A. Azevedo-de-amorim, M. Gaboardi, E. Arias, and J. Hsu, Really natural linear indexed type-checking, Symposium on Implementation and Application of Functional Programming Languages (IFL), vol.5, pp.1-5, 2014.

A. Azevedo-de-amorim, M. Gaboardi, and J. Hsu, A semantic account of metric preservation, ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages (POPL), pp.545-556, 2017.

G. Barthe, F. Dupressoir, S. Faust, B. Grégoire, F. Standaert et al., Parallel Implementations of Masking Schemes and the Bounded Moment Leakage Model. IACR Cryptology ePrint Archive, 2016.
URL : https://hal.archives-ouvertes.fr/hal-01414009

G. Barthe, F. Dupressoir, B. Grégoire, C. Kunz, B. Schmidt et al., EasyCrypt: A Tutorial, Foundations of Security Analysis and Design VII (FOSAD), vol.8604, pp.146-166, 2013.
URL : https://hal.archives-ouvertes.fr/hal-01114366

G. Barthe, T. Espitau, J. Hsu, T. Sato, and P. Strub, ?-Liftings for differential privacy, International Colloquium on Automata, Languages and Programming (ICALP), vol.80, p.12, 2017.

G. Barthe, N. Fong, M. Gaboardi, B. Grégoire, J. Hsu et al., Advanced Probabilistic Couplings for Differential Privacy, ACM SIGSAC Conference on Computer and Communications Security (CCS), pp.55-67, 2016.
URL : https://hal.archives-ouvertes.fr/hal-01410196

G. Barthe, M. Gaboardi, E. Arias, J. Hsu, A. Roth et al., Higher-Order Approximate Relational Refinement Types for Mechanism Design and Differential Privacy, In ACM SIGPLAN-SIGACT Proceedings of the ACM on Programming Languages, vol.2, p.28, 2015.

G. Barthe, T. Espitau, B. Grégoire, J. Hsu, and P. , Strub Symposium on Principles of Programming Languages (POPL), pp.55-68

G. Barthe, M. Gaboardi, E. Arias, J. Hsu, A. Roth et al., Computeraided verification in mechanism design, Conference on Web and Internet Economics (WINE), 2016.
URL : https://hal.archives-ouvertes.fr/hal-01260071

G. Barthe, M. Gaboardi, B. Grégoire, J. Hsu, and P. Strub, Proving Differential Privacy via Probabilistic Couplings, IEEE Symposium on Logic in Computer Science (LICS), pp.749-758, 2016.
URL : https://hal.archives-ouvertes.fr/hal-01411097

G. Barthe, B. Grégoire, J. Hsu, and P. Strub, Coupling proofs are probabilistic product programs, ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages (POPL), 2017.
URL : https://hal.archives-ouvertes.fr/hal-01649028

G. Barthe, B. Grégoire, and S. Zanella-béguelin, Formal Certification of Code-Based Cryptographic Proofs, ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages (POPL), pp.90-101, 2009.

G. Barthe, B. Köpf, F. Olmedo, and S. Z. Béguelin, Probabilistic relational reasoning for differential privacy, ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages (POPL), pp.97-110, 2012.

G. Barthe and F. Olmedo, Beyond Differential Privacy: Composition Theorems and Relational Logic for f-divergences between Probabilistic Programs, International Colloquium on Automata, Languages and Programming (ICALP), vol.7966, pp.641-657, 1976.

O. Bousquet and A. Elisseeff, Stability and Generalization, Journal of Machine Learning Research, vol.2, pp.499-526, 2002.

R. Bubley and M. Dyer, Path coupling: A technique for proving rapid mixing in Markov chains, IEEE Symposium on Foundations of Computer Science (FOCS), pp.223-231, 1997.

S. Chaudhuri, S. Gulwani, and R. Lublinerman, Continuity analysis of programs, ACM SIGPLANSIGACT Symposium on Principles of Programming Languages (POPL), pp.57-70, 2010.

P. Narendra-m-dixit, . Srivastava, and . Vishnoi, A finite population model of molecular evolution: Theory and computation, Journal of Computational Biology, vol.19, pp.1176-1202, 2012.

H. Eldib, C. Wang, M. I. Mostafa, P. Taha, and . Schaumont, Quantitative Masking Strength: Quantifying the Power Side-Channel Resistance of Software Code, IEEE Transansactions on CAD of Integrated Circuits and Systems, vol.34, pp.1558-1568, 2015.

A. Elisseeff, T. Evgeniou, and M. Pontil, Stability of Randomized Learning Algorithms, Journal of Machine Learning Research, vol.6, pp.55-79, 2005.

M. Gaboardi, A. Haeberlen, J. Hsu, A. Narayan, and B. C. Pierce, Linear dependent types for differential privacy, ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages (POPL), pp.357-370, 2013.
DOI : 10.1145/2429069.2429113

URL : https://hal.archives-ouvertes.fr/hal-00909340

M. Hardt, B. Recht, and Y. Singer, Train faster, generalize better: Stability of stochastic gradient descent, International Conference on Machine Learning (ICML), vol.48, pp.1225-1234, 2016.

L. Daniel, A. G. Hartl, and . Clark, Principles of Population Genetics, 2006.

J. Hsu, Probabilistic Couplings for Probabilistic Reasoning, Ph.D. Dissertation. University of Pennsylvania, 2017.

X. Huang, M. Kwiatkowska, S. Wang, and M. Wu, Safety Verification of Deep Neural Networks, International Conference on Computer Aided Verification (CAV), vol.10426, pp.3-29, 2017.

T. Jansen, Analyzing Evolutionary Algorithms: The Computer Science Perspective, 2013.

M. Jerrum, A Very Simple Algorithm for Estimating the Number of k-Colorings of a Low-Degree Graph, Random Structures and Algorithms, vol.7, pp.157-166, 1995.

J. Benjamin-lucien-kaminski, C. Katoen, F. Matheja, and . Olmedo, Weakest Precondition Reasoning for Expected Run-Times of Probabilistic Programs, European Symposium on Programming (ESOP), vol.9632, pp.364-389, 2016.

G. Katz, C. W. Barrett, D. L. Dill, K. Julian, and M. J. Kochenderfer, Reluplex: An Efficient SMT Solver for Verifying Deep Neural Networks, International Conference on Computer Aided Verification (CAV), vol.10426, pp.97-117, 2017.
DOI : 10.1007/978-3-319-63387-9_5

URL : http://arxiv.org/pdf/1702.01135

. Dexter-kozen, Semantics of probabilistic programs, IEEE Symposium on Foundations of Computer Science (FOCS), pp.101-114, 1979.

. Dexter-kozen, A Probabilistic PDL, J. Comput. System Sci, vol.30, pp.162-178, 1985.

T. Lindvall, Lectures on the coupling method, 2002.

C. Morgan, A. Mciver, and K. Seidel, Probabilistic Predicate Transformers, ACM Transactions on Programming Languages and Systems, vol.18, pp.325-353, 1996.

I. Panageas, P. Srivastava, and N. K. Vishnoi, Evolutionary Dynamics in Finite Populations Mix Rapidly, ACM-SIAM Symposium on Discrete Algorithms (SODA), pp.480-497, 2016.

J. Reed and . Benjamin-c-pierce, Distance Makes the Types Grow Stronger: A Calculus for Differential Privacy, ACM SIGPLAN International Conference on Functional Programming (ICFP), 2010.

T. Sato, Approximate Relational Hoare Logic for Continuous Random Samplings, Conference on the Mathematical Foundations of Programming Semantics (MFPS), 2016.

D. Selsam, P. Liang, and D. L. Dill, Developing Bug-Free Machine Learning Systems With Formal Mathematics, International Conference on Machine Learning (ICML), vol.70, pp.3047-3056, 2017.

O. Shamir, Without-Replacement Sampling for Stochastic Gradient Methods: Convergence Results and Application to Distributed Optimization, 2016.

H. Thorisson, Coupling, Stationarity, and Regeneration, 2000.

C. Villani, Optimal transport: Old and new, 2008.

K. Nisheeth and . Vishnoi, The Speed of Evolution, ACM-SIAM Symposium on Discrete Algorithms (SODA), pp.1590-1601, 2015.

D. Winograd-cort, A. Haeberlen, A. Roth, and B. C. Pierce, A framework for adaptive differential privacy, ACM SIGPLAN International Conference on Functional Programming (ICFP), vol.10, pp.1-10, 2017.