In a recent work, Chen, Hoza, Lyu, Tal and Wu (FOCS 2023) showed an improved error reduction framework for the derandomization of regular read-once branching programs (ROBPs). Their result is based on a clever modification to the inverse Laplacian perspective of space-bounded derandomization, which was originally introduced by Ahmadinejad, Kelner, Murtagh, Peebles, Sidford and Vadhan (FOCS 2020). In this work, we give an alternative error reduction framework for regular ROBPs. Our new framework is based on a binary recursive formula from the work of Chattopadhyay and Liao (CCC 2020), that they used to construct weighted pseudorandom generators (WPRGs) for general ROBPs. Based on our new error reduction framework, we give alternative proofs to the following results for regular ROBPs of length $n$ and width $w$, both of which were proved in the work of Chen et al. using their error reduction: $\bullet$ There is a WPRG with error $\varepsilon$ that has seed length $\tilde{O}(\log(n)(\sqrt{\log(1/\varepsilon)}+\log(w))+\log(1/\varepsilon)).$ $\bullet$ There is a (non-black-box) deterministic algorithm which estimates the expectation of any such program within error $\pm\varepsilon$ with space complexity $\tilde{O}(\log(nw)\cdot\log\log(1/\varepsilon)).$ (This was first proved in the work of Ahmadinejad et al., but the proof by Chen et al. is simpler.) Because of the binary recursive nature of our new framework, both of our proofs are based on a straightforward induction that is arguably simpler than the Laplacian-based proof in the work of Chen et al.

翻译：在近期的工作中，Chen、Hoza、Lyu、Tal 和 Wu（FOCS 2023）针对正则一次读取分支程序（ROBP）的去随机化提出了一种改进的误差缩减框架。他们的结果基于对空间有界去随机化的逆拉普拉斯视角的巧妙修改，该视角最初由 Ahmadinejad、Kelner、Murtagh、Peebles、Sidford 和 Vadhan（FOCS 2020）引入。在本工作中，我们为正则 ROBP 提供了另一种误差缩减框架。我们的新框架基于 Chattopadhyay 和 Liao（CCC 2020）工作中提出的二进制递归公式，该公式被用于构造一般 ROBP 的加权伪随机生成器（WPRG）。基于我们的新误差缩减框架，我们对以下关于长度为 $n$、宽度为 $w$ 的正则 ROBP 的结果给出了替代证明，这两个结果均已在 Chen 等人的工作中使用其误差缩减方法得到证明： $\bullet$ 存在一个误差为 $\varepsilon$ 的 WPRG，其种子长度为 $\tilde{O}(\log(n)(\sqrt{\log(1/\varepsilon)}+\log(w))+\log(1/\varepsilon))$。 $\bullet$ 存在一个（非黑盒）确定性算法，能够在空间复杂度 $\tilde{O}(\log(nw)\cdot\log\log(1/\varepsilon))$ 下估算任意此类程序的期望值，误差在 $\pm\varepsilon$ 范围内。（该结果最初由 Ahmadinejad 等人证明，但 Chen 等人的证明更简洁。） 由于我们新框架的二进制递归特性，两个证明均基于直接的归纳法，其简洁性可论证地优于 Chen 等人工作中基于拉普拉斯方法的证明。