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Efficient Online Linear Optimization with Approximation Algorithms

机译:高效在线线性优化与近似算法

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摘要

We revisit the problem of extit{online linear optimization} in case the setof feasible actions is accessible through an approximated linear optimizationoracle with a factor $lpha$ multiplicative approximation guarantee. Thissetting is in particular interesting since it captures natural onlineextensions of well-studied extit{offline} linear optimization problems whichare NP-hard, yet admit efficient approximation algorithms. The goal here is tominimize the $lpha$extit{-regret} which is the natural extension of thestandard extit{regret} in extit{online learning} to this setting. We present new algorithms with significantly improved oracle complexity forboth the full information and bandit variants of the problem. Mainly, for bothvariants, we present $lpha$-regret bounds of $O(T^{-1/3})$, were $T$ is thenumber of prediction rounds, using only $O(log{T})$ calls to the approximationoracle per iteration, on average. These are the first results to obtain bothaverage oracle complexity of $O(log{T})$ (or even poly-logarithmic in $T$) and$lpha$-regret bound $O(T^{-c})$ for a constant $c>0$, for both variants.
机译:我们重新审视 texit {在线线性优化}的问题,以防通过近似线性优化可以通过具有因子$ alpha $乘法近似保证的近似线性优化来访问的Setof可操作操作。这种启动尤其有趣,因为它捕获了学习良好的 yryit {离线}线性优化问题的自然onlineEnions {offline}线性优化问题,尚未承认有效的近似算法。这里的目标是Tominimize $ alpha $ textit {-regret},它是TheStandard Texit {遗憾}的自然扩展 Textit {在线学习}到此设置。我们展示了新的算法,具有显着改善的Oracle复杂性,禁止问题的完整信息和强盗变体。主要是,对于两者来说,我们呈现$ alpha $ -regret界限$ o(t ^ { - 1/3})$,它是$ t $的预测轮,只使用$ o( log {t})每次迭代,$调用近似值。这些是获得$ O( log {t})$(甚至是$ t $)的o(甚至是多对数)和$ alpha $ -regret $ o(t ^ { - c})的第一个结果$常用$ C> 0 $,适用于两个变体。

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    Dan Garber;

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  • 年度 2021
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