# Proving Expected Sensitivity of Probabilistic Programs with Randomized Variable-Dependent Termination Time

## TL;DR

This paper introduces a sound, automated, martingale-based method for proving the expected sensitivity of probabilistic programs with randomized, input-dependent termination times, extending previous fixed-iteration approaches.

## Contribution

It extends expected sensitivity analysis to probabilistic while loops with randomized termination, providing a compositional, automated, martingale-based proof technique.

## Key findings

- Effective on classical examples like Gambler's Ruin and stochastic gradient descent
- Automated approach handles various probabilistic programs from literature
- Demonstrates soundness and compositionality of the method

## Abstract

The notion of program sensitivity (aka Lipschitz continuity) specifies that changes in the program input result in proportional changes to the program output. For probabilistic programs the notion is naturally extended to expected sensitivity. A previous approach develops a relational program logic framework for proving expected sensitivity of probabilistic while loops, where the number of iterations is fixed and bounded. In this work, we consider probabilistic while loops where the number of iterations is not fixed, but randomized and depends on the initial input values. We present a sound approach for proving expected sensitivity of such programs. Our sound approach is martingale-based and can be automated through existing martingale-synthesis algorithms. Furthermore, our approach is compositional for sequential composition of while loops under a mild side condition. We demonstrate the effectiveness of our approach on several classical examples from Gambler's Ruin, stochastic hybrid systems and stochastic gradient descent. We also present experimental results showing that our automated approach can handle various probabilistic programs in the literature.