Complexity Bounds for MCMC via Diffusion Limits

We connect known results about diffusion limits of Markov chain Monte Carlo (MCMC) algorithms to the Computer Science notion of algorithm complexity. Our main result states that any diffusion limit of a Markov process implies a corresponding complexity bound (in an appropriate metric). We then combine this result with previously-known MCMC diffusion limit results to prove that under appropriate assumptions, the Random-Walk Metropolis (RWM) algorithm in $d$ dimensions takes $O(d)$ iterations to converge to stationarity, while the Metropolis-Adjusted Langevin Algorithm (MALA) takes $O(d^{1\mathrm{/}3})$ iterations to converge to stationarity.