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Learning Randomized Reductions

Ferhat Erata, Orr Paradise, Thanos Typaldos, Timos Antonopoulos, ThanhVu Nguyen, Shafi Goldwasser, Ruzica Piskac

TL;DR

This work presents Bitween, a learning-based framework for automatically discovering randomized self-reductions (RSRs) of a function $f$ from correlated samples, by formalizing RSRs with query and recovery classes $(Q,P)$ and a PAC-like learning notion under multiple sample-access models. The Vanilla Bitween backend uses regression-based learning to identify RSRs within a fixed query set, outperforming symbolic regression and MILP baselines, while Agentic Bitween leverages large language models to dynamically generate novel query functions and verify properties, achieving richer RSR discovery. The authors introduce RSR-Bench, a benchmark of 80 mathematical and ML-related functions, to evaluate discovery methods under rigorous verification, and demonstrate significant improvements in verifiable RSRs and function coverage. Collectively, the paper provides a rigorous theoretical framework and a scalable automation pipeline for discovering randomized reductions with potential impact on complexity theory and cryptography.

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.

Learning Randomized Reductions

TL;DR

This work presents Bitween, a learning-based framework for automatically discovering randomized self-reductions (RSRs) of a function from correlated samples, by formalizing RSRs with query and recovery classes and a PAC-like learning notion under multiple sample-access models. The Vanilla Bitween backend uses regression-based learning to identify RSRs within a fixed query set, outperforming symbolic regression and MILP baselines, while Agentic Bitween leverages large language models to dynamically generate novel query functions and verify properties, achieving richer RSR discovery. The authors introduce RSR-Bench, a benchmark of 80 mathematical and ML-related functions, to evaluate discovery methods under rigorous verification, and demonstrate significant improvements in verifiable RSRs and function coverage. Collectively, the paper provides a rigorous theoretical framework and a scalable automation pipeline for discovering randomized reductions with potential impact on complexity theory and cryptography.

Abstract

A self-corrector for a function takes a black-box oracle computing 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 at a given point can be recovered by computing 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 (, , , , ) 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.

Paper Structure

This paper contains 20 sections, 16 equations, 5 figures, 7 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Motivating Example
  3. Related Work
  4. Theoretical Foundations
  5. Bitween: Learning Randomized Self-Reductions
  6. Empirical Evaluation
  7. Conclusion
  8. Theory: RSR Learning vs PAC Learning
  9. Proofs
  10. General definitions
  11. Vanilla Bitween
  12. Experimental Hyperparameters
  13. Common Settings Across All Backends
  14. Backend-Specific Configurations
  15. Bitween Agent Instructions
  16. ...and 5 more sections

Figures (5)

  • Figure 1: An approximate implementation of the sigmoid activation function used as oracle and a graph representing randomly selected points by Bitween which are then used to discover an RSR.
  • Figure 2: (Left) Performance heatmap of average verified RSRs across mathematical function categories. The dotted red line represents the boundary from symbolic to neural methods. (Right) Multi-dimensional performance comparison for different methods. Verified Rate and RSR Rate is the percentage of the total properties proposed by the method that are verified and verified (manually found) RSRs, respectively. Function RSR Coverage is the percentage of individual functions (benchmarks) for which the method returned at least one verified RSR.
  • Figure 3: A-Bitween's intensive tool usage across all the benchmarks. Particularly useful proved the $sequential\_thinking\_tool$ for guiding the exploration and helping the LLM agent.
  • Figure 4: (Left) Runtime performance across methods and benchmarks highlighting the speed of V-Bitween-LR, V-Bitween-MILP and the overhead of A-Bitween variants. (Right) Token usage patterns correlate with the LLM's ability to efficiently utilize them.
  • Figure 5: Overview of V-Bitween.

Theorems & Definitions (9)

  • Definition 1: Randomized self-reduction
  • Definition 3: Sample access
  • Remark 4
  • Definition 5: Learning RSR
  • Claim 6
  • Claim 7
  • proof : Proof of \ref{['claim:pac-implies-rsr']}
  • proof : Proof sketch of \ref{['claim:rsr-not-implies-pac']}
  • Definition 8: Randomized reduction