Non-Euclidean High-Order Smooth Convex Optimization
This work addresses a foundational problem in optimization theory, providing near-optimal algorithms for non-Euclidean high-order convex optimization, which is incremental but resolves a key open question.
The paper tackles the problem of optimizing convex objectives with Hölder continuous q-th derivatives using a q-th order oracle, developing algorithms that work for general norms and show near-optimality in high dimensions, resolving an open question in parallel convex optimization for the first-order case.
We develop algorithms for the optimization of convex objectives that have Hölder continuous $q$-th derivatives by using a $q$-th order oracle, for any $q \geq 1$. Our algorithms work for general norms under mild conditions, including the $\ell_p$-settings for $1\leq p\leq \infty$. We can also optimize structured functions that allow for inexactly implementing a non-Euclidean ball optimization oracle. We do this by developing a non-Euclidean inexact accelerated proximal point method that makes use of an \emph{inexact uniformly convex regularizer}. We show a lower bound for general norms that demonstrates our algorithms are nearly optimal in high-dimensions in the black-box oracle model for $\ell_p$-settings and all $q \geq 1$, even in randomized and parallel settings. This new lower bound, when applied to the first-order smooth case, resolves an open question in parallel convex optimization.