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

• URL: http://arxiv.org/abs/2512.14879v1
• Date: Tue, 16 Dec 2025 19:50:03 GMT
• Title: Entropy-Reservoir Bregman Projection: An Information-Geometric Unification of Model Collapse
• Authors: Jingwei Chen,
• Abstract summary: We present EntropyReser Bregman Projection- ERBP, an information-geometric framework that unifies these phenomena.<n>Our theory yields a necessary condition for collapse, (ii) a sufficient condition that guarantees a non-language entropy floor, and (iii) closed-form rates that depend on sample size.
• Score: 3.533187668612022
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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.

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