Retraining as Approximate Bayesian Inference

Deployed prediction systems are often retrained on fixed calendars, even when model staleness and retraining burden vary over time. This short communication formulates retraining for Bayesian prediction systems as a cost-sensitive predictive-regret decision. The central monitoring state is posterior learning debt, defined as the Kullback--Leibler divergence from a reference shadow posterior to the deployed frozen posterior. In the decision layer, a retraining cost is compared with the expected one-period predictive regret of waiting. A continuous-severity version retrains when calibrated expected regret exceeds the retraining cost, while the familiar two-state excess-loss rule is a special case. The empirical study is an exact-state proof-of-concept in a synthetic conjugate simulation with warm-started deployed and shadow normal-inverse-gamma posteriors, separate update, monitoring, and evaluation batches, lagged deployment actions, expanded baseline grids, and score-unit sensitivity. Under the primary 75th-percentile score-unit scaling, an age-adjusted debt-threshold policy improves on tuned calendar retraining in all 72 non-stable scenario cells and on tuned CUSUM in 58 of 72 cells, with mean relative objectives 0.677 and 0.975, respectively. Debt-utility and hybrid-utility policies also improve strongly over tuned calendar retraining, but they do not dominate tuned CUSUM. Median and mean score-unit sensitivities show the same main calendar result, while the CUSUM comparison remains policy-dependent. The contribution is a transparent decision layer for deployed Bayesian prediction systems, not a universal replacement for drift detection.

Production machine learning systems degrade under concept drift, yet practitioners have little principled guidance on when to retrain. Retraining is costly, retraining budgets are finite, and a retrained model does not take effect instantly: training and deployment latency leave a stale model serving predictions while the data continues to move. We present a controlled empirical study of three practical model-refresh policies (periodic retraining, error-threshold triggering, and statistical drift-triggered retraining with ADWIN) against a no-retrain baseline, evaluated under a unified system model that makes retraining budgets and training-plus-deployment latency explicit. Across 3,933 experiment runs spanning three drift regimes, three budget levels, up to five latency levels, three datasets, and two learning modes, we find that the single most consequential design decision is not the retraining policy but whether the deployed model learns incrementally. With per-sample incremental updates, and for the linear online learner with immediate labels studied here, no policy differs from the no-retrain baseline by a practically significant margin in any of 54 paired comparisons, even at extreme latency. Without incremental updates, policy choice separates outcomes by 15-55 percentage points of post-drift accuracy, and simple periodic retraining significantly outperforms both reactive policies under abrupt and gradual drift, while reactive policies retain an advantage only under recurring drift. We document systematic failure modes of reactive policies and a latency-budget queueing interaction that silently halves effective retraining budgets, and release the full simulator, dataset pipelines, and per-run artifacts for reproducibility.

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.