Sharp Tail Bounds Beyond Twice the Mean

What the math gives to ML

The paper gives a sharp worst-case upper-tail bound for sums of independent nonnegative variables with bounded means, identifying sparse Bernoulli spikes at the threshold as the extremal distribution. The transferable asset is a distributionally robust certificate: rare, full-sized spikes are harder to control than smooth fluctuations when only nonnegativity, independence, and means are known. A neural network can use the bound as a differentiable regularizer for independently sampled branches, activation loads, or compute allocations. The method is most appropriate when component means are analytically controlled or conservatively upper-bounded.