numerical stability

Nearly every candidate can recite softmax(QKᵀ/√d_k)V. What the interviewer is after is the variance derivation and the precise training failure the scale heads off. This is the response that tells rote recall apart from real grasp.

A deceptively easy ML-coding ask. Writing exp/sum is trivial; the signal is the max-subtraction trick and the log-sum-exp form that stop it overflowing. Here is the stable implementation and the reason the naive version breaks.

The textbook variance formula suffers catastrophic precision loss on streaming data and cannot update incrementally. Welford's one-pass update is the norm for online feature normalization. Here it is.