial-time algorithm that approximates the maximum of a low degree multivariate polynomial with non-negative coefficients over the Euclidean unit sphere. Beyond being of interest in its own right, this is related to an open question in quantum information theory, and our techniques have already led to improved results in this area (Brandão and Harrow, STOC '13). 2) We give a polynomial-time algorithm that, given a d dimensional subspace of R^n that (almost) contains the characteristic function of a set of size n/k, finds a vector vvv in the subspace satisfying ∣v∣44>c(k/d1/3)∣v∣22|v|_4^4 > c(k/d^{1/3}) |v|_2^2∣v∣44​>c(k/d1/3)∣v∣22​, where ∣v∣p=(Eivip)1/p|v|_p = (E_i v_i^p)^{1/p}∣v∣p​=(Ei​vip​)1/p. Aside from being a natural relaxation, this is also motivated by a connection to the Small Set Expansion problem shown by Barak et al. (STOC 2012) and our results yield a certain improvement for that problem. 3) We use this notion of L_4 vs. L_2 sparsity to obtain a polynomial-time algorithm with substantially improved guarantees for recovering a planted μμμ-sparse vector v in a random d-dimensional subspace of R^n. If v has mu n nonzero coordinates, we can recover it with high probability whenever μ<O(min⁡(1,n/d2))μ< O(\min(1,n/d^2))μ<O(min(1,n/d2)), improving for d<n2/3d < n^{2/3}d<n2/3 prior methods which intrinsically required μ<O(1/(d))μ< O(1/\sqrt(d))μ<O(1/(​d)).
We develop an argmax principle for analyzing sum-of-squares relaxations of optimization problems over the unit sphere. Given a feasible pseudo-expectation, we form a polynomial of high-order pseudo-moments, such as Φk(u)=E~⟨x,u⟩2kΦ_k(u)=\widetilde{\mathbb E}\langle x,u\rangle^{2k}Φk​(u)=E⟨x,u⟩2k. Our guiding principle is that its maximizers are rounding candidates: their local and global optimality conditions reveal the reweighed pseudo-expectation inequalities governing SoS convergence. This viewpoint unifies several problems previously analyzed by rather different techniques. We obtain three results. First, for Best Separable State, we give a degree-O(n/ε)O(\sqrt{n/ε})O(n/ε​) SoS analysis for approximating hsep(P)h_{\mathrm{sep}}(P)hsep​(P) in the perfect-completeness regime, improving and simplifying Barak, Kothari and Steurer (STOC'17). The dependence is essentially tight for inverse-linear gap under the Exponential-Time Hypothesis, matching hardness from QMA(2)\mathrm{QMA}(2)QMA(2) protocols. Second, for the matrix 2→42\to42→4 norm, degree-O(n/ε)O(\sqrt n/ε)O(n​/ε) SoS gives a multiplicative (1+ε)(1+ε)(1+ε) approximation. Barak et al. (STOC'12) previously gave a comparable-time constant-gap decision algorithm; our result gives a multiplicative guarantee and extends to a family of p→qp\to qp→q norms with even qqq. Finally, for degree-ddd polynomial optimization, we recover the convergence theorem of Bhattiprolu et al. (FOCS'17) with a shorter, more direct proof: degree-kkk SoS gives approximation ratio Od((n/k)d/2−1)O_d((n/k)^{d/2-1})Od​((n/k)d/2−1). The paper introduces no new relaxation. Instead, the high-moment argmax gives a common way to read an SoS solution, unifying previously separate convergence analyses and yielding sharper bounds or simpler proofs.
We study quantum algorithms for approximating Lasserre's hierarchy values for polynomial optimization. Let f,g1,…,gmf,g_1,\ldots,g_mf,g1​,…,gm​ be real polynomials in nnn variables and f⋆f^\starf⋆ the infimum of fff over the semialgebraic set S(g)={x:gi(x)≥0}S(g)=\{x: g_i(x)\ge 0\}S(g)={x:gi​(x)≥0}. Let λkλ_kλk​ be the value of the order-kkk Lasserre relaxation. Assume either (i) f⋆=λkf^\star=λ_kf⋆=λk​ and the optimum is attained in the ℓ1\ell_1ℓ1​-ball of radius 1/21/21/2, or (ii) S(g)S(g)S(g) lies in the simplex {x≥0:∑jxj≤1/2}\{x\ge 0: \sum_j x_j\le 1/2\}{x≥0:∑j​xj​≤1/2}, and the constraints define this simplex. After an appropriate coefficient rescaling, we give a quantum algorithm based on matrix multiplicative weights that approximates λkλ_kλk​ to accuracy ε>0\varepsilon>0ε>0 with runtime, for fixed kkk, O(nkε−4+nk/2ε−5),O ⁣(sg ⁣[nkε−4+ ⁣(nk+ ⁣∑i=1mnk−di)1/2 ⁣ε−5]), O(n^k\varepsilon^{-4}+n^{k/2}\varepsilon^{-5}),\qquad O\!\left(s_g\!\left[n^k\varepsilon^{-4}+\!\left(n^{k}+\!\sum_{i=1}^m n^{k-d_i}\right)^{1/2}\!\varepsilon^{-5}\right]\right), O(nkε−4+nk/2ε−5),O(sg​[nkε−4+(nk+∑i=1m​nk−di​)1/2ε−5]), where sgs_gsg​ bounds the sparsity of the coefficient-matching matrices associated with the constraints. Classical matrix multiplicative-weights methods scale as O(n3kpoly(1/ε))O(n^{3k}\mathrm{poly}(1/\varepsilon))O(n3kpoly(1/ε)) even in the unconstrained case. As an example, we obtain an O(nε−4+nε−5)O(n\varepsilon^{-4}+\sqrt{n}\varepsilon^{-5})O(nε−4+n​ε−5) quantum algorithm for portfolio optimization, improving over the classical O(nω+1log⁡(1/ε))O(n^{ω+1}\log(1/\varepsilon))O(nω+1log(1/ε)) bound with ω≈2.373ω\approx2.373ω≈2.373. Our approach builds on and sharpens the analysis of Apeldoorn and Gilyén for the SDPs arising in polynomial optimization. We also show how to implement the required block encodings without QRAM. Under the stated assumptions, our method achieves a super-quadratic speedup in the problem dimension for computing Lasserre relaxations.
The sum-of-squares hierarchy of semidefinite programs has become a common tool for algorithm design in theoretical computer science, including problems in quantum information. In this work we study a connection between a Hermitian version of the SoS hierarchy, related to the quantum de Finetti theorem, and geometric quantization of compact Kähler manifolds (such as complex projective space CPd\mathbb{C}P^{d}CPd, the set of all pure states in a (d+1)(d + 1)(d+1)-dimensional Hilbert space). We show that previously known HSoS rounding algorithms can be recast as quantizing an objective function to obtain a finite-dimensional matrix, finding its top eigenvector, and then (possibly nonconstructively) rounding it by using a version of the Husimi quasiprobability distribution. Dually, we recover most known quantum de Finetti theorems by doing the same steps in the reverse order: a quantum state is first approximated by its Husimi distribution, and then quantized to obtain a separable state approximating the original one. In cases when there is a transitive group action on the manifold we give some new proofs of existing de Finetti theorems, as well as some applications including a new version of Renner's exponential de Finetti theorem proven using the Borel--Weil--Bott theorem, and hardness of approximation results and optimal degree-2 integrality gaps for the basic SDP relaxation of \textsc{Quantum Max-ddd-Cut} (for arbitrary ddd). We also describe how versions of these results can be proven when there is no transitive group action. In these cases we can deduce some error bounds for the HSoS hierarchy on complex projective varieties which are smooth.
We study the best separable state problem (BSS), which asks for the maximum acceptance probability of a quantum measurement over unentangled states. In classical terms, the goal is to maximize ⟨(x⊗y),M(x⊗y)⟩\langle(x \otimes y), M (x \otimes y)\rangle⟨(x⊗y),M(x⊗y)⟩ over unit vectors x,yx,yx,y where 0⪯M⪯I0 \preceq M \preceq I0⪯M⪯I; we call this value BSS(M)\mathrm{BSS}(M)BSS(M). We study BSS\mathrm{BSS}BSS in the "perfect completeness" regime, where given MMM such that BSS(M)=1\mathrm{BSS}(M) = 1BSS(M)=1 the goal is to find the best possible solution x,yx,yx,y -- this generalizes the problem of finding a rank-one matrix as close as possible to a given subspace of Rn×n\mathbb{R}^{n \times n}Rn×n guaranteed to contain a rank-one matrix. The strongest known algorithmic guarantees for this problem are: (1) an algorithm which finds a solution with value 1−ε1-\varepsilon1−ε in time exp⁡(n(log⁡n)O(1)/ε2)\exp(\sqrt{n} (\log n)^{O(1)} / \varepsilon^2)exp(n​(logn)O(1)/ε2), due to Barak, Kothari, and Steurer, and (2) an algorithm which finds a solution with value q/nq/nq/n in time roughly nO(q)n^{O(q)}nO(q), due to Bhattiprolu, Ghosh, Guruswami, Lee, and Tulsiani. We give a much simpler approach to rounding the SoS relaxation, generalizing the canonical "global correlation rounding" technique, and obtain a better running time. Given MMM with BSS(M)=1\mathrm{BSS}(M) = 1BSS(M)=1, our algorithm finds a solution with value 1−ε1-ε1−ε in time nO(n/ε)n^{O(\sqrt{n/\varepsilon})}nO(n/ε​), and a solution of value q/nq/nq/n in time nO(q)n^{O(\sqrt q)}nO(q​). Using the same techniques, we prove a new variant of the "pinning lemma", a measure-decomposition theorem widely used in LP/SDP rounding, high-dimensional probability, and statistical physics, which we believe is of independent interest.
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