Sander Gribling

Simon Apers, Sander Gribling, Dániel Szilágyi

Hamiltonian Monte Carlo (HMC) is a Markov chain algorithm for sampling from a high-dimensional distribution with density $e^{-f(x)}$, given access to the gradient of $f$. A particular case of interest is that of a $d$-dimensional Gaussian distribution with covariance matrix $Σ$, in which case $f(x) = x^\top Σ^{-1} x$. We show that HMC can sample from a distribution that is $\varepsilon$-close in total variation distance using $\widetilde{O}(\sqrtκ d^{1/4} \log(1/\varepsilon))$ gradient queries, where $κ$ is the condition number of $Σ$. Our algorithm uses long and random integration times for the Hamiltonian dynamics. This contrasts with (and was motivated by) recent results that give an $\widetildeΩ(κd^{1/2})$ query lower bound for HMC with fixed integration times, even for the Gaussian case.

Sander Gribling, Aaron Sidford, Chenyi Zhang

We provide algorithms that compute $ε$-estimates of the $\ell_p$-Lewis weights of a matrix $A \in \mathbb{R}^{m \times n}$ for $p \geq 4$ using $O(p^2 \log(m/ε))$ rounds of leverage score computation, where $\ell_p$-Lewis weights and leverage scores are both standard measures of row importance. This improves upon the state-of-the-art round complexity of $O(p^3 \log(m/ε))$ due to Fazel, Lee, Padmanabha, and Sidford (2022). We obtain our results by carefully applying a local variant of relatively smooth gradient descent to primal and dual forms of the $\ell_p$-Lewis weight optimization problem and providing tools to convert between different notions of approximate $\ell_p$-Lewis weights.