Multilevel-Langevin pathwise average for Gibbs approximation
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We propose and study a new multilevel method for the numerical approximation of a Gibbs distribution π on R d , based on (over-damped) Langevin diffusions. This method both inspired by [PP18] and [GMS + 20] relies on a multilevel occupation measure, i.e. on an appropriate combination of R occupation measures of (constant-step) discretized schemes of the Langevin diffusion with respective steps γr = γ02 –r , r = 0,. .. , R. For a given diffusion, we first state a result under general assumptions which guarantees an ϵ-approximation (in a L 2-sense) with a cost proportional to ϵ –2 (i.e. proportional to a Monte-Carlo method without bias) or ϵ –2 | log ϵ| 3 under less contractive assumptions. This general result is then applied to over-damped Langevin diffusions in a strongly convex setting, with a study of the dependence in the dimension d and in the spectrum of the Hessian matrix D 2 U of the potential U : R d → R involved in the Gibbs distribution. This leads to strategies with cost in O(dϵ –2 log 3 (dϵ –2)) and in O(dϵ –2) under an additional condition on the third derivatives of U. In particular, in our last main result, we show that, up to universal constants, an appropriate choice of the diffusion coefficient and of the parameters of the procedure leads to a cost controlled by (λ U 1) 2 λ 3 U dϵ –2 (whereλU and λ U respectively denote the supremum and the infimum of the largest and lowest eigenvalue of D 2 U). In our numerical illustrations, we show that our theoretical bounds are confirmed in practice and finally propose an opening to some theoretical or numerical strategies in order to increase the robustness of the procedure when the largest and smallest eigenvalues of D 2 U are respectively too large or too small.
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