  Adam is an adaptive gradient method that has experienced widespread adoption
due to its fast and reliable training performance. Recent approaches have not
offered significant improvement over Adam, often because they do not innovate
upon one of its core features: normalization by the root mean square (RMS) of
recent gradients. However, as noted by Kingma and Ba (2015), any number of
$L^p$ normalizations are possible, with the RMS corresponding to the specific
case of $p=2$. In our work, we theoretically and empirically characterize the
influence of different $L^p$ norms on adaptive gradient methods for the first
time. We show mathematically how the choice of $p$ influences the size of the
steps taken, while leaving other desirable properties unaffected. We evaluate
Adam with various $L^p$ norms on a suite of deep learning benchmarks, and find
that $p > 2$ consistently leads to improved learning speed and final
performance. The choices of $p=3$ or $p=6$ also match or outperform
state-of-the-art methods in all of our experiments.
