A Variational Framework for LLM Generator-Regulator Games

Quanyan Zhu

stat.OTJun 16, 2026

A Variational Framework for LLM Generator-Regulator Games

Authors: Quanyan Zhu

Organizations: Department of Electrical and Computer Engineering, Tandon School of Engineering, New York University, Brooklyn, NY, USA.

Abstract

This paper develops a variational framework for regulated language generation. Starting from autoregressive token sampling, we derive the induced distribution over complete messages and relate it to an entropy-regularized Gibbs law. Regulation is modeled as an optimal discriminator whose convex-dual value is an f-divergence, and the generator-regulator interaction is formulated as a saddle-point problem. The framework applies to moderation, censorship, AI deception detection, compliance auditing, phishing defense, and manipulation control, where regulation concerns a distribution over possible messages rather than a single output. The equilibrium clarifies the tradeoff among utility, entropy, regulatory alignment, and finite-length detectability. Two finite-vocabulary case studies, censorship filtering and phishing defense, illustrate how the theory can be evaluated through utility, entropy, divergence, receiver-side scores, and detection probability.

Explore similar work

Language Generation with Replay: A Learning-Theoretic View of Model Collapse
Mar 12, 2026Giorgio Racca, Michal Valko, Amartya SanyalCollapse
Improving LLMs via Validator-to-Generator Alignment
Jul 2, 2026Juan Diego Rodriguez, Jocelyn Zhang, Katrin Erk +1
Escaping Mode Collapse in LLM Generation via Geometric Regulation
May 1, 2026Xin Du, Kumiko Tanaka-IshiiLarge Language Model Generation Autoregressive Generation

Mar 12, 2026 · cs.LGJ/K move · Enter open · S save

Language Generation with Replay: A Learning-Theoretic View of Model Collapse

Giorgio Racca, Michal Valko, Amartya Sanyal

University of Copenhagen · Isara Labs

As scaling laws push the training of frontier large language models (LLMs) toward ever-growing data requirements, training pipelines are approaching a regime where much of the publicly available online text may be consumed. At the same time, widespread LLM usage increases the volume of machine-generated content on the web; together, these trends raise the likelihood of generated text re-entering future training corpora, increasing the associated risk of performance degradation often called model collapse. In practice, model developers address this concern through data cleaning, watermarking, synthetic-data policies, or, in some cases, blissful ignorance. However, the problem of model collapse in generative models has not been examined from a learning-theoretic perspective: we study it through the theoretical lens of the language generation in the limit framework, introducing a replay adversary that augments the example stream with the generator's own past outputs. Our main contribution is a fine-grained learning-theoretic characterization of when replay fundamentally limits generation: while replay is benign for the strongest notion of uniform generation, it provably creates separations for the weaker notions of non-uniform generation and generation in the limit. Interestingly, our positive results mirror heuristics widely used in practice, such as data cleaning, watermarking, and output filtering, while our separations show when these ideas can fail.

Collapse

Open paper Ask about it PDF

Abstract

Explore similar work

Language Generation with Replay: A Learning-Theoretic View of Model Collapse

Improving LLMs via Validator-to-Generator Alignment

Escaping Mode Collapse in LLM Generation via Geometric Regulation