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

On the sum of the L1 influences of bounded functions

Let $f\colon \{-1,1\}^n \to [-1,1]$ have degree $d$ as a multilinear polynomial. It is well-known that the total influence of $f$ is at most $d$. Aaronson and Ambainis asked whether the total $L_1$ influence of $f$ can also be bounded as a function of $d$. Ba\v{c}kurs and Bavarian answered this question in the affirmative, providing a bound of $O(d^3)$ for general functions and $O(d^2)$ for homogeneous functions. We improve on their results by providing a bound of $d^2$ for general functions and $O(d\log d)$ for homogeneous functions. In addition, we prove a bound of $d/(2 \pi)+o(d)$ for monotone functions, and provide a matching example.Comment: 16 pages; accepted for publication in the Israel Journal of Mathematic

The intersection of two halfspaces has high threshold degree

The threshold degree of a Boolean function f:{0,1}^n->{-1,+1} is the least degree of a real polynomial p such that f(x)=sgn p(x). We construct two halfspaces on {0,1}^n whose intersection has threshold degree Theta(sqrt n), an exponential improvement on previous lower bounds. This solves an open problem due to Klivans (2002) and rules out the use of perceptron-based techniques for PAC learning the intersection of two halfspaces, a central unresolved challenge in computational learning. We also prove that the intersection of two majority functions has threshold degree Omega(log n), which is tight and settles a conjecture of O'Donnell and Servedio (2003). Our proof consists of two parts. First, we show that for any nonconstant Boolean functions f and g, the intersection f(x)^g(y) has threshold degree O(d) if and only if ||f-F||_infty + ||g-G||_infty < 1 for some rational functions F, G of degree O(d). Second, we settle the least degree required for approximating a halfspace and a majority function to any given accuracy by rational functions. Our technique further allows us to make progress on Aaronson's challenge (2008) and contribute strong direct product theorems for polynomial representations of composed Boolean functions of the form F(f_1,...,f_n). In particular, we give an improved lower bound on the approximate degree of the AND-OR tree.Comment: Full version of the FOCS'09 pape

On the sum-of-squares degree of symmetric quadratic functions

We study how well functions over the boolean hypercube of the form $f_k(x)=(|x|-k)(|x|-k-1)$ can be approximated by sums of squares of low-degree polynomials, obtaining good bounds for the case of approximation in $\ell_{\infty}$-norm as well as in $\ell_1$-norm. We describe three complexity-theoretic applications: (1) a proof that the recent breakthrough lower bound of Lee, Raghavendra, and Steurer on the positive semidefinite extension complexity of the correlation and TSP polytopes cannot be improved further by showing better sum-of-squares degree lower bounds on $\ell_1$-approximation of $f_k$; (2) a proof that Grigoriev's lower bound on the degree of Positivstellensatz refutations for the knapsack problem is optimal, answering an open question from his work; (3) bounds on the query complexity of quantum algorithms whose expected output approximates such functions.Comment: 33 pages. Second version fixes some typos and adds reference