For power-law word frequencies, gradient descent needs about d^{1-ε} iterations to reach relative error ε on a linear bigram model, while sign descent needs about √(d/ε), explaining Adam's advantage on heavy-tailed text data.
Normalizing the loss and using the same approach as in Proposition B.1 gives Ld(t) − L∗ d Ld(0) − L∗ d ≤ ϵ2 2 , after t ≥ ˜O(d)
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Scaling Laws for Gradient Descent and Sign Descent for Linear Bigram Models under Zipf's Law
For power-law word frequencies, gradient descent needs about d^{1-ε} iterations to reach relative error ε on a linear bigram model, while sign descent needs about √(d/ε), explaining Adam's advantage on heavy-tailed text data.