Pseudorandomness via the discrete Fourier transform

We present a new approach to constructing unconditional pseudorandom generators against classes of functions that involve computing a linear function of the inputs. We give an explicit construction of a pseudorandom generator that fools the discrete Fourier transforms of linear functions with seed-length that is nearly logarithmic (up to polyloglog factors) in the input size and the desired error parameter. Our result gives a single pseudorandom generator that fools several important classes of tests computable in logspace that have been considered in the literature, including halfspaces (over general domains), modular tests and combinatorial shapes. For all these classes, our generator is the first that achieves near logarithmic seed-length in both the input length and the error parameter. Getting such a seed-length is a natural challenge in its own right, which needs to be overcome in order to derandomize RL - a central question in complexity theory. Our construction combines ideas from a large body of prior work, ranging from a classical construction of [NN93] to the recent gradually increasing independence paradigm of [KMN11, CRSW13, GMRTV12], while also introducing some novel analytic machinery which might find other applications

Moment-Matching Polynomials

We give a new framework for proving the existence of low-degree, polynomial approximators for Boolean functions with respect to broad classes of non-product distributions. Our proofs use techniques related to the classical moment problem and deviate significantly from known Fourier-based methods, which require the underlying distribution to have some product structure. Our main application is the first polynomial-time algorithm for agnostically learning any function of a constant number of halfspaces with respect to any log-concave distribution (for any constant accuracy parameter). This result was not known even for the case of learning the intersection of two halfspaces without noise. Additionally, we show that in the "smoothed-analysis" setting, the above results hold with respect to distributions that have sub-exponential tails, a property satisfied by many natural and well-studied distributions in machine learning. Given that our algorithms can be implemented using Support Vector Machines (SVMs) with a polynomial kernel, these results give a rigorous theoretical explanation as to why many kernel methods work so well in practice

Bounded Independence Fools Degree-2 Threshold Functions

Let x be a random vector coming from any k-wise independent distribution over {-1,1}^n. For an n-variate degree-2 polynomial p, we prove that E[sgn(p(x))] is determined up to an additive epsilon for k = poly(1/epsilon). This answers an open question of Diakonikolas et al. (FOCS 2009). Using standard constructions of k-wise independent distributions, we obtain a broad class of explicit generators that epsilon-fool the class of degree-2 threshold functions with seed length log(n)*poly(1/epsilon). Our approach is quite robust: it easily extends to yield that the intersection of any constant number of degree-2 threshold functions is epsilon-fooled by poly(1/epsilon)-wise independence. Our results also hold if the entries of x are k-wise independent standard normals, implying for example that bounded independence derandomizes the Goemans-Williamson hyperplane rounding scheme. To achieve our results, we introduce a technique we dub multivariate FT-mollification, a generalization of the univariate form introduced by Kane et al. (SODA 2010) in the context of streaming algorithms. Along the way we prove a generalized hypercontractive inequality for quadratic forms which takes the operator norm of the associated matrix into account. These techniques may be of independent interest.Comment: Using v1 numbering: removed Lemma G.5 from the Appendix (it was wrong). Net effect is that Theorem G.6 reduces the m^6 dependence of Theorem 8.1 to m^4, not m^

Algorithms and lower bounds for de Morgan formulas of low-communication leaf gates

The class $FORMULA[s] \circ \mathcal{G}$ consists of Boolean functions computable by size-$s$ de Morgan formulas whose leaves are any Boolean functions from a class $\mathcal{G}$. We give lower bounds and (SAT, Learning, and PRG) algorithms for $FORMULA[n^{1.99}]\circ \mathcal{G}$, for classes $\mathcal{G}$ of functions with low communication complexity. Let $R^{(k)}(\mathcal{G})$ be the maximum $k$-party NOF randomized communication complexity of $\mathcal{G}$. We show: (1) The Generalized Inner Product function $GIP^k_n$ cannot be computed in $FORMULA[s]\circ \mathcal{G}$ on more than $1/2+\varepsilon$ fraction of inputs for $$s = o \! \left ( \frac{n^2}{ \left(k \cdot 4^k \cdot {R}^{(k)}(\mathcal{G}) \cdot \log (n/\varepsilon) \cdot \log(1/\varepsilon) \right)^{2}} \right).$$ As a corollary, we get an average-case lower bound for $GIP^k_n$ against $FORMULA[n^{1.99}]\circ PTF^{k-1}$. (2) There is a PRG of seed length $n/2 + O\left(\sqrt{s} \cdot R^{(2)}(\mathcal{G}) \cdot\log(s/\varepsilon) \cdot \log (1/\varepsilon) \right)$ that $\varepsilon$-fools $FORMULA[s] \circ \mathcal{G}$. For $FORMULA[s] \circ LTF$, we get the better seed length $O\left(n^{1/2}\cdot s^{1/4}\cdot \log(n)\cdot \log(n/\varepsilon)\right)$. This gives the first non-trivial PRG (with seed length $o(n)$) for intersections of $n$ half-spaces in the regime where $\varepsilon \leq 1/n$. (3) There is a randomized $2^{n-t}$-time $\#$SAT algorithm for $FORMULA[s] \circ \mathcal{G}$, where $$t=\Omega\left(\frac{n}{\sqrt{s}\cdot\log^2(s)\cdot R^{(2)}(\mathcal{G})}\right)^{1/2}.$$ In particular, this implies a nontrivial #SAT algorithm for $FORMULA[n^{1.99}]\circ LTF$. (4) The Minimum Circuit Size Problem is not in $FORMULA[n^{1.99}]\circ XOR$. On the algorithmic side, we show that $FORMULA[n^{1.99}] \circ XOR$ can be PAC-learned in time $2^{O(n/\log n)}$

A PRG for Lipschitz Functions of Polynomials with Applications to Sparsest Cut

We give improved pseudorandom generators (PRGs) for Lipschitz functions of low-degree polynomials over the hypercube. These are functions of the form psi(P(x)), where P is a low-degree polynomial and psi is a function with small Lipschitz constant. PRGs for smooth functions of low-degree polynomials have received a lot of attention recently and play an important role in constructing PRGs for the natural class of polynomial threshold functions. In spite of the recent progress, no nontrivial PRGs were known for fooling Lipschitz functions of degree O(log n) polynomials even for constant error rate. In this work, we give the first such generator obtaining a seed-length of (log n)\tilde{O}(d^2/eps^2) for fooling degree d polynomials with error eps. Previous generators had an exponential dependence on the degree. We use our PRG to get better integrality gap instances for sparsest cut, a fundamental problem in graph theory with many applications in graph optimization. We give an instance of uniform sparsest cut for which a powerful semi-definite relaxation (SDP) first introduced by Goemans and Linial and studied in the seminal work of Arora, Rao and Vazirani has an integrality gap of exp(\Omega((log log n)^{1/2})). Understanding the performance of the Goemans-Linial SDP for uniform sparsest cut is an important open problem in approximation algorithms and metric embeddings and our work gives a near-exponential improvement over previous lower bounds which achieved a gap of \Omega(log log n)

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