Online balancing of vectors with small coordinates

What the math gives to ML

The paper gives a constructive online discrepancy principle: signs can be selected sequentially so that every coordinate of every prefix sum remains bounded with exponentially small failure probability when each arriving vector has sufficiently small coordinates. The transferable asset is a bounded-state controller for additive neural memories whose updates have sign symmetry. A practical adaptation is a signed fast-weight or recurrent state in which each incoming update chooses a sign by comparing a compact smooth potential before and after the two possible updates; experiments can test whether this prevents state saturation and improves long-context stability. The guarantee applies only when updates are norm-bounded and coordinate-wise diffuse, so the implementation must explicitly normalize updates.