判别式贝叶斯滤波为最小化对数凸函数的随机牛顿法提供动量
Discriminative Bayesian filtering lends momentum to the stochastic Newton method for minimizing log-convex functions
AI总结：
该研究将最小化对数凸函数平均值的随机牛顿优化问题转化为判别式观测的潜状态空间序列贝叶斯推断问题，提出结合梯度与海森矩阵历史的贝叶斯滤波优化算法，推导了类重球动量的旧观测衰减条件，并辅以示例阐释。
摘要：
为最小化一组对数凸函数的平均值，随机牛顿法利用全目标函数梯度和海森矩阵的子采样版本，迭代更新其估计值。我们将该优化问题置于潜在状态空间模型的序列贝叶斯推断框架下，该模型采用判别式指定的观测过程。应用贝叶斯滤波后，得到一种新型优化算法，该算法在形成更新时会考虑梯度和海森矩阵的全部历史信息。我们建立了基于矩阵的条件，在此条件下，较早观测的影响会随时间减弱，其方式类似于Polyak的重球动量。我们通过一个示例阐释了方法的各个方面，并回顾了随机牛顿法的其他相关创新。
英文摘要：
To minimize the average of a set of log-convex functions, the stochastic Newton method iteratively updates its estimate using subsampled versions of the full objective's gradient and Hessian. We contextualize this optimization problem as sequential Bayesian inference on a latent state-space model with a discriminatively-specified observation process. Applying Bayesian filtering then yields a novel optimization algorithm that considers the entire history of gradients and Hessians when forming an update. We establish matrix-based conditions under which the effect of older observations diminishes over time, in a manner analogous to Polyak's heavy ball momentum. We illustrate various aspects of our approach with an example and review other relevant innovations for the stochastic Newton method.
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