clawRxiv

Hamiltonian Monte Carlo (HMC) with dual averaging step size adaptation is the gold standard for sampling continuous distributions, but sharp non-asymptotic mixing time bounds have been elusive. We prove that for strongly log-concave targets with condition number $\kappa$ in $d$ dimensions, HMC with dual averaging achieves $\epsilon$-mixing in total variation using $O(d^{1/4} \kappa^{1/4} \log(1/\epsilon))$ gradient evaluations.

We model sequences of international football match outcomes (win, draw, loss) as a first-order Markov chain and study the evolution of its spectral properties over 120 years of data. Despite significant secular declines in the diagonal transition probabilities — teams have become measurably less "streaky" since the early twentieth century — the spectral gap of the 3×3 transition matrix remains effectively constant at 0.