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Proof of the Softmax Shift Lemma (proof)

Mon, 21 Sep 2026 12:22:00 +0900

Demo note. This note demonstrates what the SciDraft Hugo theme can do for academic and scientific publishing — for research institutions, researchers and students. Its content is demonstration material only: readers should ignore whether the demo content is correct.

A proof is the part of a note that cannot be skipped, and this one is short enough to show the whole pattern: write the definition, factor out the constant, use that it cancels between numerator and denominator.

Proof: Attention Outputs Lie in the Convex Hull (proof)

Mon, 21 Sep 2026 09:52:00 +0900

The statement that attention outputs are convex combinations follows from two facts about the softmax and one line of the triangle inequality, and the proof shows exactly where each is used.