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.
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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.