Convergence Of The Unadjusted Langevin Algorithm For Discontinuous Gradients
arxiv(2023)
摘要
We demonstrate that for strongly log-convex densities whose potentials are
discontinuous on manifolds, the ULA algorithm converges with stepsize bias of
order $1/2$ in Wasserstein-p distance. Our resulting bound is then of the same
order as the convergence of ULA for gradient Lipschitz potential. Additionally,
we show that so long as the gradient of the potential obeys a growth bound
(therefore imposing no regularity condition), the algorithm has stepsize bias
of order $1/4$. We therefore unite two active areas of research: i) the study
of numerical methods for SDEs with discontinuous coefficients and ii) the study
of the non-asymptotic bias of the ULA algorithm (and variants). In particular
this is the first result of the former kind we are aware of on an unbounded
time interval.
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