Amortized mean-shift interacting particles

A new method for Bayesian integral estimation, amortized mean-shift interacting particles, produces weighted node sets from posterior samples in a single forward pass, eliminating the per-observation optimization that previously limited the technique's practicality [1]. The standard approach for evaluating posterior expectations, tail probabilities, and risks in Bayesian inverse problems is a Monte-Carlo average, but its error decays only as the square root of the sample size, making high accuracy computationally prohibitive when each sample requires a partial-differential-equation forward model [2]. Mean-shift interacting particles address this by returning a small set of signed-weight nodes — a deterministic quadrature whose weighted averages estimate the target integrals with far fewer evaluations [2]. The original formulation, however, required a per-observation optimization that, in its most accurate form, read the posterior score at every step, effectively returning the computational cost it was designed to save [3]. The amortized version replaces that optimization with a set-equivariant network that emits the quadrature directly from an observation and a set of independent posterior samples [4]. Training requires only joint parameter-observation samples and a posterior to draw from — a conditional normalizing flow, a physics-based posterior, or an empirical conditional measure — and the map learns to integrate that posterior from samples alone, evaluating neither its density nor its score [4]. Because the network reads its reference only through samples, any posterior the user can sample is admissible [3]. The method improves on independent samples through two mechanisms: reweighting the samples by a closed-form optimum is provably no worse than the equal weights of Monte-Carlo at every budget, while moving the nodes provides the larger empirical gain [2]. The underlying mean-shift interacting particles algorithm extends the classical mean shift procedure, widely used for identifying modes in kernel density estimators, and can be interpreted as a preconditioned gradient descent on the maximum mean discrepancy [5]. Across closed-form, sampled, learned, and physics-based posteriors — including a groundwater field with a thousand coefficients — a single trained map integrates below the Monte-Carlo floor at every node budget [1]. A posterior-whitened, dimension-aware kernel removes the high-dimensional barrier that previously broke the construction [2]. The result is a Pareto improvement on the Monte-Carlo estimator itself, meaning the construction does not compete with drawing more samples but strictly dominates the standard approach [1].