A note on Probably Certifiably Correct algorithms

Many optimization problems of interest are known to be intractable, and while there are often heuristics that are known to work on typical instances, it is usually not easy to determine a posteriori whether the optimal solution was found. In this short note, we discuss algorithms that not only solve the problem on typical instances, but also provide a posteriori certificates of optimality, probably certifiably correct (PCC) algorithms. As an illustrative example, we present a fast PCC algorithm for minimum bisection under the stochastic block model and briefly discuss other examples

Macroeconomic coordination and commercial integration in MERCOSUR

Commercial integration in the Mercosur area has increased substantially in the last few years and it is expected to continue to grow rapidly in the near future. However, given the historical record of policy management in the region, especially in Brazil and Argentina, the main partners of this integration initiative, it is not clear whether macroeconomic policies will provide the required conditions of sustainability for such a rapid trade expansion. This paper discusses the relationship between macroeconomic coordination and commercial integration in the context of Mercosur. After examining the impact of policy instability on trade flows within the region in recent years, it evaluates the prospects for closer coordination of macroeconomic policies in the future.

State-government bailouts in Brazil

As a result of the consolidation of the democracy after the end of the military regime in the mid-1980s, Brazil has gone through a period of remarkable decentralization both in fiscal and political terms. The move towards decentralized management and control of public finances has been followed by a series of bailouts of state governments by the federal government. The lack of effective control on borrowing, coupled with reputational effects originating from these repeated bailout operations, reduced fiscal discipline and created an explosive accumulation of debts in Brazilian states during the last decade. The main purpose of this paper is to assess the determinants of state debt bailouts in Brazil and their relationship with states’ fiscal discipline during the 1990s. After providing a brief overview of intergovernmental fiscal relationships in the Brazilian economy, the paper describes state debt developments from the mid-1980s on, with special emphasis on the 1989, 1993 and 1997 state debt bailouts. Then it discusses the determinants of state debt bailouts in Brazil along the lines of a conceptual framework which recognizes that the essence of the bailout question is the issue of moral hazard and also presents empirical evidence that the occurrence of bailouts is associated with lower fiscal discipline in Brazilian states during the 1990s.

Sharp nonasymptotic bounds on the norm of random matrices with independent entries

We obtain nonasymptotic bounds on the spectral norm of random matrices with independent entries that improve significantly on earlier results. If $X$ is the $n\times n$ symmetric matrix with $X_{ij}\sim N(0,b_{ij}^2)$, we show that $$\mathbf{E}\Vert X\Vert \lesssim\max_i\sqrt{\sum_jb_{ij}^2}+\max _{ij}\vert b_{ij}\vert \sqrt{\log n}.$$ This bound is optimal in the sense that a matching lower bound holds under mild assumptions, and the constants are sufficiently sharp that we can often capture the precise edge of the spectrum. Analogous results are obtained for rectangular matrices and for more general sub-Gaussian or heavy-tailed distributions of the entries, and we derive tail bounds in addition to bounds on the expected norm. The proofs are based on a combination of the moment method and geometric functional analysis techniques. As an application, we show that our bounds immediately yield the correct phase transition behavior of the spectral edge of random band matrices and of sparse Wigner matrices. We also recover a result of Seginer on the norm of Rademacher matrices.Comment: Published at http://dx.doi.org/10.1214/15-AOP1025 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Computational Hardness of Certifying Bounds on Constrained PCA Problems

Given a random n×n symmetric matrix W drawn from the Gaussian orthogonal ensemble (GOE), we consider the problem of certifying an upper bound on the maximum value of the quadratic form x⊤Wx over all vectors x in a constraint set S⊂Rn. For a certain class of normalized constraint sets S we show that, conditional on certain complexity-theoretic assumptions, there is no polynomial-time algorithm certifying a better upper bound than the largest eigenvalue of W. A notable special case included in our results is the hypercube S={±1/n−−√}n, which corresponds to the problem of certifying bounds on the Hamiltonian of the Sherrington-Kirkpatrick spin glass model from statistical physics. Our proof proceeds in two steps. First, we give a reduction from the detection problem in the negatively-spiked Wishart model to the above certification problem. We then give evidence that this Wishart detection problem is computationally hard below the classical spectral threshold, by showing that no low-degree polynomial can (in expectation) distinguish the spiked and unspiked models. This method for identifying computational thresholds was proposed in a sequence of recent works on the sum-of-squares hierarchy, and is believed to be correct for a large class of problems. Our proof can be seen as constructing a distribution over symmetric matrices that appears computationally indistinguishable from the GOE, yet is supported on matrices whose maximum quadratic form over x∈S is much larger than that of a GOE matrix.ISSN:1868-896

Open problem: Tightness of maximum likelihood semidefinite relaxations

We have observed an interesting, yet unexplained, phenomenon: Semidefinite programming (SDP) based relaxations of maximum likelihood estimators (MLE) tend to be tight in recovery problems with noisy data, even when MLE cannot exactly recover the ground truth. Several results establish tightness of SDP based relaxations in the regime where exact recovery from MLE is possible. However, to the best of our knowledge, their tightness is not understood beyond this regime. As an illustrative example, we focus on the generalized Procrustes problem