Last-Iterate Convergence Rate of Normalized Gradient Descent under Hölder Smoothness
Normalized gradient descent is a widely studied adaptive optimization method. Most existing analyses focus on the best iterate or a weighted average of the iterates, whereas practical implementations typically return the last iterate. In this paper, we study the last-iterate convergence of normalized gradient descent for convex, $(ν,M_ν)$-Hölder-smooth objectives. For a constant stepsize, we establish an upper bound of $\mathcal{O}\bigl((\log^2(T)/T)^{(1+ν)/2}\bigr)$, which contains a logarithmic overhead relative to the known $\mathcal{O}\bigl(T^{-(1+ν)/2}\bigr)$ guarantees for the best and weighted-average iterates. For $ν= 0$, this overhead is known to be unavoidable. We complement thi...
| Authors | Yuki Takezawa, Eduard Gorbunov |
|---|---|
| Primary category | math.OC |
| Published | 5 October 2026 |
| Last updated at source | 5 October 2026 |
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