Entropic Mirror Monte Carlo

TL;DR

This work tackles the challenge of efficiently sampling from multimodal, high-dimensional targets in settings where the normalizing constant of the target is unknown. It introduces Entropic Mirror Monte Carlo (EM2C), which combines the contractive Entropic Mirror Descent update with a Markovian exploration kernel to produce a sequence of proposals that converge to the target distribution while effectively exploring complex geometries. The authors establish contraction results for the idealized EM2C mappings, analyze the impact of using the Unadjusted Langevin kernel (including a controlled bias term), and provide a practical Monte Carlo implementation that uses either mixture-model projections via EM or normalizing-flow projections. Numerical experiments on Gaussian mixtures and two-dimensional multimodal geometries demonstrate improved mode recovery and alignment with the target distribution, supporting EM2C’s practical potential for challenging Bayesian and high-dimensional sampling tasks.

Abstract

Importance sampling is a Monte Carlo method which designs estimators of expectations under a target distribution using weighted samples from a proposal distribution. When the target distribution is complex, such as multimodal distributions in highdimensional spaces, the efficiency of importance sampling critically depends on the choice of the proposal distribution. In this paper, we propose a novel adaptive scheme for the construction of efficient proposal distributions. Our algorithm promotes efficient exploration of the target distribution by combining global sampling mechanisms with a delayed weighting procedure. The proposed weighting mechanism plays a key role by enabling rapid resampling in regions where the proposal distribution is poorly adapted to the target. Our sampling algorithm is shown to be geometrically convergent under mild assumptions and is illustrated through various numerical experiments.

Figures (12)

• Figure 1: Intermediate sequence of proposal distributions build with the Entropic Mirror Descent for sampling from the bimodal target distribution $\pi = 0.5 \cdot \mathcal{N}(0,1) + 0.5 \cdot \mathcal{N}(10,1)$ (black solid line) and starting with distribution $\mu_0$ (blue solid line). Top row: theoretical sequence $\{\mu_t\}_{t \geq 0}$ (red dashed line). Bottom row: updates (green dashed line) when using a bimodal variational distributions build upon $5000$ samples from $\{\mu_t\}_{t \geq 0}$.
• Figure 2: Marginal Gaussian mixture distribution $\widetilde{\pi}_i$ associated with the benchmark target distributions $\pi_{i,d}$ as defined in \ref{['eq:target-mixt']}.
• Figure 3: Evolution of EM2C proposal distributions for the GM4 target in dimension $d=10$. Rows correspond to $\lambda \in \{0.5, 0.8, 1.0\}$, columns to EM2C iterations. For visualization, we display the two-dimensional projection onto coordinates $(x_0, x_1)$. Gray and red points denote samples from $\pi$ and $\tilde{\mu}_t$, respectively.
• Figure 4: Evolution of $\mathrm{SW}_2(\mu_t,\pi)$ for the GM4 target ($d=10$), shown for RW and ULA kernels and $\lambda\in\{0.5,0.8,1.0\}$.
• Figure 5: Qualitative comparison of sampling methods on Two-dimensional multimodal targets (top row: dual moons, bottom row: two rings). Gray points denote samples from the target distribution.

Theorems & Definitions (6)

• Proposition 1
• Theorem 2
• Proposition 3
• proof
• Definition 1
• Definition 2