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.

References

This remarkable result, which saves a factor of $\sqrt\kappa$ over the basic rate for gradient descent, has been dubbed the acceleration phenomenon, and it remains an intriguing open question to establish such a phenomenon for sampling.

— Statistical optimal transport  (2407.18163 - Chewi et al., 2024) in Section: Sampling, Subsection: Some recent developments (Algorithms)

The results suggest the following informal conjecture. Fix a consistent explicit, possibly randomized, discretization of eq:KLD whose number of pointwise evaluations of U and \nabla U per step is bounded independently of \kappa, and let \mathfrak t be any fixed-parameter tuning rule for this discretization.

eq:KLD:

dXt=Vt dt,dVt=−∇U(Xt) dt−γVt dt+2γ dWt.dX_t=V_t\,dt,\qquad dV_t=-\nabla U(X_t)\,dt-\gamma V_t\,dt+\sqrt{2\gamma}\,dW_t.

— Provable Non-Acceleration of Standard Strang Splittings of Kinetic Langevin Dynamics  (2608.25279 - Bou-Rabee, 26 Aug 2026) in Section 6, “Consequences, scope and open problems”

Whether such internal randomization permits a ballistic mixing bound, uniform over the class eq:classUkappa, from arbitrary point-mass initial states remains open.

eq:classUkappa:

$_\kappa^d =\left\{U\in C^\infty(^d):\ Id_d\preceq\nabla^2U(x)\preceq\kappaId_d \text{ for every }x\right\}. $

— Provable Non-Acceleration of Standard Strang Splittings of Kinetic Langevin Dynamics  (2608.25279 - Bou-Rabee, 26 Aug 2026) in Section 6, “Consequences, scope and open problems”

The result also leaves open warm-start acceleration, target-dependent parameters, time-dependent splittings, and analyses that compare a discretization to the accelerated exact semigroup without demanding global pointwise contractivity.

— Provable Non-Acceleration of Standard Strang Splittings of Kinetic Langevin Dynamics  (2608.25279 - Bou-Rabee, 26 Aug 2026) in Section 6, “Consequences, scope and open problems”

Gradient-adjusted underdamped Langevin dynamics \citep{ZuoOsherLi2025}, building on the Hessian-free high-resolution dynamics of Li, Zha and Tao \citep{LiZhaTao2022}, augments the kinetic Langevin drift with additional gradient terms and achieves total variation mixing on the ballistic scale \sqrt\kappa on Gaussian targets; extending the proved acceleration beyond the Gaussian case remains open.

— Provable Non-Acceleration of Standard Strang Splittings of Kinetic Langevin Dynamics  (2608.25279 - Bou-Rabee, 26 Aug 2026) in Section 6, “Consequences, scope and open problems”