  This paper proposes a new easy-to-implement parameter-free gradient-based
optimizer: DoWG (Distance over Weighted Gradients). We prove that DoWG is
efficient -- matching the convergence rate of optimally tuned gradient descent
in convex optimization up to a logarithmic factor without tuning any
parameters, and universal -- automatically adapting to both smooth and
nonsmooth problems. While popular algorithms following the AdaGrad framework
compute a running average of the squared gradients to use for normalization,
DoWG maintains a new distance-based weighted version of the running average,
which is crucial to achieve the desired properties. To complement our theory,
we also show empirically that DoWG trains at the edge of stability, and
validate its effectiveness on practical machine learning tasks.
