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A pre-expectation calculus for probabilistic sensitivity

Authors:

Alejandro Aguirre,

Benjamin Lucien Kaminski,

Joost-Pieter Katoen,

Christoph MathejaAuthors Info & Claims

Proceedings of the ACM on Programming Languages, Volume 5, Issue POPL

Article No.: 52, Pages 1 - 28

https://doi.org/10.1145/3434333

Published: 04 January 2021 Publication History

Abstract

Sensitivity properties describe how changes to the input of a program affect the output, typically by upper bounding the distance between the outputs of two runs by a monotone function of the distance between the corresponding inputs. When programs are probabilistic, the distance between outputs is a distance between distributions. The Kantorovich lifting provides a general way of defining a distance between distributions by lifting the distance of the underlying sample space; by choosing an appropriate distance on the base space, one can recover other usual probabilistic distances, such as the Total Variation distance. We develop a relational pre-expectation calculus to upper bound the Kantorovich distance between two executions of a probabilistic program. We illustrate our methods by proving algorithmic stability of a machine learning algorithm, convergence of a reinforcement learning algorithm, and fast mixing for card shuffling algorithms. We also consider some extensions: using our calculus to show convergence of Markov chains to the uniform distribution over states and an asynchronous extension to reason about pairs of program executions with different control flow.

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Cited By

Chatterjee KGoharshady ENovotný PŽikelić Đ(2024)Equivalence and Similarity Refutation for Probabilistic ProgramsProceedings of the ACM on Programming Languages10.1145/36564628:PLDI(2098-2122)Online publication date: 20-Jun-2024
https://dl.acm.org/doi/10.1145/3656462
Chatterjee KGoharshady AMeggendorfer TŽikelić Đ(2024)Quantitative Bounds on Resource Usage of Probabilistic ProgramsProceedings of the ACM on Programming Languages10.1145/36498248:OOPSLA1(362-391)Online publication date: 29-Apr-2024
https://dl.acm.org/doi/10.1145/3649824
Bao JTrivedi NPathak DHsu JRoy S(2024)Data-driven invariant learning for probabilistic programsFormal Methods in System Design10.1007/s10703-024-00466-xOnline publication date: 3-Dec-2024
https://doi.org/10.1007/s10703-024-00466-x
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Index Terms

A pre-expectation calculus for probabilistic sensitivity
1. Mathematics of computing
  1. Probability and statistics
2. Theory of computation
  1. Logic
    1. Logic and verification
  2. Semantics and reasoning
    1. Program reasoning

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Published In

cover image Proceedings of the ACM on Programming Languages

Proceedings of the ACM on Programming Languages Volume 5, Issue POPL

January 2021

1789 pages

EISSN:2475-1421

DOI:10.1145/3445980

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Copyright © 2021 Owner/Author.

This work is licensed under a Creative Commons Attribution International 4.0 License.

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Association for Computing Machinery

New York, NY, United States

Publication History

Published: 04 January 2021

Published in PACMPL Volume 5, Issue POPL

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Cited By

Chatterjee KGoharshady ENovotný PŽikelić Đ(2024)Equivalence and Similarity Refutation for Probabilistic ProgramsProceedings of the ACM on Programming Languages10.1145/36564628:PLDI(2098-2122)Online publication date: 20-Jun-2024
https://dl.acm.org/doi/10.1145/3656462
Chatterjee KGoharshady AMeggendorfer TŽikelić Đ(2024)Quantitative Bounds on Resource Usage of Probabilistic ProgramsProceedings of the ACM on Programming Languages10.1145/36498248:OOPSLA1(362-391)Online publication date: 29-Apr-2024
https://dl.acm.org/doi/10.1145/3649824
Bao JTrivedi NPathak DHsu JRoy S(2024)Data-driven invariant learning for probabilistic programsFormal Methods in System Design10.1007/s10703-024-00466-xOnline publication date: 3-Dec-2024
https://doi.org/10.1007/s10703-024-00466-x
Moosbrugger MMüllner JBartocci EKovács L(2024)Polar: An Algebraic Analyzer for (Probabilistic) LoopsPrinciples of Verification: Cycling the Probabilistic Landscape10.1007/978-3-031-75783-9_8(179-200)Online publication date: 13-Nov-2024
https://doi.org/10.1007/978-3-031-75783-9_8
Feng SChen MSu HKaminski BKatoen JZhan N(2023)Lower Bounds for Possibly Divergent Probabilistic ProgramsProceedings of the ACM on Programming Languages10.1145/35860517:OOPSLA1(696-726)Online publication date: 6-Apr-2023
https://dl.acm.org/doi/10.1145/3586051
Moosbrugger MMüllner JKovács L(2023)Automated Sensitivity Analysis for Probabilistic LoopsiFM 202310.1007/978-3-031-47705-8_2(21-39)Online publication date: 6-Nov-2023
https://doi.org/10.1007/978-3-031-47705-8_2
Zhou ZHuang ZMisailovic S(2023)AquaSense: Automated Sensitivity Analysis of Probabilistic Programs via Quantized InferenceAutomated Technology for Verification and Analysis10.1007/978-3-031-45332-8_16(288-301)Online publication date: 19-Oct-2023
https://doi.org/10.1007/978-3-031-45332-8_16
Sun YFu HChatterjee KGoharshady A(2023)Automated Tail Bound Analysis for Probabilistic Recurrence RelationsComputer Aided Verification10.1007/978-3-031-37709-9_2(16-39)Online publication date: 17-Jul-2023
https://doi.org/10.1007/978-3-031-37709-9_2
Vasilenko EVazou NBarthe G(2022)Safe couplings: coupled refinement typesProceedings of the ACM on Programming Languages10.1145/35476436:ICFP(596-624)Online publication date: 31-Aug-2022
https://dl.acm.org/doi/10.1145/3547643
Aguirre AKatsumata SKura S(2022)Weakest preconditions in fibrationsMathematical Structures in Computer Science10.1017/S096012952200033032:4(472-510)Online publication date: 28-Oct-2022
https://doi.org/10.1017/S0960129522000330
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