Adam's O(1/T^1/4) convergence is proven from a sign-like descent perspective, but the dimension-free claim depends on restrictive coordinate-wise assumptions.
SGD with AdaGrad stepsizes: Full adaptivity with high probability to unknown parameters, unbounded gradients and affine variance
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Simple Convergence Proof of Adam From a Sign-like Descent Perspective
Adam's O(1/T^1/4) convergence is proven from a sign-like descent perspective, but the dimension-free claim depends on restrictive coordinate-wise assumptions.