Improved Pseudorandomness for Unordered Branching Programs through Local Monotonicity

We present an explicit pseudorandom generator with seed length $\tilde{O}((\log n)^{w+1})$ for read-once, oblivious, width $w$ branching programs that can read their input bits in any order. This improves upon the work of Impagliazzo, Meka and Zuckerman where they required seed length $n^{1/2+o(1)}$.

A central ingredient in  our work is the following bound that we prove on the Fourier spectrum of branching programs. For any width $w$ read-once, oblivious branching program $B:\{0,1\}^n \rightarrow \{0,1\}$, any $k \in \{1,\ldots,n\}$, $$\sum_{S\subseteq[n]: |S|=k}|\widehat{B}(S)| \le O(\log n)^{wk}.$$ This settles a conjecture posed by Reingold, Steinke, and Vadhan.

Our analysis crucially uses a notion of local monotonicity on the edge labeling of the branching program. We carry critical parts of our proof under the assumption of local monotonicity and show how to deduce our results for unrestricted branching programs.

Based on joint work with Pooya Hatami, Omer Reingold and Avishay Tal.