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This research advances the theoretical foundations of first-order optimization by resolving an open problem regarding $G$-Lipschitz convex functions over $\ell_p$-balls. The authors introduce a novel stable center concept and reduce complex optimization tasks to chasing nested convex sets, achieving nearly optimal oracle rates in high dimensions. These algorithmic improvements are critical for enhancing the efficiency of machine learning solvers that rely on rigorous convergence guarantees.
Stable Movement for Nondual Lipschitz Convex Optimization: Efficiency and Nearly Optimal Oracle Rates
arXiv:2609.20701v1 Announce Type: cross Abstract: 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, efficiently realizing the nearly optimal rates of (MBG+26), thereby resolving the nonsmooth end of the COLT 2015 open problem (Guz15b). In particular, the rate is $\widetilde{O}(GR/T)$ for Euclidean Lipschitzness over an $\ell_1$-ball of radius $R$ ($p=1,q=2$). Our solution consists of reducing convex Lipschitz optimization to the chasing nested convex sets problem in sublevel sets of an evolving bundle (LNN95; BBE+20): 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 near optimally. For nested subsets of $R B_{p}^{d}$, we introduce a novel notion of stable center whose movement is bounded by $\widetilde{O}_{p,q}(RT^{1-1/p+(1/q-1/2)_{+}})$ in the $\ell_{q}$-norm after $T$ steps, which we show 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 in polynomial time for our optimization algorithm in the real-arithmetic model.
Abstract: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, efficiently realizing the nearly optimal rates of (MBG+26), thereby resolving the nonsmooth end of the COLT 2015 open problem (Guz15b). In particular, the rate is $\widetilde{O}(GR/T)$ for Euclidean Lipschitzness over an $\ell_1$-ball of radius $R$ ($p=1,q=2$). Our solution consists of reducing convex Lipschitz optimization to the chasing nested convex sets problem in sublevel sets of an evolving bundle (LNN95; BBE+20): 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 near optimally. For nested subsets of $R B_{p}^{d}$, we introduce a novel notion of stable center whose movement is bounded by $\widetilde{O}_{p,q}(RT^{1-1/p+(1/q-1/2)_{+}})$ in the $\ell_{q}$-norm after $T$ steps, which we show 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 in polynomial time for our optimization algorithm in the real-arithmetic model.
Optimization and Control (math.OC); Machine Learning (cs.LG)
arXiv:2609.20701 [math.OC]
(or arXiv:2609.20701v1 [math.OC] for this version)
https://doi.org/10.48550/arXiv.2609.20701 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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