# An Information-Theoretic Criterion for Efficient Data Synthesis

arXiv: https://arxiv.org/abs/2605.16379 · HTML: https://mlsanity.com/paper/2605.16379/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in LLM synthetic data generation, one of 13 directions in Synthetic data and self-improvement for LLMs.
- Builds on 23 of its 34 references that are in the map.
- First posted on arXiv 11 May 2026.

## Placed in

- Synthetic data and self-improvement for LLMs / [LLM synthetic data generation](https://mlsanity.com/synth/llm-synthetic-data-generation/)

## Builds on (most cited first, 23 of 23)

- [Distilling the Knowledge in a Neural Network](https://mlsanity.com/paper/1503.02531/) (2015)
- [Scaling Laws for Neural Language Models](https://mlsanity.com/paper/2001.08361/) (2020)
- [DeepSeek-R1: Incentivizing Reasoning Capability in LLMs via Reinforcement Learning](https://mlsanity.com/paper/2501.12948/) (2025)
- [Constitutional AI: Harmlessness from AI Feedback](https://mlsanity.com/paper/2212.08073/) (2022)
- [Competition-Level Code Generation with AlphaCode](https://mlsanity.com/paper/2203.07814/) (2022)
- [On-Policy Distillation of Language Models: Learning from Self-Generated Mistakes](https://mlsanity.com/paper/2306.13649/) (2023)
- [AlphaEvolve: A coding agent for scientific and algorithmic discovery](https://mlsanity.com/paper/2506.13131/) (2025)
- [Self-Rewarding Language Models](https://mlsanity.com/paper/2401.10020/) (2024)
- [Policy Distillation](https://mlsanity.com/paper/1511.06295/) (2015)
- [DeepSeek-Prover-V2: Advancing Formal Mathematical Reasoning via Reinforcement Learning for Subgoal Decomposition](https://mlsanity.com/paper/2504.21801/) (2025)
- [Self-Consuming Generative Models Go MAD](https://mlsanity.com/paper/2307.01850/) (2023)
- [OpenCodeInterpreter: Integrating Code Generation with Execution and Refinement](https://mlsanity.com/paper/2402.14658/) (2024)
- [KernelBench: Can LLMs Write Efficient GPU Kernels?](https://mlsanity.com/paper/2502.10517/) (2025)
- [Is Model Collapse Inevitable? Breaking the Curse of Recursion by Accumulating Real and Synthetic Data](https://mlsanity.com/paper/2404.01413/) (2024)
- [A Tale of Tails: Model Collapse as a Change of Scaling Laws](https://mlsanity.com/paper/2402.07043/) (2024)
- [Seed-Prover: Deep and Broad Reasoning for Automated Theorem Proving](https://mlsanity.com/paper/2507.23726/) (2025)
- [Gold-medalist Performance in Solving Olympiad Geometry with AlphaGeometry2](https://mlsanity.com/paper/2502.03544/) (2025)
- [Seed-Prover 1.5: Mastering Undergraduate-Level Theorem Proving via Learning from Experience](https://mlsanity.com/paper/2512.17260/) (2025)
- [AutoSAT: Automatically Optimize SAT Solvers via Large Language Models](https://mlsanity.com/paper/2402.10705/) (2024)
- [Discovering heuristics in a complex SAT solver with large language models](https://mlsanity.com/paper/2507.22876/) (2025)
- [Reinforcement Learning for Chain of Thought Compression with One-Domain-to-All Generalization](https://mlsanity.com/paper/2601.06052/) (2025)
- [JudgeRLVR: Judge First, Generate Second for Efficient Reasoning](https://mlsanity.com/paper/2601.08468/) (2026)
- [Discovering Expert-Level Nash Equilibrium Algorithms with Large Language Models](https://mlsanity.com/paper/2508.11874/) (2025)

## Abstract (arXiv)

Synthetic data becomes crucial for large language model training, but its effectiveness is highly inconsistent. We provide an information-theoretic account of this inconsistency: synthetic data improves a model only when the generation-training loop is information-open, i.e., shaped by external signals (verifiers, environments, or rubrics) that inject task-relevant information beyond the model's current distribution. When the loop is information-closed (relying on the model's own outputs without such signals), the data processing inequality ensures that task-relevant information can only decrease, making collapse a predicted outcome. Among information-open pipelines, both efficiency and generalization hinge on the meta-level of supervision: a coarser signal such as binary correctness treats all acceptable outputs as equivalent, so the behavior it teaches is not tied to any particular domain or surface form and generalizes naturally across tasks and domains. These observations lead to a guiding thesis: learning preferentially converges to the most information-efficient signal component available, which accelerates learning when that component is the intended one, but causes reward hacking when a spurious pattern happens to be simpler.