On the adequacy of untuned warmup for adaptive optimization

Adaptive optimization algorithms such as Adam (Kingma & Ba, 2014) are widely used in deep learning. The stability of such algorithms is often improved with a warmup schedule for the learning rate. Motivated by the difficulty of choosing and tuning warmup schedules, Liu et al. (2019) propose automatic variance rectification of Adam's adaptive learning rate, claiming that this rectified approach ("RAdam") surpasses the vanilla Adam algorithm and reduces the need for expensive tuning of Adam with warmup. In this work, we point out various shortcomings of this analysis. We then provide an alternative explanation for the necessity of warmup based on the magnitude of the update term, which is of greater relevance to training stability. Finally, we provide some "rule-of-thumb" warmup schedules, and we demonstrate that simple untuned warmup of Adam performs more-or-less identically to RAdam in typical practical settings. We conclude by suggesting that practitioners stick to linear warmup with Adam, with a sensible default being linear warmup over $2 / (1 - \beta_2)$ training iterations.