Hardt's 'Stable Signal' principle proves retraining converges despite feedback loops
Even when AI models strongly distort their own training data, retraining finds the signal.
Moritz Hardt (MPI for Intelligent Systems / Berkeley) has released a new theoretical paper, *Retraining Seeks Stable Signals*, that tackles a fundamental challenge in machine learning: performativity. When deployed models influence the data they are later trained on—a feedback loop common in recommendation systems, automated pricing, or content ranking—the usual assumption of i.i.d. data breaks down. The standard fix is to simply keep retraining on new data, but until now it was unclear why this process should ever stabilise, especially when the model's influence is large.
Hardt's key insight is the 'stable signal principle': as long as the prediction target contains a small, model-independent component (e.g., the true quality of a product, independent of what the model predicts), then regularised repeated risk minimisation will converge geometrically toward that signal—even if the model's effect on the data is arbitrarily large. Regularisation, typically used to prevent overfitting, here acts as a force to control performativity rather than generalisation. The paper extends the analysis to affine retraining operators, time-varying effects, and nonlinear responses, and provides a new explanation for why language models remain stable when trained on model-generated data (addressing the 'model collapse' concern).
- Proves that regularised retraining converges geometrically to a stable signal even when model influence on data is arbitrarily large.
- Regularisation is reframed as a tool to control performative feedback, not just to prevent overfitting.
- Explains stability of language model training from synthetic data via the stable signal principle.
Why It Matters
Provides a theoretical foundation for why retraining works in practice, critical for all AI systems that learn from their own outputs.