Ryan O'Donnell, Rocco A. Servedio, Li-Yang Tan
We give a pseudorandom generator that fools $m$-facet polytopes over $\{0,1\}^n$ with seed length $\mathrm{polylog}(m) \cdot \log n$. The previous best seed length had superlinear dependence on $m$. An immediate consequence is a deterministic quasipolynomial time algorithm for approximating the number of solutions to any $\{0,1\}$-integer program.
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A weight-$t$ halfspace is a Boolean function $f(x)=$sign$(w_1 x_1 + \cdots + w_n x_n - θ)$ where each $w_i$ is an integer in $\{-t,\dots,t\}.$ We give an explicit pseudorandom generator that $δ$-fools any intersection of $k$ weight-$t$ halfspaces with seed length poly$(\log n, \log k,t,1/δ)$. In particular, our result gives an explicit PRG that fools any intersection of any quasipoly$(n)$ number of halfspaces of any poly$\log(n)$ weight to any $1/$poly$\log(n)$ accuracy using seed length poly$\log(n).$ Prior to this work no explicit PRG with non-trivial seed length was known even for fooling i…
We give a pseudorandom generator that fools degree-$d$ polynomial threshold functions over $n$-dimensional Gaussian space with seed length $\mathrm{poly}(d)\cdot \log n$. All previous generators had a seed length with at least a $2^d$ dependence on $d$. The key new ingredient is a Local Hyperconcentration Theorem, which shows that every degree-$d$ Gaussian polynomial is hyperconcentrated almost everywhere at scale $d^{-O(1)}$.
We construct pseudorandom generators that fool functions of halfspaces (threshold functions) under a very broad class of product distributions. This class includes not only familiar cases such as the uniform distribution on the discrete cube, the uniform distribution on the solid cube, and the multivariate Gaussian distribution, but also includes any product of discrete distributions with probabilities bounded away from 0. Our first main result shows that a recent pseudorandom generator construction of Meka and Zuckerman [MZ09], when suitably modifed, can fool arbitrary functions of d halfspac…
We study the natural question of constructing pseudorandom generators (PRGs) for low-degree polynomial threshold functions (PTFs). We give a PRG with seed-length log n/eps^{O(d)} fooling degree d PTFs with error at most eps. Previously, no nontrivial constructions were known even for quadratic threshold functions and constant error eps. For the class of degree 1 threshold functions or halfspaces, we construct PRGs with much better dependence on the error parameter eps and obtain a PRG with seed-length O(log n + log^2(1/eps)). Previously, only PRGs with seed length O(log n log^2(1/eps)/eps^2) w…
Weighted Pseudorandom Generators for Read-Once Branching Programs via Weighted Pseudorandom Reductions
研究读一次分支程序的加权伪随机生成器及去随机化，通过迭代加权伪随机归约，给出标准ROBP和置换ROBP的\(\varepsilon -\)WPRG种子长度，以及正则ROBP的去随机化方法，达到最优空间复杂度。
In a recent work, O'Donnell, Servedio and Tan (STOC 2019) gave explicit pseudorandom generators (PRGs) for arbitrary $m$-facet polytopes in $n$ variables with seed length poly-logarithmic in $m,n$, concluding a sequence of works in the last decade, that was started by Diakonikolas, Gopalan, Jaiswal, Servedio, Viola (SICOMP 2010) and Meka, Zuckerman (SICOMP 2013) for fooling linear and polynomial threshold functions, respectively. In this work, we consider a natural extension of PRGs for intersections of positive spectrahedrons. A positive spectrahedron is a Boolean function $f(x)=[x_1A^1+\cdot…
We construct pseudorandom generators of seed length $\tilde{O}(\log(n)\cdot \log(1/ε))$ that $ε$-fool ordered read-once branching programs (ROBPs) of width $3$ and length $n$. For unordered ROBPs, we construct pseudorandom generators with seed length $\tilde{O}(\log(n) \cdot \mathrm{poly}(1/ε))$. This is the first improvement for pseudorandom generators fooling width $3$ ROBPs since the work of Nisan [Combinatorica, 1992]. Our constructions are based on the `iterated milder restrictions' approach of Gopalan et al. [FOCS, 2012] (which further extends the Ajtai-Wigderson framework [FOCS, 1985]),…
The problem of constructing pseudorandom generators that fool halfspaces has been studied intensively in recent times. For fooling halfspaces over the hypercube with polynomially small error, the best construction known requires seed-length O(log^2 n) (MekaZ13). Getting the seed-length down to O(log(n)) is a natural challenge in its own right, which needs to be overcome in order to derandomize RL. In this work we make progress towards this goal by obtaining near-optimal generators for two important special cases: 1) We give a near optimal derandomization of the Chernoff bound for independent, …
We show a new PRG construction fooling depth-$d$, size-$m$ $\mathsf{AC}^0$ circuits within error $\varepsilon$, which has seed length $O(\log^{d-1}(m)\log(m/\varepsilon)\log\log(m))$. Our PRG improves on previous work (Trevisan and Xue 2013, Servedio and Tan 2019, Kelley 2021) from various aspects. It has optimal dependence on $\frac{1}{\varepsilon}$ and is only one ``$\log\log(m)$'' away from the lower bound barrier. For the case of $d=2$, the seed length tightly matches the best-known PRG for CNFs (De et al. 2010, Tal 2017). There are two technical ingredients behind our new result; both of …
A central question in derandomization is whether randomized logspace (RL) equals deterministic logspace (L). To show that RL=L, it suffices to construct explicit pseudorandom generators (PRGs) that fool polynomial-size read-once (oblivious) branching programs (roBPs). Starting with the work of Nisan, pseudorandom generators with seed-length $O(\log^2 n)$ were constructed. Unfortunately, improving on this seed-length in general has proven challenging and seems to require new ideas. A recent line of inquiry has suggested focusing on a particular limitation of the existing PRGs, which is that the…
Fooling Thresholds of Halfspaces
本文为半空间阈值构造显式伪随机生成器，证明多胞体生成器可欺骗此类函数，并给出种子长度界，同时导出噪声敏感性与学习算法。
A sliding-window algorithm of window size $t$ is an algorithm whose current operation depends solely on the last $t$ symbols read. We construct pseudorandom generators (PRGs) for low-space randomized sliding-window algorithms that have access to a binary randomness source. More specifically, we lift these algorithms to the non-uniform setting of branching programs and study them as a subclass thereof that we call sliding-window branching programs (SWBPs), accordingly. For general SWBPs, given a base PRG $G_\mathrm{base}$ with seed length $d_\mathrm{base}$ that $\varepsilon_\mathrm{base}$-fools…
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 ra…
Goldreich suggested candidates of one-way functions and pseudorandom generators included in $\mathsf{NC}^0$. It is known that randomly generated Goldreich's generator using $(r-1)$-wise independent predicates with $n$ input variables and $m=C n^{r/2}$ output variables is not pseudorandom generator with high probability for sufficiently large constant $C$. Most of the previous works assume that the alphabet is binary and use techniques available only for the binary alphabet. In this paper, we deal with non-binary generalization of Goldreich's generator and derives the tight threshold for linear…
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