Revision #1 Authors: Eshan Chattopadhyay, Jyun-Jie Liao

Accepted on: 6th December 2023 22:21

Downloads: 172

Keywords:

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:

* There is a WPRG with error $\varepsilon$ that has seed length $$\tilde{O}(\log(n)(\sqrt{\log(1/\varepsilon)}+\log(w))+\log(1/\varepsilon)).$$

* 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.

In fact, because of its simplicity, our proof of the second result directly gives a slightly stronger claim: our algorithm computes a $\varepsilon$-singular value approximation (a notion of approximation introduced in a recent work by Ahmadinejad, Peebles, Pyne, Sidford and Vadhan (FOCS 2023)) of the random walk matrix of the given ROBP in space $\tilde{O}(\log(nw)\cdot\log\log(1/\varepsilon))$. It is not clear how to get this stronger result from the previous proofs.

Fixed some typos and incorrect references.

TR23-130 Authors: Eshan Chattopadhyay, Jyun-Jie Liao

Publication: 10th September 2023 11:03

Downloads: 405

Keywords:

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:

* There is a WPRG with error $\varepsilon$ that has seed length $$\tilde{O}(\log(n)(\sqrt{\log(1/\varepsilon)}+\log(w))+\log(1/\varepsilon)).$$

* 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.

In fact, because of its simplicity, our proof of the second result directly gives a slightly stronger claim: our algorithm computes a $\varepsilon$-singular value approximation (a notion of approximation introduced in a recent work by Ahmadinejad, Peebles, Pyne, Sidford and Vadhan (FOCS 2023)) of the random walk matrix of the given ROBP in space $\tilde{O}(\log(nw)\cdot\log\log(1/\varepsilon))$. It is not clear how to get this stronger result from the previous proofs.