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.

Nonlocal Transition Kernel for Efficient Learning of Restricted Boltzmann Machines

Learning restricted Boltzmann machines (RBMs) is computationally challenging because it requires expectations whose exact evaluation is generally intractable. The expectations are typically evaluated using a sampling approximation based on blocked Gibbs sampling (BGS), which is a local Markov chain Monte Carlo transition kernel. However, the locality of BGS can lead to poor sampling quality when the RBM has high energy barriers, thereby degrading learning performance. Deep tempering (DT), which performs parallel tempering over a sequence of learnable RBMs including the training RBM, alleviates this locality issue. However, DT algorithmically requires multiple steps to move through the RBM sequence to achieve a nonlocal transition. In this paper, we propose a transition kernel defined over the RBM sequence used in DT. The proposed kernel has a round-trip structure over the sequence, enabling nonlocal moves within a single transition while leaving the RBM sequence invariant. Numerical experiments show that the proposed kernel performs nonlocal transitions more frequently and achieves higher sampling quality with fewer transitions than BGS and DT. We further verify that learning based on the proposed kernel is more stable and mitigates the training failures observed with BGS- and DT-based learning.

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.