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Register a new userRounding Sum-of-Squares Relaxations
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Boaz Barak
Publication date
2013
fields
Informatics Engineering
and research's language is
English
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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 (Brand~{a}o 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 $mu$-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 $mu < O(min(1,n/d^2))$, improving for $d < n^{2/3}$ prior methods which intrinsically required $mu < O(1/sqrt(d))$.
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We develop efficient algorithms for estimating low-degree moments of unknown distributions in the presence of adversarial outliers. The guarantees of our algorithms improve in many cases significantly over the best previous ones, obtained in recent works of Diakonikolas et al, Lai et al, and Charikar et al. We also show that the guarantees of our algorithms match information-theoretic lower-bounds for the class of distributions we consider. These improved guarantees allow us to give improved algorithms for independent component analysis and learning mixtures of Gaussians in the presence of outliers. Our algorithms are based on a standard sum-of-squares relaxation of the following conceptually-simple optimization problem: Among all distributions whose moments are bounded in the same way as for the unknown distribution, find the one that is closest in statistical distance to the empirical distribution of the adversarially-corrupted sample.
We give a new approach to the dictionary learning (also known as sparse coding) problem of recovering an unknown $ntimes m$ matrix $A$ (for $m geq n$) from examples of the form [ y = Ax + e, ] where $x$ is a random vector in $mathbb R^m$ with at most $tau m$ nonzero coordinates, and $e$ is a random noise vector in $mathbb R^n$ with bounded magnitude. For the case $m=O(n)$, our algorithm recovers every column of $A$ within arbitrarily good constant accuracy in time $m^{O(log m/log(tau^{-1}))}$, in particular achieving polynomial time if $tau = m^{-delta}$ for any $delta>0$, and time $m^{O(log m)}$ if $tau$ is (a sufficiently small) constant. Prior algorithms with comparable assumptions on the distribution required the vector $x$ to be much sparser---at most $sqrt{n}$ nonzero coordinates---and there were intrinsic barriers preventing these algorithms from applying for denser $x$. We achieve this by designing an algorithm for noisy tensor decomposition that can recover, under quite general conditions, an approximate rank-one decomposition of a tensor $T$, given access to a tensor $T$ that is $tau$-close to $T$ in the spectral norm (when considered as a matrix). To our knowledge, this is the first algorithm for tensor decomposition that works in the constant spectral-norm noise regime, where there is no guarantee that the local optima of $T$ and $T$ have similar structures. Our algorithm is based on a novel approach to using and analyzing the Sum of Squares semidefinite programming hierarchy (Parrilo 2000, Lasserre 2001), and it can be viewed as an indication of the utility of this very general and powerful tool for unsupervised learning problems.
Estimation is the computational task of recovering a hidden parameter $x$ associated with a distribution $D_x$, given a measurement $y$ sampled from the distribution. High dimensional estimation problems arise naturally in statistics, machine learning, and complexity theory. Many high dimensional estimation problems can be formulated as systems of polynomial equations and inequalities, and thus give rise to natural probability distributions over polynomial systems. Sum-of-squares proofs provide a powerful framework to reason about polynomial systems, and further there exist efficient algorithms to search for low-degree sum-of-squares proofs. Understanding and characterizing the power of sum-of-squares proofs for estimation problems has been a subject of intense study in recent years. On one hand, there is a growing body of work utilizing sum-of-squares proofs for recovering solutions to polynomial systems when the system is feasible. On the other hand, a general technique referred to as pseudocalibration has been developed towards showing lower bounds on the degree of sum-of-squares proofs. Finally, the existence of sum-of-squares refutations of a polynomial system has been shown to be intimately connected to the existence of spectral algorithms. In this article we survey these developments.
We study planted problems---finding hidden structures in random noisy inputs---through the lens of the sum-of-squares semidefinite programming hierarchy (SoS). This family of powerful semidefinite programs has recently yielded many new algorithms for planted problems, often achieving the best known polynomial-time guarantees in terms of accuracy of recovered solutions and robustness to noise. One theme in recent work is the design of spectral algorithms which match the guarantees of SoS algorithms for planted problems. Classical spectral algorithms are often unable to accomplish this: the twist in these new spectral algorithms is the use of spectral structure of matrices whose entries are low-degree polynomials of the input variables. We prove that for a wide class of planted problems, including refuting random constraint satisfaction problems, tensor and sparse PCA, densest-k-subgraph, community detection in stochastic block models, planted clique, and others, eigenvalues of degree-d matrix polynomials are as powerful as SoS semidefinite programs of roughly degree d. For such problems it is therefore always possible to match the guarantees of SoS without solving a large semidefinite program. Using related ideas on SoS algorithms and low-degree matrix polynomials (and inspired by recent work on SoS and the planted clique problem by Barak et al.), we prove new nearly-tight SoS lower bounds for the tensor and sparse principal component analysis problems. Our lower bounds for sparse principal component analysis are the first to suggest that going beyond existing algorithms for this problem may require sub-exponential time.
In order to obtain the best-known guarantees, algorithms are traditionally tailored to the particular problem we want to solve. Two recent developments, the Unique Games Conjecture (UGC) and the Sum-of-Squares (SOS) method, surprisingly suggest that this tailoring is not necessary and that a single efficient algorithm could achieve best possible guarantees for a wide range of different problems. The Unique Games Conjecture (UGC) is a tantalizing conjecture in computational complexity, which, if true, will shed light on the complexity of a great many problems. In particular this conjecture predicts that a single concrete algorithm provides optimal guarantees among all efficient algorithms for a large class of computational problems. The Sum-of-Squares (SOS) method is a general approach for solving systems of polynomial constraints. This approach is studied in several scientific disciplines, including real algebraic geometry, proof complexity, control theory, and mathematical programming, and has found applications in fields as diverse as quantum information theory, formal verification, game theory and many others. We survey some connections that were recently uncovered between the Unique Games Conjecture and the Sum-of-Squares method. In particular, we discuss new tools to rigorously bound the running time of the SOS method for obtaining approximate solutions to hard optimization problems, and how these tools give the potential for the sum-of-squares method to provide new guarantees for many problems of interest, and possibly to even refute the UGC.
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