3 research outputs found
Tail Probabilities for Randomized Program Runtimes via Martingales for Higher Moments
Programs with randomization constructs is an active research topic,
especially after the recent introduction of martingale-based analysis methods
for their termination and runtimes. Unlike most of the existing works that
focus on proving almost-sure termination or estimating the expected runtime, in
this work we study the tail probabilities of runtimes-such as "the execution
takes more than 100 steps with probability at most 1%." To this goal, we devise
a theory of supermartingales that overapproximate higher moments of runtime.
These higher moments, combined with a suitable concentration inequality, yield
useful upper bounds of tail probabilities. Moreover, our vector-valued
formulation enables automated template-based synthesis of those
supermartingales. Our experiments suggest the method's practical use.Comment: 38 page
Analysis of Bayesian Networks via Prob-Solvable Loops
Prob-solvable loops are probabilistic programs with polynomial assignments
over random variables and parametrised distributions, for which the full
automation of moment-based invariant generation is decidable. In this paper we
extend Prob-solvable loops with new features essential for encoding Bayesian
networks (BNs). We show that various BNs, such as discrete, Gaussian,
conditional linear Gaussian and dynamic BNs, can be naturally encoded as
Prob-solvable loops. Thanks to these encodings, we can automatically solve
several BN related problems, including exact inference, sensitivity analysis,
filtering and computing the expected number of rejecting samples in
sampling-based procedures. We evaluate our work on a number of BN benchmarks,
using automated invariant generation within Prob-solvable loop analysis
Concentration-Bound Analysis for Probabilistic Programs and Probabilistic Recurrence Relations
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 boundsComment: 28 page