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 independently of 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过程计算贝叶斯后验均值的无偏方法，该方法将先进的分裂方法与增强的梯度近似相结合。我们的方法通过在多级蒙特卡洛框架中耦合不同离散化层级的马尔可夫链，避免了Metropolis校正步骤。理论分析表明，我们所提出的估计量是无偏的，具有有限方差，并满足中心极限定理。对于估计d维Lipschitz函数期望值达到精度$\epsilon>0$，该方法仅需$\mathcal{O}(d^{1/4}\epsilon^{-2})$次期望梯度计算，且无需假设热启动条件。我们展示了使用近似梯度和随机梯度时的类似收敛界，并证明该方法的计算成本与数据集规模无关。所提出的方法在MNIST数据集上的多项逻辑回归问题以及足球比分的泊松回归模型中进行了测试。实验表明，即使使用非精确梯度，每个有效样本所需的梯度计算次数也与维度无关。对于乘积分布，我们给出了与维度无关的方差界。我们的结果表明，所提出的无偏算法可比“金标准”随机哈密顿蒙特卡洛方法更为高效。