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Discretisation of Langevin diffusion in the weak log-concave case

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  • Crespo, Marelys

Abstract

The Euler discretisation of Langevin diffusion, also known as Unadjusted Langevin Algorithm, is commonly used in machine learning for sampling from a given distribution µ ∝ e−U . In this paper we investigate a potential U : Rd −→ R which is a weakly convex function and has Lipschitz gradient. We parameterize the weak convexity with the help of the Kurdyka-Lojasiewicz (KL) inequality, that permits to handle a vanishing curvature settings, which is far less restrictive when compared to the simple strongly convex case. We prove that the final horizon of simulation to obtain an ε approximation (in terms of entropy) is of the order ε−1d1+2(1+r)2Poly(log(d), log(ε−1)), where the parameter r is involved in the KL inequality and varies between 0 (strongly convex case) and 1 (limiting Laplace situation).

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  • Crespo, Marelys, 2024. "Discretisation of Langevin diffusion in the weak log-concave case," TSE Working Papers 24-1506, Toulouse School of Economics (TSE).
    • Handle: RePEc:tse:wpaper:129118
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    References listed on IDEAS

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    1. Gadat, Sébastien & Panloup, Fabien & Pellegrini, C., 2020. "On the cost of Bayesian posterior mean strategy for log-concave models," TSE Working Papers 20-1155, Toulouse School of Economics (TSE), revised Feb 2022.
    2. Gareth O. Roberts & Jeffrey S. Rosenthal, 1998. "Optimal scaling of discrete approximations to Langevin diffusions," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 60(1), pages 255-268.
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