We study the generalization error of statistical learning algorithms in a
non-i.i.d. setting, where the training data is sampled from a stationary mixing
process. We develop an analytic framework for this scenario based on a
reduction to online learning with delayed feedback. In particular, we show that
the existence of an online learning algorithm with bounded regret (against a
fixed statistical learning algorithm in a specially constructed game of online
learning with delayed feedback) implies low generalization error of said
statistical learning method even if the data sequence is sampled from a mixing
time series. The rates demonstrate a trade-off between the amount of delay in
the online learning game and the degree of dependence between consecutive data
points, with near-optimal rates recovered in a number of well-studied settings
when the delay is tuned appropriately as a function of the mixing time of the
process.

通过将训练数据从一个混合过程进行采样，我们研究了统计学习算法在非独立同分布的环境中的泛化误差，并基于延迟反馈的在线学习提出了一个分析框架。特别地，我们展示了即使数据序列是从混合时间序列采样得到的情况下，通过存在一个具有有界遗憾的在线学习算法（针对一个固定的统计学习算法，在一个特殊构建的延迟反馈的在线学习博弈中），所述统计学习方法的泛化误差较低。这些速率展示了在线学习博弈中的延迟量和连续数据点之间的依赖程度之间的权衡，当延迟适当地调整为过程的混合时间的函数时，可以在许多研究良好的情形下恢复近乎最优的速率。

通过延迟的在线到 PAC 转换推导混合过程的归纳界

Generalization bounds for mixing processes via delayed online-to-PAC  conversions

The Riemannian geometry of covariance matrices has been essential to several
successful applications, in computer vision, biomedical signal and image
processing, and radar data processing. For these applications, an important
ongoing challenge is to develop Riemannian-geometric tools which are adapted to
structured covariance matrices. The present paper proposes to meet this
challenge by introducing a new class of probability distributions, Gaussian
distributions of structured covariance matrices. These are Riemannian analogs
of Gaussian distributions, which only sample from covariance matrices having a
preassigned structure, such as complex, Toeplitz, or block-Toeplitz. The
usefulness of these distributions stems from three features: (1) they are
completely tractable, analytically or numerically, when dealing with large
covariance matrices, (2) they provide a statistical foundation to the concept
of structured Riemannian barycentre (i.e. Fr\'echet or geometric mean), (3)
they lead to efficient statistical learning algorithms, which realise, among
others, density estimation and classification of structured covariance
matrices. The paper starts from the observation that several spaces of
structured covariance matrices, considered from a geometric point of view, are
Riemannian symmetric spaces. Accordingly, it develops an original theory of
Gaussian distributions on Riemannian symmetric spaces, of their statistical
inference, and of their relationship to the concept of Riemannian barycentre.
Then, it uses this original theory to give a detailed description of Gaussian
distributions of three kinds of structured covariance matrices, complex,
Toeplitz, and block-Toeplitz. Finally, it describes algorithms for density
estimation and classification of structured covariance matrices, based on
Gaussian distribution mixture models.

本文提出了一种新的概率分布 —— 结构化协方差矩阵的高斯分布，以应对在计算机视觉、生物医学信号和图像处理、雷达数据处理中使用 Riemannian 几何技术处理结构化协方差矩阵的挑战。通过在 Riemannian 对称空间上发展高斯分布的原理和应用，提供了一种建模方法和统计学基础，并提出了密度估计和分类等有效的统计学习算法。