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移动目标？我们研究在含噪声观测及潜在模型误设定条件下，对多维时变参数的追踪问题。基于梯度的滤波器通过假定的目标函数梯度更新时变参数。自然的滤波目标为假定观测密度的对数函数，这催生了广泛使用的分数驱动滤波器族。与优化文献类似，此类滤波器存在两种形式：显式滤波器在预测参数处计算梯度，而隐式滤波器则在更新后参数处计算梯度。针对两类滤波器，我们推导出滤波参数轨迹指数稳定性的新颖充分条件，证明稳定性可独立于数据生成过程得到保证。在数据生成过程满足温和附加矩条件时，我们还获得关于伪真实参数轨迹的有限样本与渐近均方误差界。对于隐式滤波器，这些保证在弱参数约束下成立；对于显式滤波器，则需额外满足分数函数的利普希茨连续性及充分小的学习率。仿真实验支持我们的理论发现，并表明隐式梯度滤波器在精度与稳定性两方面均优于显式梯度滤波器。