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We present new constructions of pseudorandom generators (PRGs) for two of the most widely studied non-uniform circuit classes in complexity theory. Our main result is a construction of the first non-trivial PRG for linear threshold (LTF) circuits of arbitrary constant depth and super-linear size. This PRG fools circuits with depth d∈N and n1+δ wires, where δ=2−O(d) , using seed length O(n1−δ) and with error 2−nδ . This tightly matches the best known lower bounds for this circuit class. As a consequence of our result, all the known hardness for LTF circuits has now effectively been translated into pseudorandomness. This brings the extensive effort in the last decade to construct PRGs and deterministic circuit-analysis algorithms for this class to the point where any subsequent improvement would yield breakthrough lower bounds. Our second contribution is a PRG for De Morgan formulas of size s whose seed length is s1/3+o(1)⋅polylog(1/ϵ) for error ϵ . In particular, our PRG can fool formulas of sub-cubic size s=n3−Ω(1) with an exponentially small error ϵ=exp(−nΩ(1)) . This significantly improves the inverse-polynomial error of the previous state-of-the-art for such formulas by Impagliazzo, Meka, and Zuckerman (FOCS 2012, JACM 2019), and again tightly matches the best currently-known lower bounds for this class. In both settings, a key ingredient in our constructions is a pseudorandom restriction procedure that has tiny failure probability, but simplifies the function to a non-natural “hybrid computational model” that combines several computational models. 

There are only a few known general approaches for constructing explicit pseudorandom generators (PRGs). The "iterated restrictions" approach, pioneered by Ajtai and Wigderson [Ajtai and Wigderson, 1989], has provided PRGs with seed length polylog n or even Õ(log n) for several restricted models of computation. Can this approach ever achieve the optimal seed length of O(log n)? In this work, we answer this question in the affirmative. Using the iterated restrictions approach, we construct an explicit PRG for read-once depth-2 AC⁰[⊕] formulas with seed length O(log n) + Õ(log(1/ε)). In particular, we achieve optimal seed length O(log n) with near-optimal error ε = exp(-Ω̃(log n)). Even for constant error, the best prior PRG for this model (which includes read-once CNFs and read-once 𝔽₂-polynomials) has seed length Θ(log n ⋅ (log log n)²) [Chin Ho Lee, 2019]. A key step in the analysis of our PRG is a tail bound for subset-wise symmetric polynomials, a generalization of elementary symmetric polynomials. Like elementary symmetric polynomials, subset-wise symmetric polynomials provide a way to organize the expansion of ∏_{i=1}^m (1 + y_i). Elementary symmetric polynomials simply organize the terms by degree, i.e., they keep track of the number of variables participating in each monomial. Subset-wise symmetric polynomials keep track of more data: for a fixed partition of [m], they keep track of the number of variables from each subset participating in each monomial. Our tail bound extends prior work by Gopalan and Yehudayoff [Gopalan and Yehudayoff, 2014] on elementary symmetric polynomials.