Tail Probabilities for Randomized Program Runtimes via Martingales for Higher Moments

Satoshi Kura, Natsuki Urabe, Ichiro Hasuo

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.

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