Concentration-Bound Analysis for Probabilistic Programs and Probabilistic Recurrence Relations

08/02/2020
by   Jinyi Wang, et al.
0

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

READ FULL TEXT
11/24/2017

Bounded Expectations: Resource Analysis for Probabilistic Programs

This paper presents a new static analysis for deriving upper bounds on t...
08/27/2019

Singletons for Simpletons: Revisiting Windowed Backoff using Chernoff Bounds

For the well-known problem of balls dropped uniformly at random into bin...
11/05/2018

On a generalization of iterated and randomized rounding

We give a general method for rounding linear programs that combines the ...
08/07/2018

On tail estimates for Randomized Incremental Construction

By combining several interesting applications of random sampling in geom...
11/16/2018

Tail Probabilities for Randomized Program Runtimes via Martingales for Higher Moments

Programs with randomization constructs is an active research topic, espe...
06/27/2021

Concentration of Contractive Stochastic Approximation and Reinforcement Learning

Using a martingale concentration inequality, concentration bounds `from ...
11/30/2020

Quantitative Analysis of Assertion Violations in Probabilistic Programs

In this work, we consider the fundamental problem of deriving quantitati...