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The Polyak-Lojasiewicz Criterion (criterion)

Mon, 21 Sep 2026 11:42:00 +0900

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A criterion is a condition that decides whether an argument or a method applies, and this one is the usual substitute for convexity in the analysis of gradient descent: it bounds the loss by the size of its gradient, so a small gradient implies a small excess loss.

Sylvester's Criterion for Positive Definiteness (criterion)

Mon, 21 Sep 2026 09:42:00 +0900

Positive definiteness is the condition that decides whether a quadratic form curves upwards, and Sylvester’s criterion turns it into a finite computation: look at the determinants of the leading blocks of the matrix, from the top left outwards.