Weighted Polynomial Approximations: Limits for Learning and Pseudorandomness

Polynomial approximations to boolean functions have led to many positive results in computer science. In particular, polynomial approximations to the sign function underly algorithms for agnostically learning halfspaces, as well as pseudorandom generators for halfspaces. In this work, we investigate the limits of these techniques by proving inapproximability results for the sign function. Firstly, the polynomial regression algorithm of Kalai et al. (SIAM J. Comput. 2008) shows that halfspaces can be learned with respect to log-concave distributions on $\mathbb{R}^n$ in the challenging agnostic learning model. The power of this algorithm relies on the fact that under log-concave distributions, halfspaces can be approximated arbitrarily well by low-degree polynomials. We ask whether this technique can be extended beyond log-concave distributions, and establish a negative result. We show that polynomials of any degree cannot approximate the sign function to within arbitrarily low error for a large class of non-log-concave distributions on the real line, including those with densities proportional to $\exp(-|x|^{0.99})$. Secondly, we investigate the derandomization of Chernoff-type concentration inequalities. Chernoff-type tail bounds on sums of independent random variables have pervasive applications in theoretical computer science. Schmidt et al. (SIAM J. Discrete Math. 1995) showed that these inequalities can be established for sums of random variables with only $O(\log(1/\delta))$-wise independence, for a tail probability of $\delta$. We show that their results are tight up to constant factors. These results rely on techniques from weighted approximation theory, which studies how well functions on the real line can be approximated by polynomials under various distributions. We believe that these techniques will have further applications in other areas of computer science.Comment: 22 page

A Nearly Optimal Lower Bound on the Approximate Degree of AC0^0

The approximate degree of a Boolean function $f \colon \{-1, 1\}^n \rightarrow \{-1, 1\}$ is the least degree of a real polynomial that approximates $f$ pointwise to error at most $1/3$. We introduce a generic method for increasing the approximate degree of a given function, while preserving its computability by constant-depth circuits. Specifically, we show how to transform any Boolean function $f$ with approximate degree $d$ into a function $F$ on $O(n \cdot \operatorname{polylog}(n))$ variables with approximate degree at least $D = \Omega(n^{1/3} \cdot d^{2/3})$. In particular, if $d= n^{1-\Omega(1)}$, then $D$ is polynomially larger than $d$. Moreover, if $f$ is computed by a polynomial-size Boolean circuit of constant depth, then so is $F$. By recursively applying our transformation, for any constant $\delta > 0$ we exhibit an AC$^0$ function of approximate degree $\Omega(n^{1-\delta})$. This improves over the best previous lower bound of $\Omega(n^{2/3})$ due to Aaronson and Shi (J. ACM 2004), and nearly matches the trivial upper bound of $n$ that holds for any function. Our lower bounds also apply to (quasipolynomial-size) DNFs of polylogarithmic width. We describe several applications of these results. We give: * For any constant $\delta > 0$, an $\Omega(n^{1-\delta})$ lower bound on the quantum communication complexity of a function in AC$^0$. * A Boolean function $f$ with approximate degree at least $C(f)^{2-o(1)}$, where $C(f)$ is the certificate complexity of $f$. This separation is optimal up to the $o(1)$ term in the exponent. * Improved secret sharing schemes with reconstruction procedures in AC$^0$.Comment: 40 pages, 1 figur

The Direct Productivity Impact of Infrastructure Investment: Dynamic Panel Data Evidence From Sub Saharan Africa

The paper aimed at isolating the direct productivity of economic infrastructure using a production function approach. Based on an extension of endogenous growth theory with public finance, infrastructure could have either a negative or positive effect on economic growth. The empirical analysis utilises a panel of 19 countries from Sub Saharan Africa (SSA). With SSA infrastructure being less developed both in terms of quantity and quality, the a priori expectation was that all types of infrastructure have a positive and signifcant effect on aggregate income level. It is found that, like static estimation techniques, dynamic panel data (DPD) estimation techniques could also produce counterintuitive results if endogeneity of infrastructure is not accounted for. Positive and significant direct productive effects of infrastructure (total roads, electricity generation capacity, and telephones) were obtained using the Pooled Mean Group (PMG) estimator (a form of DPD analysis) after instrumentation for infrastructure. Representing infrastructure with an index constructed from the three infrastructure types also produced similar results. The results are confirmed with the use of the System General Method of Moments (SYS GMM) which constructs instruments for infrastructure using appropriate lags of the variables in first differences and in levels. Thus, it would appear that the negative and counterintuitive productivity results that are sometimes obtained in the literature could be partly due to limitations in methodologies that do not appropriately account for time varying fixed effects and the endogeneity of infrastructure in the economic growth process, especially for developing countries. Control variables for the macroeconomic environment and level of political and civil rights are also found to have a positive and significant effect on aggregate output.

Developing health-related quality-of-life instruments for use in Asia: the issues.

About half of the world's population live in Asia. Mandarin (the official language of China), Hindi and Japanese are among the ten languages spoken by the largest number of primary speakers. The numbers of Tamil and Malay speakers are expected to grow rapidly in the next few decades. Most health-related quality-of-life (HR-QOL) instruments currently used in Asia are translations and/or adaptations of instruments developed in North America and Western Europe. We illustrate and discuss several major issues in the development of HR-QOL instruments for use in Asia. We have seen insufficient quality in translation and semantic equivalence, which is not a uniquely Asian problem. This problem will be alleviated by putting recently proposed guidelines for translation and adaptation of patient-reported outcomes into practice and formally conducting equivalence studies. For copyright or other reasons it is rare to see major adaptations, such as exclusion of a domain in the original instrument or inclusion of a new domain, made to existing instruments. Evidence is limited and mixed as to whether there are differences in the concepts of HR-QOL between Asian and North American/Western European cultures that are important enough to justify such major adaptations, or the development of indigenous instruments, as opposed to the translation/adaptation of existing instruments. There are substantial cultural differences concerning what questions are appropriate to ask and answer. Many HR-QOL instruments are designed for self-completion. This mode of administration is often not feasible in Asia because of low literacy rates and the presence of many different regional languages. Alternative administration methods and analytic strategies that allow for pooling data collected by different modes are needed. The availability of HR-QOL instruments in various Asian countries seems to reflect the status of economic development of the countries rather than their disease burden. For instance, many important HR-QOL instruments are available in Japanese but not in Hindi or Tamil

Regional Fisheries Livelihoods Programme for South and Southeast Asia (RFLP) Activity 1.5 (2011): Systems and Procedures for Participatory Monitoring of Management Measures Developed, Introduced and Implemented-catch Monitoring

Coastal fisheries are very essential for supporting the livelihoods of many rural poor in the coastal areas, particularly coastal community fisheries members. They serve as sources of food, employment and income generation for many coastal families. "Coastal Community Fisheries Catch Monitoring" Project which was conducted from April to November 2011 provides some data which indicates the importance of small-scale fisheries. The project was financially supported by the Regional Fisheries Livelihoods Programme, Cambodian component (RFLP/CMB) and was activity 1.5 of the approved RFLP CMB 2011 activity work plan and budget. For this catch monitoring study, 26 small-scale subsistence fishers, including 05 women, from five community fisheries (CFi?s) in the RFLP CMB area of geographic coverage from the four coastal provinces of Cambodia were selected and following appropriate training collected specific catch data and recorded it in fisher's logbook on a daily basis for the purpose of getting a better understanding of catch per unit of effort (CEPU), the health of inshore fish stock and the contribution of aquatic products to small-scale fisher households along the coast of Cambodia. The key data items recorded included total catchweight, catch weight by species, total catch sale price, fish price of the main species and total lengths of some key species. The study involved designing logbooks, training the selected 26 fishers as data collectors on data collection methods, collecting data from all the selected fishers, designing a database and entering all the collected data into the database, checking for errors and analyzing the collected data for final reporting and preparing report

Bias Correction in the Dynamic Panel Data Model with a Nonscalar Disturbance Covariance Matrix

By using asymptotic expansion techniques approximation formulae are developed for the bias of ordinary and generalized Least Squares Dummy Variable (LSDV) estimators in dynamic panel data models. Earlier results on bias approximation in first-order stable dynamic panel data models are extended to higher-order dynamic models with general disturbance covariance structure. The focus is on estimation of both short- and long-run coefficients. The results show that proper modelling of the disturbance covariance structure is indispensable. The bias approximations are used to construct bias corrected estimators which are then applied to quarterly data from 14 European Union countries. Money demand functions for M1, M2 and M3 are estimated for the EU area as a whole for the period 1991:I-1995:IV. The empirical results show that in general plausible long-run effects are obtained by the bias corrected estimators. Moreover, bias correction can be substantial underlining the importance of more refined estimation techniques. Also the efficiency gains by exploiting the heteroscedasticity and cross-correlation patterns between countries are considerable.

The weak instrument problem of the system GMM estimator in dynamic panel data models

The system GMM estimator for dynamic panel data models combines moment conditions for the model in first differences with moment conditions for the model in levels. It has been shown to improve on the GMM estimator in the first differenced model in terms of bias and root mean squared error. However, we show in this paper that in the covariance stationary panel data AR(1) model the expected values of the concentration parameters in the differenced and levels equations for the crosssection at time t are the same when the variances of the individual heterogeneity and idiosyncratic errors are the same. This indicates a weak instrument problem also for the equation in levels. We show that the 2SLS biases relative to that of the OLS biases are then similar for the equations in differences and levels, as are the size distortions of the Wald tests. These results are shown in a Monte Carlo study to extend to the panel data system GMM estimator.Dynamic panel data, system GMM, weak instruments