概率规划的集中界分析与概率递归关系
Concentration-Bound Analysis for Probabilistic Programs and Probabilistic Recurrence Relations
摘要：
分析概率程序和随机算法是计算机科学中的经典问题。在随机过程分析中的第一个基本问题是考虑期望或平均值，另一个基本问题是考虑浓度界限，即表明偏离平均值的大概率很小。同样，在概率程序和随机算法中，期望终止时间/运行时间及其集中度的分析也是基本问题，本文主要研究概率程序的集中度和随机算法的概率递归性.对于概率规划，获得浓度界的基本技术是考虑鞅并应用经典的Azuma不等式。对于随机算法的概率递归，卡普的经典“食谱”方法，这是类似于递归的主定理，是标准的方法来获得浓度界。在这项工作中，我们提出了一种新的方法，通过合成指数上鞅的概率规划和概率递归关系的浓度界。对于概率程序，我们提出了在几种情况下合成这种上鞅的算法。我们还表明，我们的方法可以得到更好的浓度范围比简单地应用经典的Azuma不等式在文献中考虑的各种概率程序。对于概率递归，我们的方法可以得到更严格的界限比卡普的成熟的方法对经典算法。此外，我们表明，我们的方法可以得到的界限可比的最佳界快速排序，由McDiarmid和海沃德提出。我们还提出了一个原型实现，可以自动推断这些界限
英文摘要：
Analyzing probabilistic programs and randomized algorithms are classical problems in computer science. The first basic problem in the analysis of stochastic processes is to consider the expectation or mean, and another basic problem is to consider concentration bounds, i.e. showing that large deviations from the mean have small probability. Similarly, in the context of probabilistic programs and randomized algorithms, the analysis of expected termination time/running time and their concentration bounds are fundamental problems.In this work, we focus on concentration bounds for probabilistic programs and probabilistic recurrences of randomized algorithms. For probabilistic programs, the basic technique to achieve concentration bounds is to consider martingales and apply the classical Azuma's inequality. For probabilistic recurrences of randomized algorithms, Karp's classical "cookbook" method, which is similar to the master theorem for recurrences, is the standard approach to obtain concentration bounds. In this work, we propose a novel approach for deriving concentration bounds for probabilistic programs and probabilistic recurrence relations through the synthesis of exponential supermartingales. For probabilistic programs, we present algorithms for synthesis of such supermartingales in several cases. We also show that our approach can derive better concentration bounds than simply applying the classical Azuma's inequality over various probabilistic programs considered in the literature. For probabilistic recurrences, our approach can derive tighter bounds than the Karp's well-established methods on classical algorithms. Moreover, we show that our approach could derive bounds comparable to the optimal bound for quicksort, proposed by McDiarmid and Hayward. We also present a prototype implementation that can automatically infer these bounds
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