Log-Derivative Trick for Parametric Markov Chains

arXiv:2610.36655v1 Announce Type: new Abstract: Optimizing expected rewards and reachability probabilities in parametric Markov chains (pMCs) is central to the analysis and synthesis of probabilistic systems with loops. While traditional symbolic synthesis approaches--such as parameter lifting and rational function generation--suffer from scalability bottlenecks, gradient-based optimization emerges as an effective alternative. However, existing gradient-based techniques still rely on symbolic gradient computation--such as solving large linear systems--over the global state space, inheriting scalability limits. Finite-horizon approximation is possible but introduces uncontrollable bias. In this paper, we introduce a scalable Monte Carlo gradient estimator based on a measure-theoretic generalization of the log-derivative trick. We prove that our estimator provides unbiased gradient estimates under almost-sure termination and standard regularity conditions. Crucially, our method avoids global state-space materialization and linear system solving entirely: it operates purely along sampled trajectories, requiring only black-box evaluations of local transition probabilities (or densities) and their derivatives. Unlike existing pMC techniques that are fundamentally restricted to finite discrete systems, our measure-theoretic framework naturally extends to continuous and general $\sigma$-finite state spaces. Empirical evaluations on synthetic discrete pMCs and continuous-state case studies demonstrate that our method scales to state spaces far beyond the reach of existing solvers, significantly reducing execution time without introducing bias.