Theory Seminar - Pseudorandom Generators from Polarizing Random Walks

Abstract: We propose a new framework for constructing pseudorandom generators for n-variate Boolean functions. It is based on two new notions. First, we introduce fractional pseudorandom generators, which are pseudorandom distributions taking values in [−1,1]^n . Next, we use a fractional pseudorandom generator as steps of a random walk in [−1,1]^n  that converges to {-1,1}^n . We prove that this random walk converges fast (in time logarithmic in n) due to polarization. As an application, we construct pseudorandom generators for Boolean functions with bounded Fourier tails. We use this to obtain a pseudorandom generator for functions with sensitivity s, whose seed length is polynomial in s. Other examples include functions computed
by branching programs of various sorts or by bounded depth circuits.

Based on joint work with Pooya Hatami, Kaave Hosseini,  and Shachar Lovett.