Minimax Mixing Time of the Metropolis-Adjusted Langevin Algorithm for Log-Concave Sampling
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We study the mixing time of the Metropolis-adjusted Langevin algorithm (MALA) for sampling from a log-smooth and strongly log-concave distribution. We establish its optimal minimax mixing time under a warm start. Our main contribution is two-fold. First, for a d-dimensional log-concave density with condition number κ, we show that MALA with a warm start mixes in Õ(κ√(d)) iterations up to logarithmic factors. This improves upon the previous work on the dependency of either the condition number κ or the dimension d. Our proof relies on comparing the leapfrog integrator with the continuous Hamiltonian dynamics, where we establish a new concentration bound for the acceptance rate. Second, we prove a spectral gap based mixing time lower bound for reversible MCMC algorithms on general state spaces. We apply this lower bound result to construct a hard distribution for which MALA requires at least Ω̃(κ√(d)) steps to mix. The lower bound for MALA matches our upper bound in terms of condition number and dimension. Finally, numerical experiments are included to validate our theoretical results.
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