On Approximating Four Covering and Packing Problems

In this paper, we consider approximability issues of the following four problems: triangle packing, full sibling reconstruction, maximum profit coverage and 2-coverage. All of them are generalized or specialized versions of set-cover and have applications in biology ranging from full-sibling reconstructions in wild populations to biomolecular clusterings; however, as this paper shows, their approximability properties differ considerably. Our inapproximability constant for the triangle packing problem improves upon the previous results; this is done by directly transforming the inapproximability gap of Haastad for the problem of maximizing the number of satisfied equations for a set of equations over GF(2) and is interesting in its own right. Our approximability results on the full siblings reconstruction problems answers questions originally posed by Berger-Wolf et al. and our results on the maximum profit coverage problem provides almost matching upper and lower bounds on the approximation ratio, answering a question posed by Hassin and Or.Comment: 25 page

Determinacy, indeterminacy and dynamic misspecification in linear rational expectations models

This paper proposes a testing strategy for the null hypothesis that a multivariate linear rational expectations (LRE) model has a unique stable solution (determinacy) against the alternative of multiple stable solutions (indeterminacy). Under a proper set of identification restrictions, determinacy is investigated by a misspecification-type approach in which the result of the overidentifying restrictions test obtained from the estimation of the LRE model through a version of generalized method of moments is combined with the result of a likelihood-based test for the cross-equation restrictions that the LRE places on its finite order reduced form under determinacy. This approach (i) circumvents the nonstandard inferential problem that a purely likelihood-based approach implies because of the presence of nuisance parameters that appear under the alternative but not under the null, (ii) does not involve inequality parametric restrictions and nonstandard asymptotic distributions, and (iii) gives rise to a joint test which is consistent against indeterminacy almost everywhere in the space of nuisance parameters, i.e. except for a point of zero measure which gives rise to minimum state variable solutions, and is also consistent against the dynamic misspecification of the LRE model. Monte Carlo simulations show that the testing strategy delivers reasonable size coverage and power in finite samples. An empirical illustration focuses on the determinacy/indeterminacy of a New Keynesian monetary business cycle model for the US.Determinatezza, Indeterminatezza, Massima verosimiglianza, Metodo generalizzato dei momenti, Modello lineare con aspettative, Identificazione, Variabili Strumentali, VAR,VARMA Determinacy, Generalized method of moments, Indeterminacy, LRE model, Identification, Instrumental Variables, Maximum Likelihood, VAR, VARMA

Near-Optimal Straggler Mitigation for Distributed Gradient Methods

Modern learning algorithms use gradient descent updates to train inferential models that best explain data. Scaling these approaches to massive data sizes requires proper distributed gradient descent schemes where distributed worker nodes compute partial gradients based on their partial and local data sets, and send the results to a master node where all the computations are aggregated into a full gradient and the learning model is updated. However, a major performance bottleneck that arises is that some of the worker nodes may run slow. These nodes a.k.a. stragglers can significantly slow down computation as the slowest node may dictate the overall computational time. We propose a distributed computing scheme, called Batched Coupon's Collector (BCC) to alleviate the effect of stragglers in gradient methods. We prove that our BCC scheme is robust to a near optimal number of random stragglers. We also empirically demonstrate that our proposed BCC scheme reduces the run-time by up to 85.4% over Amazon EC2 clusters when compared with other straggler mitigation strategies. We also generalize the proposed BCC scheme to minimize the completion time when implementing gradient descent-based algorithms over heterogeneous worker nodes

A simple recipe for making accurate parametric inference in finite sample

Constructing tests or confidence regions that control over the error rates in the long-run is probably one of the most important problem in statistics. Yet, the theoretical justification for most methods in statistics is asymptotic. The bootstrap for example, despite its simplicity and its widespread usage, is an asymptotic method. There are in general no claim about the exactness of inferential procedures in finite sample. In this paper, we propose an alternative to the parametric bootstrap. We setup general conditions to demonstrate theoretically that accurate inference can be claimed in finite sample