The Exact Worst-Case Tail Probability under Bounded Kurtosis

What the math gives to ML

This paper gives an exact, distribution-free upper envelope for one-sided outliers when only mean, variance, and a fourth-moment bound are available, without assuming bounded skewness. The transferable asset is not merely a sharper inequality but the explicit phase transition: beyond a computable threshold, the worst-case tail has a closed form and can be inverted into an exact target quantile. Neural-network training can use this to replace ad hoc percentile or global-norm clipping thresholds with kurtosis-calibrated clipping for activations or gradients. The main caveat is that the kurtosis must be estimated online, so experiments should separately test estimation noise and the idealized known-kurtosis setting.