Stable but Wrong: When Learning Stabilizes Away from the Truth

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.

Reinforcement learning (RL) of large language models is notoriously sensitive to small differences between training and inference engines, often referred to as the training-inference mismatch (TIM). However, completely eliminating TIM is impractical, as it would come at a major cost to rollout efficiency. In this paper, we show that the instability of RL under TIM is primarily caused by drift: a persistent bias between training and inference engines that accumulates with every training step. We derive an additive "score centering" correction term that stabilizes RL under TIM by canceling drift. When training models from 0.6B to 30B parameters, score centering alone matches or outperforms methods based on importance sampling under quantization, with the gap growing as the mismatch becomes more severe. Because the correction is additive, score centering also composes with importance sampling -- their composition outperforms pure importance-sampling baselines in our staleness experiments.

Outcome-based reinforcement learning can produce models with similar task performance but very different ways of communicating about their mistakes. We study failure disclosure: whether a model admits that an attempted solution failed rather than staying silent or presenting it as successful. Across repeated outcome-only GRPO training runs, failure disclosure varies far more than task accuracy. The pattern extends to a second reasoning task and stabilized PPO, persists at 7B, and also appears in an instruction-conditioned 32B setting. We also find that small floating-point and sampling differences during training can redirect reporting behavior even when the task objective and earlier training history are held fixed. Additional tests show that failure disclosure is not a single decision: Checking the answer, entering a report, and completing the admission can separate, and the weak point depends on the task and response format. Further, experiments with neutral controls show more broadly that behaviors left weakly constrained by training are especially likely to vary across runs, of which failure disclosure is an example. We can reduce variability in failure disclosure by discouraging the model from drifting from its starting policy on failed, well-formed responses. This makes reporting substantially more consistent, though its effect on task performance depends on the setting. Stable task accuracy therefore does not guarantee stable safety-relevant behavior: Researchers should measure these behaviors directly across runs and design training methods that keep them reliable.