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Near-Optimal Acceleration for Smooth $\ell_p$ / $\ell_q$ Nondual Convex First-Order Oracle Optimization

arXiv · AI, language, vision and robotics · article · Sep 18, 2026 · UTC

We study the optimization of convex objectives with $(L,κ-1)$-Hölder-continuous gradients in $\ell_q$ over $R B_p^d$, $1<κ\le 2$. (MG26) provides selectors with a movement bound for the problem of chasing high-dimensional convex nested sets for every $p<q$ and generally reduces Lipschitz convex optimization to bounds on the movement of selectors. We couple that movement with Hölder descent yielding a polynomial-runtime first-order method whose feasible output, in the high-dimensional regime $T\le d$ and for $p<\min\{q,2\}$, has error $$ \widetilde O_{κ,p,q}\!\left( \frac{LR^κ}{T^{κ(1+1/p-(1/q-

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