Norm-matched zeroth-order adaptation preserves the isotropic retention floor while contracting only the anisotropic component, producing a quadratic forgetting gap that favors ZO precisely when the first-order direction has above-average retention curvature.
Hence tr(HM)≤λ max(M) tr(H) =d ¯λ λmax(M).(90) This upper bound is attained by choosing H=d ¯λuu ⊤, where u is a unit top eigenvector of M
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Why Zeroth-Order Adaptation May Forget Less: A Randomized Shaping Theory
Norm-matched zeroth-order adaptation preserves the isotropic retention floor while contracting only the anisotropic component, producing a quadratic forgetting gap that favors ZO precisely when the first-order direction has above-average retention curvature.