arXivDaily arXiv每日学术速递 周一至周五更新
arXiv 2007.02738cs.LGcs.DSstat.ML
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覆盖函数的结构化样本优化

Optimization from Structured Samples for Coverage Functions

Microsoft Research Asia(微软亚洲研究院);Institute of Computing Technology, Chinese Academy of Sciences(中国科学院计算技术研究所);University of Chinese Academy of Sciences(中国科学院大学)

Wei Chen, Xiaoming Sun, Jialin Zhang, Zhijie Zhang

摘要:

我们重新审视了样本优化(OPS)模型,它研究了直接从样本数据优化目标函数的问题。先前的结果表明,我们不能使用$\{S_i, f(S_i)\}_{i=1}^t$形式的多项式多个独立样本来获得最大覆盖问题的恒定近似比(Reynanski等人,2017),即使覆盖函数是$(1 - ε)$ -PMAC可使用这些样本学习的(Badanidiyuru等人,2012),这意味着大多数函数值可以以高概率近似地学习得非常好。在这项工作中,为了规避OPS的不可能结果,我们提出了一个更强大的模型,称为结构化样本优化(OPSS)的覆盖函数,其中的数据样本编码的功能的结构信息。我们表明,在样本分布的三个一般假设下,我们可以设计有效的OPSS算法,实现恒定的近似最大覆盖问题。在这些假设下,我们进一步证明了一个常数下界,当不考虑计算效率时,这是紧的。此外,我们还表明,如果我们删除任何一个的三个假设,OPSS的最大覆盖问题没有常数近似。

英文摘要:

We revisit the optimization from samples (OPS) model, which studies the problem of optimizing objective functions directly from the sample data. Previous results showed that we cannot obtain a constant approximation ratio for the maximum coverage problem using polynomially many independent samples of the form $\{S_i, f(S_i)\}_{i=1}^t$ (Balkanski et al., 2017), even if coverage functions are $(1 - ε)$-PMAC learnable using these samples (Badanidiyuru et al., 2012), which means most of the function values can be approximately learned very well with high probability. In this work, to circumvent the impossibility result of OPS, we propose a stronger model called optimization from structured samples (OPSS) for coverage functions, where the data samples encode the structural information of the functions. We show that under three general assumptions on the sample distributions, we can design efficient OPSS algorithms that achieve a constant approximation for the maximum coverage problem. We further prove a constant lower bound under these assumptions, which is tight when not considering computational efficiency. Moreover, we also show that if we remove any one of the three assumptions, OPSS for the maximum coverage problem has no constant approximation.

机构、总结、中文摘要均由AI分析而来,可能存在错误,请以论文正文为准。

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