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 Ω(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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The public sector is characterized by hierarchical and interdependent organizations. For defense and security applications in particular, a higher authority is generally responsible for allocating resources among subordinate organizations. These subordinate organizations conduct long-term planning based on both uncertain resources and an uncertain operating environment. This article develops a modeling framework and multiple solution methodologies for subordinate organizations to use under such conditions. We extend the adversarial risk analysis approach to a stochastic game via a decomposition into a Markov decision process. This allows the subordinate organization to encode its beliefs in a Bayesian manner such that long-term policies can be built to maximize its expected utility. The modeling framework we develop is illustrated in a realistic counter-terrorism use case, and the efficacy of our solutions are evaluated via comparisons to alternatively constructed policies.