We present an unbiased method for Bayesian posterior means based on kinetic Langevin dynamics that combines advanced splitting methods with enhanced gradient approximations. Our approach avoids Metropolis correction by coupling Markov chains at different discretization levels in a multilevel Monte Carlo approach. Theoretical analysis demonstrates that our proposed estimator is unbiased, attains finite variance, and satisfies a central limit theorem. It can achieve accuracy $\epsilon>0$ for estimating expectations of Lipschitz functions in $d$ dimensions with $\mathcal{O}(d^{1/4}\epsilon^{-2})$ expected gradient evaluations, without assuming warm start. We exhibit similar bounds using both approximate and stochastic gradients, and our method's computational cost is shown to scale logarithmically with the size of the dataset. The proposed method is tested using a multinomial regression problem on the MNIST dataset and a Poisson regression model for soccer scores. Experiments indicate that the number of gradient evaluations per effective sample is independent of dimension, even when using inexact gradients. For product distributions, we give dimension-independent variance bounds. Our results demonstrate that the unbiased algorithm we present can be much more efficient than the ``gold-standard" randomized Hamiltonian Monte Carlo.

翻译：我们提出了一种基于动力学朗之万动力学的贝叶斯后验均值无偏估计方法，该方法将先进的分裂技术与增强型梯度逼近相结合。我们的方法通过在多级蒙特卡洛框架中耦合不同离散化水平的马尔可夫链，避免了梅特罗波利斯校正步骤。理论分析表明，所提出的估计量具有无偏性、有限方差，并满足中心极限定理。对于d维空间中Lipschitz函数期望的估计，该方法可在无需热启动的条件下，以$\mathcal{O}(d^{1/4}\epsilon^{-2})$的期望梯度评估次数达到精度$\epsilon>0$。我们展示了在近似梯度和随机梯度下均有类似的界，且方法计算成本随数据集规模呈对数增长。通过MNIST数据集上的多项逻辑回归问题和足球比分泊松回归模型对方法进行测试，实验表明即使使用非精确梯度，每个有效样本的梯度评估次数与维度无关。对于乘积分布，我们给出了与维度无关的方差界。结果表明，本文提出的无偏算法可显著优于"黄金标准"随机化哈密顿蒙特卡洛方法。