We investigate the application of ensemble transform approaches to Bayesian inference of logistic regression problems. Our approach relies on appropriate extensions of the popular ensemble Kalman filter and the feedback particle filter to the cross entropy loss function and is based on a well-established homotopy approach to Bayesian inference. The arising finite particle evolution equations as well as their mean-field limits are affine-invariant. Furthermore, the proposed methods can be implemented in a gradient-free manner in case of nonlinear logistic regression and the data can be randomly subsampled similar to mini-batching of stochastic gradient descent. We also propose a closely related SDE-based sampling method which again is affine-invariant and can easily be made gradient-free. Numerical examples demonstrate the appropriateness of the proposed methodologies.

我们研究了集成变换方法在逻辑回归问题的贝叶斯推理中的应用。我们的方法依赖于流行的集成卡尔曼滤波器和反馈粒子滤波器对交叉熵损失函数的适当扩展，并且基于贝叶斯推理的成熟同伦方法。产生的有限粒子演化方程以及它们的平均场极限是仿射不变的。此外，在非线性逻辑回归的情况下，所提出的方法可以以无梯度方式实现，并且可以对数据进行随机子采样，类似于随机梯度下降的小批量。我们还提出了一种密切相关的基于 SDE 的采样方法，该方法同样是仿射不变的，并且可以很容易地实现无梯度。