[论文解读] Polyhedral aspects of Submodularity, Convexity and Concavity

Rishabh Iyer, Jeff Bilmes|arXiv (Cornell University)|Jun 24, 2015

Complexity and Algorithms in Graphs参考文献 7被引用 13

一句话总结

本文通过引入上微分、紧模面上界和拟凹扩展，建立了子模函数与凸性及凹性之间的全面对偶关系。证明了子模最大化可借助类凹最优性条件进行分析，并将Fenchel对偶性和离散分离定理扩展至凹视角，揭示了子模函数作为凸与凹行为的离散混合体。

ABSTRACT

Seminal work by Edmonds and Lovasz shows the strong connection between submodularity and convexity. Submodular functions have tight modular lower bounds, and subdifferentials in a manner akin to convex functions. They also admit poly-time algorithms for minimization and satisfy the Fenchel duality theorem and the Discrete Seperation Theorem, both of which are fundamental characteristics of convex functions. Submodular functions also show signs similar to concavity. Submodular maximization, though NP hard, admits constant factor approximation guarantees. Concave functions composed with modular functions are submodular, and they also satisfy diminishing returns property. This manuscript provides a more complete picture on the relationship between submodularity with convexity and concavity, by extending many of the results connecting submodularity with convexity to the concave aspects of submodularity. We first show the existence of superdifferentials, and efficiently computable tight modular upper bounds of a submodular function. While we show that it is hard to characterize this polyhedron, we obtain inner and outer bounds on the superdifferential along with certain specific and useful supergradients. We then investigate forms of concave extensions of submodular functions and show interesting relationships to submodular maximization. We next show connections between optimality conditions over the superdifferentials and submodular maximization, and show how forms of approximate optimality conditions translate into approximation factors for maximization. We end this paper by studying versions of the discrete seperation theorem and the Fenchel duality theorem when seen from the concave point of view. In every case, we relate our results to the existing results from the convex point of view, thereby improving the analysis of the relationship between submodularity, convexity, and concavity.

研究动机与目标

• 正式建立子模函数与凸性及凹性之间的对偶关系，尤其在优化背景中。
• 开发类似于凸分析中次微分的上微分与上梯度的多面体框架。
• 建立子模函数的拟凹扩展，并将其与子模最大化问题的近似算法联系起来。
• 将经典结果（如Fenchel对偶性与离散分离定理）扩展至子模函数的凹侧。
• 识别出子模函数的子类（例如M♮-凹函数），其凹性性质可精确成立，从而支持精确或近似算法。

提出的方法

• 引入上微分的概念，作为R^V的多面体划分，为一般子模函数提供内界与外界界。
• 利用广义上微分结构，开发可高效计算的紧模面上界。
• 提出子模函数的拟凹扩展，尤其将其与多线性扩展及其在特定方向上的凹性行为相关联。
• 基于上微分推导子模最大化问题的最优性条件，并将其与近似保证联系起来。
• 从凹视角重新诠释Fenchel对偶性与离散分离定理，表明其在受限条件下依然有效。
• 利用Minkowski和定理与多面体几何，将Lovász扩展与拟凹扩展关联，尤其在求和运算中。

实验结果

研究问题

• RQ1 能否系统地刻画一般子模函数的上微分与上梯度？它们如何被近似？
• RQ2 在子模优化中，从凹视角看，Fenchel对偶性与离散分离定理在多大程度上成立？
• RQ3 是否存在子模函数的子类（如M♮-凹函数），其上微分结构是精确的，从而支持高效最大化？
• RQ4 所提出的拟凹扩展与现有连续松弛（如多线性扩展）之间有何关系？
• RQ5 能否形式化上微分与子模最大化近似算法之间的关系，并在算法上加以利用？

主要发现

• 本文证明了子模函数具有类似于凸分析中次微分的上微分结构，从而能够对其中的凹性特征进行多面体刻画。
• 证明了子模函数存在紧模面上界，并可通过上微分的内界与外界界高效计算。
• 研究表明，Fenchel对偶性与离散分离定理可扩展至子模函数的凹侧，但仅在受限条件下成立。
• 对于M♮-凹函数，上微分结构是精确的，且满足∂^f(X) = ∂^f_Δ(2,2)(X)，从而可能支持精确的多项式时间算法。
• 近似上微分可导出子模最大化的近似算法，本文建议了上微分近似与算法性能之间的系统性关联。
• Lovász扩展满足Minkowski和性质，本文还研究了在受限条件下，所提出的拟凹扩展是否具有类似性质。