Stable Movement for Nondual Lipschitz Convex Optimization

It achieves nearly optimal rates for error after T oracle queries, resolving a COLT 2015 open problem.The method involves chasing nested convex sets in sublevel sets of an evolving bundle, with a novel stable center concept that bounds movement in the lq-norm.

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, 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.

This report is categorized under AITOP's "Papers" desk, focusing on US frontier AI labs and Silicon Valley technology leaders. Key entities and technical topics under continuous tracking include convex optimization, Lipschitz, algorithm, mathematics.

Source attribution: curated from arXiv cs.LG (@David Martínez-Rubio, Cristóbal Guzmán).