[论文解读] On the Complexity of Constrained Determinantal Point Processes

L. Elisa Celis, Amit Deshpande+3 more|arXiv (Cornell University)|Aug 1, 2016

Point processes and geometric inequalities被引用 6

一句话总结

本文解决了受限DPP采样复杂度问题，证明当约束以一元方式描述（例如小预算或划分约束）时，高效采样是可行的，而当约束以二进制编码时，该问题为#P难。本文建立了混合判别式与划分约束DPP之间的等价关系，从而为拟阵基多面体和混合特征多项式提供了新算法。

ABSTRACT

Determinantal Point Processes (DPPs) are probabilistic models that arise in quantum physics and random matrix theory and have recently found numerous applications in computer science. DPPs define distributions over subsets of a given ground set, they exhibit interesting properties such as negative correlation, and, unlike other models, have efficient algorithms for sampling. When applied to kernel methods in machine learning, DPPs favor subsets of the given data with more diverse features. However, many real-world applications require efficient algorithms to sample from DPPs with additional constraints on the subset, e.g., partition or matroid constraints that are important to ensure priors, resource or fairness constraints on the sampled subset. Whether one can efficiently sample from DPPs in such constrained settings is an important problem that was first raised in a survey of DPPs by \cite{KuleszaTaskar12} and studied in some recent works in the machine learning literature. The main contribution of our paper is the first resolution of the complexity of sampling from DPPs with constraints. We give exact efficient algorithms for sampling from constrained DPPs when their description is in unary. Furthermore, we prove that when the constraints are specified in binary, this problem is #P-hard via a reduction from the problem of computing mixed discriminants implying that it may be unlikely that there is an FPRAS. Our results benefit from viewing the constrained sampling problem via the lens of polynomials. Consequently, we obtain a few algorithms of independent interest: 1) to count over the base polytope of regular matroids when there are additional (succinct) budget constraints and, 2) to evaluate and compute the mixed characteristic polynomials, that played a central role in the resolution of the Kadison-Singer problem, for certain special cases.

研究动机与目标

• 解决在组合约束（如划分、背包或预算约束）下从DPP中采样的计算复杂度问题。
• 确定在机器学习中涉及公平性、资源限制或先验约束的情境下，受限DPP是否存在高效采样算法。
• 建立受限DPP采样与计算混合判别式之间的正式联系，后者是数学中已知的难题。
• 为在额外预算约束下对正则拟阵的基多面体进行计数、采样和优化，开发新算法。
• 为在特定情况下高效评估和计算混合特征多项式提供有效方法，这些情况与Kadison-Singer问题相关。

提出的方法

• 作者通过DPP生成函数的多项式表示，建立了从划分约束DPP中采样与计算混合判别式之间的等价关系。
• 他们使用基于$μ$-采样预言机的递归采样策略，通过$\mathrm{poly}(m)/\delta^2$次采样，以加法误差$\delta$估计边际概率$p_e = \mathbb{P}(e \notin \mathbf{S} \mid \mathbf{S} \subseteq U)$。
• 该算法采用递归剪枝过程，识别出边际不包含概率$p_e \geq 1/m^2$的元素$e$，确保每一步均有进展。
• 通过重复采样，使用$(1+\varepsilon/m)$-乘法近似来估计$p_e$，并利用切比雪夫不等式保持浓度界限。
• 该方法依赖于通过将$U$外的系数设为零来修改生成多项式$g_\mu$，从而实现条件采样。
• 最终估计值$X$被构造为近似边际概率与剩余集合$\mu(U')$的测度的乘积，误差界限由乘法近似因子推导得出。

实验结果

研究问题

• RQ1 在组合约束（如划分或预算约束）下，是否可以从DPP中实现高效采样？
• RQ2 当约束以一元编码与二进制编码表示时，受限DPP采样的计算复杂度为何？
• RQ3 能否将计算混合判别式的难题归约为从划分约束DPP中采样？
• RQ4 能否为在额外预算约束下对正则拟阵的基多面体进行计数和采样开发高效算法？
• RQ5 能否利用所提出的框架在特定情况下高效评估混合特征多项式？

主要发现

• 当约束以一元方式描述时（包括小预算或小划分约束），受限DPP的采样可在多项式时间内完成。
• 当约束以二进制编码时，该问题通过从计算混合判别式问题的归约被证明为#P难。
• 对于二进制编码约束，存在FPRAS的可能性极低，因为这将意味着解决了#P难的混合判别式问题。
• 本文开发了一种高效算法，用于在额外预算约束下对正则拟阵的基多面体进行计数、采样和优化。
• 本文提出了一种新方法，用于在特定情况下评估和计算混合特征多项式，这些情况与Kadison-Singer问题的解决相关。
• 该算法以高概率使用$\mathrm{poly}(m, 1/\varepsilon)$次采样，实现对目标和的$(1+2\varepsilon)$-乘法近似。