Relax the technical assumptions underlying TunaMH and TunaMH-SGLD

Relax the Lipschitz-type bounded-difference conditions on per-datum potentials U_i(θ; x) used by TunaMH and TunaMH‑SGLD so that an acceptance–rejection minibatch MCMC with correct stationary distribution can be established under weaker or more general assumptions.

Background

TunaMH and the proposed TunaMH‑SGLD achieve minibatch Metropolis–Hastings corrections by using a Poisson-based estimator of the target ratio under a Lipschitz-like condition |U_i(θ') − U_i(θ)| ≤ c_i M(θ, θ'). These technical assumptions ensure nonnegativity and control of the estimator needed for correctness and efficiency.

The authors present TunaMH‑SGLD as a solution to the long-standing issue of adding an acceptance step to SGLD using minibatches, but note that relaxing the technical assumptions remains unresolved and explicitly mark it as an open question.