Retraining Seeks Stable Signals

2026-07-17

AI Summary

This work addresses the feedback loops that arise after model deployment due to performativity—wherein the model’s predictions influence the data distribution—particularly under strong interventions where the convergence behavior of retraining remains poorly understood. The paper introduces the “stable signal principle,” positing that the prediction target contains an intrinsic component independent of the model (e.g., inherent item quality), and leverages this insight to analyze the dynamics of regularized repeated risk minimization. Theoretically, it establishes that as long as a non-zero stable signal exists, retraining converges geometrically to its direction, even when model-induced effects dominate. This reveals a novel role for regularization in mitigating performative feedback and extends the framework to nonlinear, heterogeneous, and time-varying settings—including language models—thereby explaining the observed stability of training on generated data.

Abstract

Predictive models deployed at scale influence future data, a phenomenon called performativity. And there is always one way to cope: Train the model on new data, deploy it again, and repeat. This process, called retraining or repeated risk minimization, creates a feedback loop between model and data that real-world learning systems can't avoid. Results on performative prediction shed light on this dynamic: If the model's influence on the data is small, retraining reaches a fixed point. What remains open is why fixed points should naturally exist, and what governs retraining when the model's influence is strong. In this work we develop a new perspective on retraining -- the stable signal principle -- that addresses these questions. We start from the assumption that the prediction target has at least some small model-independent component, a stable signal, such as the intrinsic quality of an item. We prove that when a nonzero stable signal exists, repeated risk minimization, suitably regularized, converges geometrically to the direction of this stable signal. This is true even if the model's influence on the target is arbitrarily large relative to the stable signal. Regularization emerges naturally as a force to control performativity, rather than to promote generalization, revealing a new facet of an old concept. We extend the analysis to a broad family of affine retraining operators under arbitrary model-induced feature changes, heterogeneous time-varying effects, and nonlinear responses. The stable signal perspective also applies to data feedback loops in language modeling, providing new explanations for the stability of language model training from model-generated data.

Problem

• performativity
• retraining
• stable signal
• feedback loop
• fixed point

Innovation

• stable signal
• performativity
• retraining
• regularization
• feedback loop