AdaGrad-type algorithms provably need a complexity quadratic in the initial gap and smoothness constants under relaxed smoothness, so they cannot match the optimal rate of clipped SGD.
We now bound the remaining constants
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Complexity Lower Bounds of Adaptive Gradient Algorithms for Non-convex Stochastic Optimization under Relaxed Smoothness
AdaGrad-type algorithms provably need a complexity quadratic in the initial gap and smoothness constants under relaxed smoothness, so they cannot match the optimal rate of clipped SGD.