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.
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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.