# Entropy-Reservoir Bregman Projection: An Information-Geometric Unification of Model Collapse

arXiv: https://arxiv.org/abs/2512.14879 · HTML: https://mlsanity.com/paper/2512.14879/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Placed in Model collapse from synthetic data, one of 13 directions in Synthetic data and self-improvement for LLMs.
- Builds on 16 of its 27 references that are in the map.
- First posted on arXiv 16 Dec 2025.

## Placed in

- Synthetic data and self-improvement for LLMs / [Model collapse from synthetic data](https://mlsanity.com/synth/model-collapse-from-synthetic-data/)
- Deep reinforcement learning methods / [RL theory and sample complexity](https://mlsanity.com/deeprl/rl-theory-and-sample-complexity/)

## Builds on (most cited first, 16 of 16)

- [Training language models to follow instructions with human feedback](https://mlsanity.com/paper/2203.02155/) (2022)
- [Distilling the Knowledge in a Neural Network](https://mlsanity.com/paper/1503.02531/) (2015)
- [Retrieval-Augmented Generation for Knowledge-Intensive NLP Tasks](https://mlsanity.com/paper/2005.11401/) (2020)
- [Rethinking the Inception Architecture for Computer Vision](https://mlsanity.com/paper/1512.00567/) (2015)
- [Soft Actor-Critic: Off-Policy Maximum Entropy Deep Reinforcement Learning with a Stochastic Actor](https://mlsanity.com/paper/1801.01290/) (2018)
- [Toolformer: Language Models Can Teach Themselves to Use Tools](https://mlsanity.com/paper/2302.04761/) (2023)
- [Deep reinforcement learning from human preferences](https://mlsanity.com/paper/1706.03741/) (2017)
- [Generative Agents: Interactive Simulacra of Human Behavior](https://mlsanity.com/paper/2304.03442/) (2023)
- [Improved Techniques for Training GANs](https://mlsanity.com/paper/1606.03498/) (2016)
- [The Curious Case of Neural Text Degeneration](https://mlsanity.com/paper/1904.09751/) (2019)
- [Asynchronous Methods for Deep Reinforcement Learning](https://mlsanity.com/paper/1602.01783/) (2016)
- [Mastering Atari, Go, Chess and Shogi by Planning with a Learned Model](https://mlsanity.com/paper/1911.08265/) (2019)
- [When Does Label Smoothing Help?](https://mlsanity.com/paper/1906.02629/) (2019)
- [Gradient Descent Provably Optimizes Over-parameterized Neural Networks](https://mlsanity.com/paper/1810.02054/) (2018)
- [The Curse of Recursion: Training on Generated Data Makes Models Forget](https://mlsanity.com/paper/2305.17493/) (2023)
- [Unrolled Generative Adversarial Networks](https://mlsanity.com/paper/1611.02163/) (2016)

## Built on by (most cited first, 1 of 1)

- [Recursive Self-Improvement in AI: From Bounded Self-Refinement to Autonomous Research Loops](https://mlsanity.com/paper/2607.07663/) (2026)

## Abstract (arXiv)

Self-referential learning -- training a model on data it generated itself -- promises boundless scalability but chronically suffers from model collapse: language models degenerate into repetitive text, GANs drop modes, and reinforcement-learning policies over-exploit. Although practitioners employ ad~hoc fixes such as real-data mixing, entropy bonuses, knowledge distillation, or retrieval-augmented generation, a single principle that explains both the failure mode and the success of these fixes has remained elusive. We present Entropy-Reservoir Bregman Projection (ERBP), an information-geometric framework that unifies these phenomena. We model the closed loop as a stochastic Bregman projection sequence in distribution space. Without external coupling, finite-sample noise forces the system to project onto an ever-shrinking empirical support, causing exponential entropy decay and eventual collapse. Introducing an Entropy Reservoir -- a high-entropy distribution mixed into each projection -- injects a controllable entropy flux that provably stabilises the dynamics. Our theory yields (i) a necessary condition for collapse, (ii) a sufficient condition that guarantees a non-trivial entropy floor, and (iii) closed-form rates that depend only on sample size and the strong-convexity/Lipschitz constants of the Bregman generator. Experiments on large-language-model self-training, Soft Actor-Critic in reinforcement learning, and GAN optimisation validate our predictions and show that disparate stabilisation heuristics correspond to specific reservoir choices and coupling coefficients. ERBP thus transforms a collection of folk remedies into a single, quantitative design rule: monitor and budget your entropy flux.