Alain Durmus

Lattice Signatures and Bimodal Gaussians

Our main result is a construction of a lattice-based digital signature scheme that represents an improvement, both in theory and in practice, over today’s most efficient lattice schemes. The novel scheme is obtained as a result of a modification of the rejection sampling algorithm that is at the heart of Lyubashevsky’s signature scheme (Eurocrypt, 2012) and several other lattice primitives. Our new rejection sampling algorithm which samples from a bimodal Gaussian distribution, combined with a modified scheme instantiation, ends up reducing the standard deviation of the resulting signatures by a factor that is asymptotically square root in the security parameter. The implementations of our signature scheme for security levels of 128, 160, and 192 bits compare very favorably to existing schemes such as RSA and ECDSA in terms of efficiency. In addition, the new scheme has shorter signature and public key sizes than all previously proposed lattice signature schemes.

Nonasymptotic convergence analysis for the unadjusted Langevin algorithm

In this paper, we study a method to sample from a target distribution

Efficient Bayesian Computation by Proximal Markov Chain Monte Carlo: When Langevin Meets Moreau

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Subgeometric rates of convergence in Wasserstein distance for Markov chains

over

Quantitative bounds of convergence for geometrically ergodic Markov chain in the Wasserstein distance with application to the Metropolis Adjusted Langevin Algorithm

\mathbb{R}^d

Optimal scaling of the random walk Metropolis algorithm under L p mean differentiability

having a positive density with respect to the Lebesgue measure, known up to a normalisation factor. This method is based on the Euler discretization of the overdamped Langevin stochastic differential equation associated with

Normalizing constants of log-concave densities

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Parallelized Stochastic Gradient Markov Chain Monte Carlo algorithms for non-negative matrix factorization

. For both constant and decreasing step sizes in the Euler discretization, we obtain non-asymptotic bounds for the convergence to the target distribution

Sampling from a strongly log-concave distribution with the Unadjusted Langevin Algorithm

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High-dimensional Bayesian inference via the Unadjusted Langevin Algorithm

in total variation distance. A particular attention is paid to the dependency on the dimension