A Profile-Separation Framework for Quantitative Convergence of No-U-Turn Samplers

Abstract

We study multinomial and biased-progressive No-U-Turn Samplers for strongly log-concave targets satisfying $mI_d\preceq \nabla^2U(x)\preceq LI_d,$ and $\|\nabla^2U(x)-\nabla^2U(y)\|_{\mathrm F}\le γL^{3/2}\|x-y\|$ with \(κ\coloneqq L/m\). We introduce profile separation, a sufficient sign condition on the stationary mean U-turn diagnostics, and combine it with diagnostic concentration, leapfrog fidelity, and whole-orbit energy control to show that on a high-probability certification event, every doubling realization reaches a common terminal depth through a genuine U-turn. If \(T_\star\) is the selected physical trajectory length and \(a_\star=\sqrt m\,T_\star\), a terminal-depth transfer argument yields restricted conductance and warm-start mixing without lazifying either kernel. Up to logarithmic warm-start and accuracy factors, the transition bounds are \[ \widetilde O\!\left( 1+a_\star^2κ^2(1+γ)^{4/3} \right) \quad\text{and}\quad \widetilde O\!\left( 1+a_\star^4κ^3(1+γ)^2 \right) \] for multinomial and biased-progressive selection, respectively. These transition bounds are unconditional. Gradient-work bounds are deterministic when the maximum-depth cap is comparable to the certified depth and otherwise take cap-aware expected and high-probability forms. The framework recovers the Gaussian dimension dependence under these work-accounting conditions, provides population-profile and exact-diagnostic verification for nonlinear product targets, a near-isotropic specialization of the practical-tree certificate, and quantifies when a fixed post-warmup metric removes linear anisotropy.