Can stochastic gradient methods track a moving target? We address the problem of tracking multivariate time-varying parameters under noisy observations and potential model misspecification. Specifically, we examine implicit and explicit score-driven (ISD and ESD) filters, which update parameter predictions using the gradient of the logarithmic postulated observation density (commonly referred to as the score). For both filter types, we derive novel sufficient conditions that ensure the exponential stability of the filtered parameter path and the existence of a finite mean squared error (MSE) bound relative to the pseudo-true parameter path. Our (non-)asymptotic MSE bounds rely on mild moment conditions on the data-generating process, while our stability results are agnostic about the true process. For the ISD filter, concavity of the postulated log density combined with simple parameter restrictions is sufficient to guarantee stability. In contrast, the ESD filter additionally requires the score to be Lipschitz continuous and the learning rate to be sufficiently small. We validate our theoretical findings through simulation studies, showing that ISD filters outperform ESD filters in terms of accuracy and stability.

翻译：随机梯度方法能否追踪移动目标？我们研究了在噪声观测和潜在模型误设条件下追踪多元时变参数的问题。具体而言，我们考察了隐式与显式得分驱动（ISD 和 ESD）滤波器，这两种滤波器利用对数假设观测密度（通常称为得分）的梯度来更新参数预测。针对两类滤波器，我们推导了新的充分条件，以确保滤波参数路径的指数稳定性，并证明相对于伪真实参数路径存在有限的均方误差（MSE）界。我们的（非）渐近 MSE 界依赖于数据生成过程的温和矩条件，而稳定性结果不依赖于真实过程。对于 ISD 滤波器，假设对数密度的凹性结合简单的参数约束即足以保证稳定性。相比之下，ESD 滤波器还需满足得分的 Lipschitz 连续性及足够小的学习率。我们通过仿真研究验证了理论结果，表明 ISD 滤波器在精度和稳定性方面优于 ESD 滤波器。