When and Why SignSGD Outperforms SGD: A Theoretical Study Based on $\ell_1$-norm Lower Bounds
Sign-based optimization algorithms, such as SignSGD and Muon, have garnered significant attention for their remarkable performance in training large foundation models. Despite this empirical success, we still lack a theoretical understanding of when and why these sign-based methods outperform vanilla SGD. The core obstacle is that under standard smoothness and finite variance conditions, SGD is known to be minimax optimal for finding stationary points measured by $\ell_2$-norms, thereby fundamentally precluding any complexity gains for sign-based methods in standard settings. To overcome this barrier, we analyze sign-based optimizers leveraging $\ell_1$-norm stationarity, $\ell_\infty$-sm...
| Authors | Hongyi Tao, Dingzhi Yu, Lijun Zhang |
|---|---|
| Primary category | cs.LG |
| Published | 7 May 2026 |
| Last updated at source | 7 May 2026 |
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