On two proofs of $d^2$ mixing of weighted Dikin walks

We study the mixing time of weighted Dikin walks for sampling from exponential distributions on polytopes and truncated positive-semidefinite (PSD) cones. Our first result gives a general total-variation mixing bound under strong self-concordance, $\barν$-symmetry, and mixed-trace regularity on the local metric. The key idea is to control the Metropolis--Hastings acceptance probability on a high-probability region rather than at every point. Applying this framework to the Lee--Sidford, Lewis-weight, and John metrics yields an $\widetilde O(d^2)$ mixing bound for sampling from polytopes, while applying it to a hybrid barrier yields an $\widetilde O(d^4)$ mixing bound for sampling from truncated PSD cones. Our second result establishes stronger $χ^2$-divergence guarantees and pointwise acceptance control using a new fourth-order bootstrap condition. For a suitably scaled Lee--Sidford metric, this yields an $\widetilde O(d^2)$ mixing bound in $χ^2$-divergence, improving on the previous $\widetilde O(d^{9/4})$ bound.

Provable Non-Acceleration of Standard Strang Splittings of Kinetic Langevin Dynamics

The OBABO and BAOAB schemes and the other standard Strang splittings of kinetic (underdamped) Langevin dynamics are widely used Markov chain Monte Carlo algorithms. Under a suitable friction scaling, the underlying diffusion relaxes on a ballistic time scale, suggesting that these discretizations, suitably tuned, sample targets with condition number $κ$ in $O(\sqrtκ)$ iterations. We prove that no fixed choice of step size and friction, based only on the curvature bounds and the dimension, achieves this acceleration: total variation mixing time lower bounds for OBABO show that ballistic cold-start mixing fails uniformly over the smooth strongly convex class, and the lower bounds extend, with the same orders, to BAOAB and the other four Strang splittings. The proof transfers non-acceleration from optimization to sampling. Eliminating velocity gives an exact noisy heavy-ball recursion, and by the non-acceleration theorem of Goujaud, Taylor and Dieuleveut, for every tuning either some Gaussian target has a mode with relaxation time at least of order $κ$, or an attracting cycle exists on a smooth potential; dilating such a potential as $U_R(x)=R^2U(x/R)$ preserves its curvature bounds and produces metastability for a number of steps exponential in $R^2$, from an initial state at Wasserstein distance $O(R)$ from equilibrium. Using contraction estimates of Leimkuhler, Paulin and Whalley and a Wasserstein-to-total-variation regularization estimate, we prove a complementary upper bound of $O(κ)$ steps, up to logarithmic factors, for a fixed-parameter OBABO tuning. Hence, among fixed-parameter OBABO tunings, the optimal condition-number dependence of cold-start total variation mixing over this class is linear, up to logarithmic factors. A direct Gaussian calculation also rules out fixed-parameter acceleration for the left-endpoint exponential integrator.

A General Framework for Metropolis-Adjusted Dikin Walks: Dimension-Square Mixing on Polytopes and Log-Det Walks on Spectrahedra

We analyze exact-metric, Metropolis-adjusted Dikin walks by keeping the proposal determinant and reverse quadratic form together. Their leading uncentered terms cancel in the complete logarithmic acceptance ratio, leaving centered fluctuations that can be controlled with second-order tools. For a polytope given by $n$ inequalities and a convex $L$-Lipschitz potential, this yields warm-start mixing in $\widetilde O((d^{2}+dL^{2}R^{2})\log(w/δ))$ steps for the regularized Lee--Sidford walk. For a spectrahedron with $n\times n$ blocks, the log-det walk mixes in $\widetilde O((ψ^\star nd+dL^{2}R^{2})\log(w/δ))$ steps, where $ψ^\star$ measures matrix leverage. The two analyses share an acceptance-to-mixing reduction. A proposal-comparison argument transfers the polytope bound to an appropriately padded $O(1/d)$-accurate metric computed from high-precision Lewis weights. For spectrahedra, given $\widehatψ\geψ^\star$, a direct-or-two-seed TensorSRHT construction gives an exact-arithmetic implementation with $ψ^\star$ replaced by $\widehatψ$ in the mixing bound.

Spectral Gaps of Hit-and-Run and Coordinate Hit-and-Run

For any convex body $\mathcal{K}\subset\mathbb{R}^{n}$ containing a unit ball, the spectral gap of Hit-and-Run is $Ω(1/(n^2 C_{\mathsf{PI}}))$, where $C_{\mathsf{PI}}$ is the Poincaré constant of the uniform distribution $π$ over $\mathcal{K}$. This implies that Hit-and-Run converges to a distribution within $χ^2$-divergence $\varepsilon$ of the uniform distribution $π$ in $O(n^2 C_{\mathsf{PI}}\log(M/\varepsilon))$ steps from any starting distribution $π_0$ with $M=χ^2(π_{0}\,\|\,π)$, thus refining the known bound of $O(n^2 R^2 \log(M/\varepsilon))$ by Lovász and Vempala (2004) in terms of the outer radius $R$; for nearly isotropic bodies, together with progress on the KLS conjecture, the complexity is $O(n^2\log n\log(M/\varepsilon))$, improving the dimension dependence from cubic to nearly quadratic while maintaining logarithmic dependence on the initial distance. It was an open problem to connect the convergence of Hit-and-Run to Poincaré/KLS constants as was done for the Ball walk by Kannan, Lovász and Simonovits (1997). Unlike Hit-and-Run, the Ball walk has an unavoidable linear dependence on (a stronger notion) of the initial warmness. We directly bound the spectral gap of the Hit-and-Run Markov chain by connecting it to functional isoperimetric constants, inspired by the recent analysis of In-and-Out. Rewriting the spectral gap in terms of dual certificates leads to the Babuška--Aziz constant studied in the analysis of PDEs; it is asymptotically bounded by the improved Poincaré constant, which we show can be bounded in terms of the usual Poincaré constant. The proof is based on duality and calculus, unlike known proofs of convergence for Hit-and-Run which are based on bounding the conductance. The same technique can be applied to Coordinate Hit-and-Run, resulting in a much improved mixing time of $O(n^3C_{\mathsf{PI}}\log(M/\varepsilon))$.

Accelerated Mixing Time of Randomized Hamiltonian Monte Carlo

We show the Randomized Hamiltonian Monte Carlo (RHMC) algorithm has accelerated mixing time guarantees for sampling from log-concave probability distributions. RHMC proceeds by repeatedly simulating the continuous-time Hamiltonian dynamics for some random integration times, and resetting the velocity to be an independent Gaussian random variable between each simulation. We show that when the target distribution is log-concave and satisfies an $α$-Talagrand inequality (for example, if the target distribution is $α$-strongly log-concave), if we use a random integration time from either the triangular or the exponential distribution with mean $Θ(α^{-1/2})$, then RHMC converges exponentially fast in KL divergence, and the total integration time to reach error $\varepsilon$ in KL divergence scales as $O(α^{-1/2} \log(\varepsilon^{-1}))$. We also show that when the target distribution is log-concave, if we use a sequence of random integration times from the triangular distribution with exponentially increasing means, then the total integration time to reach error $\varepsilon$ in KL divergence scales as $O(\varepsilon^{-1/2})$. Our analysis relies on a bound on the average KL divergence along Hamiltonian dynamics, which is inspired by an analogous result on accelerated optimization methods based on Hamiltonian dynamics.

Fast-Mixing Markov Chains without Gradients

Most approaches for accelerating Markov chain mixing either rely on incorporating expensive geometric information in the proposals, or reduce the per-step cost of sampling via surrogate densities. We propose a localisation principle that allows a surrogate-based Metropolis-Hastings proposal to exploit gradient-level geometric information of the target density, without evaluating either the target gradient or the surrogate gradient. The construction relies on regularisation and tempering of the proposal measure. We show that the expected proposal displacement coincides with the Langevin drift up to controlled error. The resulting framework, Delayed Acceptance with Regularisation and Tempering (DART), achieves an $O(κ\max\{κ, d\})$ mixing time from warm start for strongly log-concave targets with condition number $κ$ in $d$ dimensions. This matches the known $O(κd)$ rate for MALA when $d \ge κ$, and scales as $O(κ^2)$, independent of dimension, otherwise. This is, to our knowledge, the first mixing time guarantee for a surrogate-transition-based MCMC method. We demonstrate DART on a hierarchical spatial generalised linear mixed model. In this setting, the Dirichlet-Neumann averaging parametrisation, originally introduced for the efficient simulation of Gaussian processes, is repurposed to supply the surrogate, and its linear memory and log-linear arithmetic scaling in the number of observation sites carry over to inference.