动态调整滤波器更新强度，让模型自适应学习最佳变化速度。
Online Learning of Scale Parameters in Score-Driven Filters
- 将更新幅度设为可学变量，通过预测下一状态选择最优调节参数。
- 在多危机市场中表现优于固定参数，避免极端波动且稳定性更强。
- 适合需要自适应时变参数的金融时间序列建模场景。
Score-driven 滤波器通过乘以控制更新幅度的尺度参数（称作 gain）来更新时变参数。我们把 gain 视为决策变量并研究其在线学习。在给定当前状态、观测值、得分和缩放规则下，每个合法 gain 会诱导一个可达的下一状态和一步前瞻预测密度；标量 gain 对应沿直线的步长，对角 gain 则对应坐标方向上的传输速率，可能改变方向。gain 选择成为一个以 Kullback-Leibler 为目标的条件一步预测决策问题。核心观察是：加速 score-driven 迭代中使用的负得分乘积反馈，可被解读为该预测损失的随机梯度，从而提供新的变分视角。因此，自适应 gain 学习可视为在线预测问题，当前得分作为上下文来预测下一个 gain。单调可微 gain 链接在有界 gain 域上诱导镜面下降几何，而持续性则产生对参考 gain 的 Bregman 拉力。在凸性、紧致性和正则性条件下，我们建立了投影与折扣镜面更新相对于时变、基于当前信息比较器的动态后悔界。模拟实验展示了缩放、链接几何、持续性和坐标传输率的作用。一个包含股票指数波动率的样本外面板显示，有界镜面 gain 通常匹配或超越恒定 gain，同时避免了无界指数链接的极端尖峰，尤其在多重危机市场中改进最显著。
原文摘要 · Abstract (English)
Score-driven filters update a time-varying parameter by multiplying a scaled log-likelihood score by a scale parameter that controls the magnitude of the update. We name this scale parameter gain, consider it a decision variable, and study its online learning. Conditional on the current state, observation, score, and scaling rule, each admissible gain induces a reachable next state and a one-step-ahead predictive density; a scalar gain selects distance along a line, whereas a diagonal gain selects coordinatewise transmission rates and may change direction. Gain selection becomes a conditional one-step predictive decision problem with a Kullback-Leibler objective. Our central observation is that the negative product-of-scores feedback employed in accelerated score-driven recursions can be read as the stochastic gradient of this predictive loss, offering a new variational perspective. Adaptive gain learning can therefore be viewed as an online prediction problem, where the current score provides the context for predicting the next gain. Monotone differentiable gain links induce mirror-descent geometries on bounded gain domains, while persistence yields a Bregman pull towards a reference gain. Under convexity, compactness, and regularity conditions, we establish dynamic-regret bounds for projected and discounted mirror updates relative to time-varying, current-information comparators. Simulations illustrate the roles of scaling, link geometry, persistence, and coordinatewise transmission rates. An out-of-sample panel of equity-index volatilities shows that the bounded mirror gain generally matches or outperforms a constant gain, while avoiding the extreme spikes of an unbounded exponential link, with the strongest improvements observed in multi-crisis markets.
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