非对偶Lipschitz凸优化的稳定移动：效率与近乎最优的预言机速率

中文摘要 AI 辅助

我们研究在半径为R的ℓ_p球上，关于ℓ_q范数的G-Lipschitz凸函数优化中，实现一阶预言机复杂度的有效算法，其中1≤p,q≤∞。对于p<q，在T次预言机查询后，我们获得误差为Õ_{p,q}(GR/T^{1/p-(1/q-1/2)_+})，有效实现了(MBG+26)中近乎最优的速率，从而解决了COLT 2015开放问题(Guz15b)的非光滑端点。特别地，对于半径为R的ℓ_1球上的欧几里得Lipschitz性（p=1,q=2），速率为Õ(GR/T)。我们的解决方案包括将凸Lipschitz优化归约为在演化束(LNN95; BBE+20)的子水平集中追逐嵌套凸集问题：在每次查询时，我们要么找到一个函数值较低的点，要么在当前束的子水平集中产生一个深割，我们追逐该深割。选择器的稳定性与深割强制移动之间的二分法近乎最优地限制了算法的迭代次数。对于R B_p^d的嵌套子集，我们引入了一种新颖的稳定中心概念，其在T步后沿ℓ_q范数的移动被Õ_{p,q}(RT^{1-1/p+(1/q-1/2)_+})所界定，我们证明这在高维中是近乎最优的。所提出选择器的蒙特卡洛平均以高概率实现近乎最优的速率，并且在我们实算术模型的优化算法中可以在多项式时间内实现。

英文摘要

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, 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.

发表机构

• IMDEA Software Institute（IMDEA软件研究所）

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