Concentration-Bound Analysis for Probabilistic Programs and Probabilistic Recurrence Relations

Abstract

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 [Azuma 1967]. For probabilistic recurrences of randomized algorithms, Karp's classical "cookbook" method [Karp 1994], 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 well-established methods of [Karp 1994] on classical algorithms such as quick sort, quick select, and randomized diameter computation. Moreover, we show that our approach could derive bounds comparable to the optimal bound for quicksort, proposed in [McDiarmid and Hayward 1996]. We also present a prototype implementation that can automatically infer these bounds.