The Exact Risk of Instance Normalization

Instance normalization (IN), which removes each input window’s statistics before forecasting and restores them afterward, is standard in time-series forecasting, yet there is little theory saying when it helps or fails. We analyze linear forecasters, starting from a characterization of mean-only IN as least squares constrained to weights summing to one, and under ridge training as ridge centered at the uniform vector on that plane. This yields closed forms for IN’s cost and benefit under unseen level shift, with a boundary: IN wins when the squared shift exceeds the variance of the best linear estimate of window level. Three consequences follow. First, the choice need not be binary. Interpolating the two forecasters admits an exact risk decomposition around any backbone, linear or not: the optimum is interior precisely when price and benefit are both positive, its excess risk is the parallel combination of the two endpoints’, and in the linear case it is the shift-to-boundary ratio ${\tau }^2/({\tau }^2+{\delta }_{\star }^2)$, computable before any label is seen. Second, division by the window standard deviation cancels identically in linear prediction, so it cannot amplify test error; it acts through training, reweighting examples by $1/{\hat{\sigma }}_i^2$, and the scale of that reweighted loss, not weight concentration, separates two catastrophic failures from six benign cases on eight benchmarks, from training data alone. Third, finite-sample estimation noise creates shift fragility even when the population model is shift-insensitive. Three preregistered experiments on PatchTST and iTransformer, outcome texts frozen before any run, mark the mechanism’s reach and its limit: the deep statistic $\hat{G}=\hat{S}{\hat{\delta }}^2$ ranks normalization gains at all four horizons ($\rho =0.944$–$0.965$) but is shown not to predict their sign, since it estimates the benefit and carries no price term; both regimes realize on transformers; and the interior optimum is found in 8 of 8 deep cells, where, estimated without test labels, it cuts regret against the oracle by 17× relative to any binary rule.