Average-case time complexity of Gaussian–Gaussian posterior sampling

Investigate the average-case time complexity of obtaining N independent samples from the Gaussian–Gaussian Bayesian posterior using the proposed resistor–inductor thermodynamic circuit implementing overdamped Langevin dynamics, measured via convergence in Wasserstein-2 distance normalized by the posterior covariance, and determine how the expected runtime scales with the parameter dimension d and tolerance εW beyond the worst-case bound O(N ln(d εW^{-2})) established under assumptions on the prior and likelihood covariances and conditioning (‖Σπ‖≤1, ‖Σℓ‖≤1, and μπᵀΣπ^{-1}μπ, yᵀΣℓ^{-1}y ≤ Mmax).

Background

The paper derives a worst-case runtime bound for the proposed thermodynamic circuit that samples the Gaussian–Gaussian posterior, showing that N samples can be obtained in time O(N ln(d εW^{-2})) under reasonable rescaling and conditioning assumptions. The analysis is based on exponential contraction of the Wasserstein-2 distance and bounds on the energy and time needed to reach a target error.

However, the authors explicitly note that this is a worst-case analysis and that the average-case behavior has not been characterized. Establishing average-case scaling would strengthen the practical runtime guarantees and provide a more complete understanding of performance across typical problem instances.