标题： 在线伯恩斯坦-冯·米塞斯定理 显示英文标题

标题： Online Bernstein-von Mises theorem

摘要： 在线学习是一种推断范式，在这种范式中，参数会根据顺序可用的数据逐步更新，与之相对的是批量学习，后者一次性处理整个数据集。 本文假设完整数据集中的小批量数据按顺序变得可用。 贝叶斯框架在每次观察到一个小批量后更新关于未知参数的信念，这种框架天然适用于在线学习。 在每个步骤中，我们使用当前先验和新观测值来更新后验分布，更新后的后验分布则作为下一个步骤的先验分布。 然而，除非模型和先验是共轭的，否则这种递归贝叶斯更新很少具有计算可行性。 当模型是正则的，更新后的后验可以通过正态分布近似，这由伯恩斯坦-冯·米塞斯定理证明合理。 我们在每个步骤采用变分近似，并研究通过这一序列过程获得的最终后验的频率性质。 在温和的假设下，我们证明，一旦小批量大小超过一个依赖于参数维度的阈值，累积的近似误差就会变得可以忽略不计。 因此，逐步更新的后验渐进地与完整的后验不可区分。 显示英文摘要

摘要： Online learning is an inferential paradigm in which parameters are updated incrementally from sequentially available data, in contrast to batch learning, where the entire dataset is processed at once. In this paper, we assume that mini-batches from the full dataset become available sequentially. The Bayesian framework, which updates beliefs about unknown parameters after observing each mini-batch, is naturally suited for online learning. At each step, we update the posterior distribution using the current prior and new observations, with the updated posterior serving as the prior for the next step. However, this recursive Bayesian updating is rarely computationally tractable unless the model and prior are conjugate. When the model is regular, the updated posterior can be approximated by a normal distribution, as justified by the Bernstein-von Mises theorem. We adopt a variational approximation at each step and investigate the frequentist properties of the final posterior obtained through this sequential procedure. Under mild assumptions, we show that the accumulated approximation error becomes negligible once the mini-batch size exceeds a threshold depending on the parameter dimension. As a result, the sequentially updated posterior is asymptotically indistinguishable from the full posterior.