Definition. Memoryless right-equivariant update [ftip-00B7]
Definition. Memoryless right-equivariant update [ftip-00B7]
A memoryless factor-update map is a function \(\Phi :\mathbb R^{n\times k}\to \mathbb R^{n\times k}\) applied to a current gradient \(G\) without a persistent optimizer state. It is right-equivariant when
\[ \Phi (GQ)=\Phi (G)Q \qquad (G\in \mathbb R^{n\times k},\ Q\in \mathrm O(k)). \]This local definition is the input of Theorem 4.5 in The loss does not see the basis, but Adam does[singh2026lossbasis]. It does not classify stateful optimizers such as Adam.