Learning Randomized Reductions

TL;DR

This paper introduces Bitween, an automated learning framework for discovering randomized self-reductions of functions, outperforming existing methods and enabling the discovery of new properties through neuro-symbolic approaches with large language models.

Contribution

The paper presents a novel automated learning framework for RSRs using linear regression and a neuro-symbolic approach with language models, advancing the discovery process in complexity theory and cryptography.

Findings

Bitween outperforms traditional symbolic methods on RSR-Bench.

Agentic Bitween discovers new RSR properties using large language models.

The framework automates the derivation of RSRs, reducing manual effort.

Abstract

A self-corrector for a function $f$ takes a black-box oracle computing $f$ that is correct on most inputs and turns it into one that is correct on every input with high probability. Self-correctors exist for any function that is randomly self-reducible (RSR), where the value $f$ at a given point $x$ can be recovered by computing $f$ on random correlated points. While RSRs enable powerful self-correction capabilities and have applications in complexity theory and cryptography, their discovery has traditionally required manual derivation by experts. We present Bitween, a method and tool for automated learning of randomized self-reductions for mathematical functions. We make two key contributions: First, we demonstrate that our learning framework based on linear regression outperforms sophisticated methods including genetic algorithms, symbolic regression, and mixed-integer linear programming for discovering RSRs from correlated samples. Second, we introduce Agentic Bitween, a neuro-symbolic approach where large language models dynamically discover novel query functions for RSR property discovery, leveraging vanilla Bitween as a tool for inference and verification, moving beyond the fixed query functions ($x+r$, $x-r$, $x \cdot r$, $x$, $r$) previously used in the literature. On RSR-Bench, our benchmark suite of 80 scientific and machine learning functions, vanilla Bitween surpasses existing symbolic methods, while Agentic Bitween discovers new RSR properties using frontier models to uncover query functions.