We study the decentralized online regularized linear regression algorithm over random time-varying graphs. At each time step, every node runs an online estimation algorithm consisting of an innovation term processing its own new measurement, a consensus term taking a weighted sum of estimations of its own and its neighbors with additive and multiplicative communication noises and a regularization term preventing over-fitting. It is not required that the regression matrices and graphs satisfy special statistical assumptions such as mutual independence, spatio-temporal independence or stationarity. We develop the nonnegative supermartingale inequality of the estimation error, and prove that the estimations of all nodes converge to the unknown true parameter vector almost surely if the algorithm gains, graphs and regression matrices jointly satisfy the sample path spatio-temporal persistence of excitation condition. Especially, this condition holds by choosing appropriate algorithm gains if the graphs are uniformly conditionally jointly connected and conditionally balanced, and the regression models of all nodes are uniformly conditionally spatio-temporally jointly observable, under which the algorithm converges in mean square and almost surely. In addition, we prove that the regret upper bound is $O(T^{1-\tau}\ln T)$, where $\tau\in (0.5,1)$ is a constant depending on the algorithm gains.

翻译：我们研究随机时变图上的分散式在线正则化线性回归算法。在每个时间步，每个节点运行一个在线估计算法，该算法包含：处理自身新观测值的创新项、对自身及邻居估计值进行加权求和（含加性与乘性通信噪声）的共识项，以及防止过拟合的正则化项。该方法不要求回归矩阵和图满足诸如相互独立性、时空独立性或平稳性等特殊统计假设。我们推导了估计误差的非负上鞅不等式，并证明：当算法增益、图和回归矩阵联合满足样本路径时空持续激励条件时，所有节点的估计值几乎必然收敛至未知真实参数向量。特别地，若图满足一致条件联合连通性与条件平衡性，且所有节点的回归模型满足一致条件时空联合可观测性，则通过选择合适的算法增益可使该条件成立，此时算法在均方意义下及几乎必然收敛。此外，我们证明了遗憾上界为$O(T^{1-\tau}\ln T)$，其中$\tau\in(0.5,1)$为依赖于算法增益的常数。