← 研踪 PaperTrailarXiv:2609.20701 · v2
Stable Movement for Nondual Lipschitz Convex Optimization: Efficiency and Nearly Optimal Oracle Rates
首次提交 2026-09-18 01:02
版本更新 2026-11-03 18:06
雷达观测 2026-11-04 23:39
摘要
We study efficient algorithms for realizing the first-order oracle complexity of optimization of $G$-Lipschitz convex functions with respect to the $\ell_q$-norm over an $\ell_p$-ball of radius $R$, where $1\leq p,q\leq\infty$. For $p<q$, we obtain error $\widetilde{O}_{p,q}(GR/T^{1/p-(1/q-1/2)_{+}})$ after $T$ oracle queries, attaining the lower bounds in a COLT 2015 open problem (Guz15b) with efficient algorithms. In particular, the rate is $\widetilde{O}(GR/T)$ for Euclidean Lipschitzness with an $\ell_1$-ball domain of radius $R$ ($p=1,q=2$). Our solution consists of reducing convex Lipschitz optimization to the chasing nested convex sets problem (BBE+20) in sublevel sets of an evolving maximum of affine lower bounds, that is, the bundle (LNN95): at each query we either find a point with low function value or we produce a deep cut in the current sublevel of the bundle that we chase. The dichotomy between stability of selectors and forced movement by deep cuts bounds the number of iterations of the algorithm nearly optimally. For nested convex subsets of the $\ell_p$ ball $R B_p^d$, we introduce novel stable selectors whose movement is nearly optimal in high dimensions. A Monte Carlo average of the proposed selector achieves near-optimal rates with high probability and can be implemented for our optimization algorithm in polynomial time in the real-arithmetic model.
作者备注：polished writing, added a few more general results (such as a better lower bound, and a more general potential-energy center)
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v2 · 2026-11-03 18:06 · 最新观测版本
Stable Movement for Nondual Lipschitz Convex Optimization: Efficiency and Nearly Optimal Oracle Rates
We study efficient algorithms for realizing the first-order oracle complexity of optimization of $G$-Lipschitz convex functions with respect to the $\ell_q$-norm over an $\ell_p$-ball of radius $R$, where $1\leq p,q\leq\infty$. For $p<q$, we obtain error $\widetilde{O}_{p,q}(GR/T^{1/p-(1/q-1/2)_{+}})$ after $T$ oracle queries, attaining the lower bounds in a COLT 2015 open problem (Guz15b) with efficient algorithms. In particular, the rate is $\widetilde{O}(GR/T)$ for Euclidean Lipschitzness with an $\ell_1$-ball domain of radius $R$ ($p=1,q=2$). Our solution consists of reducing convex Lipschitz optimization to the chasing nested convex sets problem (BBE+20) in sublevel sets of an evolving maximum of affine lower bounds, that is, the bundle (LNN95): at each query we either find a point with low function value or we produce a deep cut in the current sublevel of the bundle that we chase. The dichotomy between stability of selectors and forced movement by deep cuts bounds the number of iterations of the algorithm nearly optimally. For nested convex subsets of the $\ell_p$ ball $R B_p^d$, we introduce novel stable selectors whose movement is nearly optimal in high dimensions. A Monte Carlo average of the proposed selector achieves near-optimal rates with high probability and can be implemented for our optimization algorithm in polynomial time in the real-arithmetic model.
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