Title: Entropy-Reservoir Bregman Projection: An Information-Geometric Unification of Model Collapse
Primary area: generative models
Keywords: self-referential learning, model collapse, entropy reservoir, Bregman projection, information geometry, generative AI

Abstract: Self-referential learning---training a model on data it generated itself---promises
		boundless scalability but chronically suffers from \emph{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 \textbf{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 \emph{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.

TL;DR: Self-training is a stochastic Bregman-projection loop whose entropy inevitably vanishes unless it is continuously mixed with a high-entropy reservoir—an insight that unifies and explains all existing anti-collapse tricks.