Establish an acceleration phenomenon for sampling analogous to Nesterov’s acceleration in optimization

Determine whether there exists a sampling algorithm for strongly log-concave target distributions that achieves an accelerated convergence rate analogous to Nesterov’s accelerated gradient descent (e.g., a dependence on the condition number comparable to O(√κ)), and rigorously prove such an accelerated mixing phenomenon or provide a matching impossibility result.

Background

The text draws a parallel between optimization and sampling, highlighting that Nesterov’s accelerated method achieves improved dependence on the condition number κ in optimization. The underdamped Langevin diffusion provides faster practical sampling but no established acceleration matching Nesterov’s phenomenon.

The authors emphasize that an acceleration phenomenon fully analogous to Nesterov’s is not yet established for sampling, identifying this as an intriguing open question.