The Best of Times, the Worst of Times: Moment-Based Analysis of Probabilistic Cost Structures

2026-07-28

AI Summary

This work addresses the challenge of analyzing the cost of randomized programs involving nonlinear operations such as max/min, which traditional moment propagation techniques struggle to handle, while full distribution propagation remains computationally infeasible. The authors propose a compositional cost analysis method that solves local recurrence equations bottom-up using hierarchical cost expressions and summarizes subproblems via surrogate distributions composed of an exact short-time prefix and a parameterized tail. This approach enables efficient computation of cost moments—including mean and higher-order moments—with provable error bounds. To the best of our knowledge, it is the first technique to support scalable, error-bounded moment analysis for randomized programs with nonlinear constructs. The framework is implemented in the tool DICKENS and validated on case studies including quantum repeater waiting times, RFID collision resolution, and fork-join completion times, demonstrating both accuracy and efficiency.

Abstract

This paper studies how to compute the moments -- mean, variance, and beyond -- of the cost (e.g., running time) of certain probabilistic programs, in which local costs combine not only additively but also via the extremal operations $\max$ and $\min$. Such costs arise naturally -- for instance, the number of rounds of a contention-resolution protocol, the waiting time of a quantum repeater, and the completion time of a fork-join computation -- but fall outside the scope of moment-based analyses developed for additive costs. The difficulty is that $\max$ and $\min$ are nonlinear: the moments of $\max(X, Y)$ are not determined by those of $X$ and $Y$, so propagating moments alone fails. In contrast, propagating full distributions would suffice, but is computationally intractable. We present a compositional cost analysis for a family of probabilistic programs whose cost structure can be represented as a hierarchical cost expression. The analysis proceeds bottom-up through the hierarchical structure, solving local recurrence equations at each node and summarizing each subproblem with a surrogate distribution. Each surrogate consists of an exact short-time prefix and a compact parametric tail. Our approach computes the mean of the cost distribution with a sound error bound, and systematically lifts to second and higher moments. In addition, precision can be increased by refining the surrogate representation, trading additional computation for tighter bounds. We implemented our method in a tool, called DICKENS, and evaluated its capabilities on three problems: quantum repeater waiting times, RFID collision resolution, and completion times of fork-join computations.

Problem

• probabilistic programs
• moment analysis
• nonlinear cost structures
• max/min operations
• cost distribution

Innovation

• moment analysis
• probabilistic programs
• extremal operations
• surrogate distribution
• hierarchical cost expression