  Universal methods for optimization are designed to achieve theoretically
optimal convergence rates without any prior knowledge of the problem's
regularity parameters or the accurarcy of the gradient oracle employed by the
optimizer. In this regard, existing state-of-the-art algorithms achieve an
$\mathcal{O}(1/T^2)$ value convergence rate in Lipschitz smooth problems with a
perfect gradient oracle, and an $\mathcal{O}(1/\sqrt{T})$ convergence rate when
the underlying problem is non-smooth and/or the gradient oracle is stochastic.
On the downside, these methods do not take into account the problem's
dimensionality, and this can have a catastrophic impact on the achieved
convergence rate, in both theory and practice. Our paper aims to bridge this
gap by providing a scalable universal gradient method - dubbed UnderGrad -
whose oracle complexity is almost dimension-free in problems with a favorable
geometry (like the simplex, linearly constrained semidefinite programs and
combinatorial bandits), while retaining the order-optimal dependence on $T$
described above. These "best-of-both-worlds" results are achieved via a
primal-dual update scheme inspired by the dual exploration method for
variational inequalities.
