Self-normalised Bennett inequalities for Hilbert-valued martingales

What the math gives to ML

The paper provides a constructive, time-uniform concentration mechanism for vector-valued martingale sums normalized by their predictable covariance rather than by a fixed scalar variance. Its transferable asset is the explicit exponential supermartingale combining a Bennett rate for a whitened norm with a log-determinant complexity penalty, giving a confidence boundary valid simultaneously over all training times. A practical neural-network adaptation is an uncertainty-aware optimizer controller that tracks minibatch gradient noise covariance and shrinks updates only when cumulative whitened noise exceeds the anytime boundary.