Can stochastic gradient methods track a moving target? We study the problem of tracking multidimensional time-varying parameters under noisy observations and possible model misspecification. Gradient-based filters update the time-varying parameters using the gradient of a postulated objective function. A natural filtering objective is the logarithm of the postulated observation density, which gives rise to the widely used class of score-driven filters. As in the optimization literature, these filters come in two forms: explicit filters evaluate the gradient at the predicted parameter, whereas implicit filters evaluate it at the updated parameter. For both filter types, we derive novel sufficient conditions for exponential stability of the filtered parameter path, showing that stability can be guaranteed independently of the data-generating process. Under mild additional moment conditions on the data-generating process, we also obtain finite-sample and asymptotic mean squared error bounds relative to the pseudo-true parameter path. For implicit filters, these guarantees hold under weak parameter restrictions. For explicit filters, they additionally require Lipschitz continuity of the score and a sufficiently small learning rate. Simulation studies support our theoretical findings and show that implicit gradient filters outperform explicit ones in both accuracy and stability.

翻译：随机梯度方法能否追踪动态目标？我们研究在含噪观测及潜在模型误设下多维时变参数的追踪问题。基于梯度的滤波器利用假定的目标函数梯度更新时变参数，其典型的滤波目标为假定观测密度的对数，由此衍生出广泛使用的得分驱动滤波器类。与优化文献类似，此类滤波器包含两种形式：显式滤波器在预测参数处进行梯度评估，而隐式滤波器则在新参数处进行梯度评估。针对两种滤波器类型，我们推导出滤波参数路径指数稳定性的新型充分条件，证明稳定性可独立于数据生成过程得以保证。在数据生成过程满足温和附加矩条件时，我们还获得了相对于伪真实参数路径的有限样本及渐近均方误差界。对隐式滤波器而言，这些保证在弱参数约束下成立；而对显式滤波器，则需额外满足得分的利普希茨连续性及充分小的学习率。仿真实验支持了我们的理论发现，表明隐式梯度滤波器在精度与稳定性上均优于显式滤波器。