SIAM Journal on Computing, Volume 49, Issue 1, Page 67-97, January 2020.
Set functions with convenient properties (such as submodularity) appear in application areas of current interest, such as algorithmic game theory, and allow for improved optimization algorithms. It is natural to ask (e.g., in the context of data driven optimization) how robust such properties are, and whether small deviations from them can be tolerated. We consider two such questions in the important special case of linear set functions. One question that we address is whether any set function that approximately satisfies the modularity equation (linear functions satisfy the modularity equation exactly) is close to a linear function. The answer to this is positive (in a precise formal sense) as shown by Kalton and Roberts [Trans. Amer. Math. Soc., 278 (1983), pp. 803--816] (and further improved by Bondarenko, Prymak, and Radchenko [J. Math. Anal. Appl., 402 (2013), pp. 234--241]). We revisit their proof idea that is based on expander graphs and provide significantly stronger upper bounds by combining it with new techniques. Furthermore, we provide improved lower bounds for this problem. Another question that we address is that of how to learn a linear function $h$ that is close to an approximately linear function $f$, while querying the value of $f$ on only a small number of sets. We present a deterministic algorithm that makes only linearly many (in the number of items) nonadaptive queries, and thus improve upon a previous algorithm of Chierichetti, Das, Dasgupta, and Kumar [Proceedings of the 56th Symposium on Foundations of Computer Science, 2015, pp. 1143--1162] that is randomized and makes more than a quadratic number of queries. Our learning algorithm is based on the Hadamard transform.

中文翻译：

---

再谈近似模块化

SIAM计算杂志，第49卷，第1期，第67-97页，2020年1月。
具有便利属性（例如子模数）的集合函数出现在当前关注的应用领域中，例如算法博弈论，并允许改进的优化算法。自然会问（例如，在数据驱动的优化的情况下）此类属性的健壮性以及是否可以容忍与这些属性之间的小偏差。我们在线性集函数的重要特例中考虑了两个这样的问题。我们要解决的一个问题是，任何近似满足模块化方程（线性函数恰好满足模块化方程）的集合函数是否接近线性函数。正如卡尔顿和罗伯茨[Trans。阿米尔。数学。Soc。，278（1983），pp.803--816]（并由Bondarenko，Prymak和Radchenko进一步改进[J. Math。Anal。Appl。，402（2013），第234--241页]。我们重新审视他们基于扩展器图的证明思想，并通过将其与新技术结合提供明显更强的上限。此外，我们为此问题提供了改进的下界。我们要解决的另一个问题是如何学习线性函数$ h $，该函数与近似线性函数$ f $接近，而仅在少数集合上查询$ f $的值。我们提出了一种确定性算法，该算法仅线性地（在项目数中）进行许多非自适应查询，因此改进了先前的Chierichetti，Das，Dasgupta和Kumar算法[第56届计算机科学基础研讨会论文集，2015年， pp。1143--1162]，它是随机的，并且进行的查询次数超过二次数。