Whether an optimizer keeps gradient descent's low-rank bias in factored models is determined by its equivariance under orthogonal gauge rotations, and Adam and other coordinate-wise rules fail this test.
E EXTENDED RELATED WORK Transformation invariance in optimizer design.The engineering literature has grappled with similar design choices at a lower level; for instance, Ling et al
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The Loss Does Not See the Basis, but Adam Does
Whether an optimizer keeps gradient descent's low-rank bias in factored models is determined by its equivariance under orthogonal gauge rotations, and Adam and other coordinate-wise rules fail this test.