用正则化随机梯度下降学习算子，解决高维逆问题中的不稳定性。
Learning Operators by Regularized Stochastic Gradient Descent with Operator-valued Kernels
- 基于算子值核的正则化随机梯度下降方法
- 实现无维度依赖的预测与估计误差界，接近最优收敛率
- 适用于结构化预测和参数化偏微分方程，理论可解释性强
我们研究一类统计逆问题，即从波兰空间到可分希尔伯特空间的回归算子估计，目标函数属于由算子值核诱导的向量值再生核希尔伯特空间。为应对此类问题的不适定性，我们在在线与有限时域两种设定下分析了正则化随机梯度下降（SGD）算法：前者采用多项式衰减的步长与正则化参数，后者使用固定值。在合适的结构与分布假设下，我们建立了预测与估计误差的维度无关上界。所得收敛速率在期望意义下近似最优，并推导出高概率估计，表明几乎必然收敛。我们的分析提出了一种在无限维设定中获得高概率保证的通用技术。通过结构化预测与参数化偏微分方程的应用实例，展示了框架的实际适用性。
原文摘要 · Abstract (English)
We consider a class of statistical inverse problems involving the estimation of a regression operator from a Polish space to a separable Hilbert space, where the target lies in a vector-valued reproducing kernel Hilbert space induced by an operator-valued kernel. To address the associated ill-posedness, we analyze regularized stochastic gradient descent (SGD) algorithms in both online and finite-horizon settings. The former uses polynomially decaying step sizes and regularization parameters, while the latter adopts fixed values. Under suitable structural and distributional assumptions, we establish dimension-independent bounds for prediction and estimation errors. The resulting convergence rates are near-optimal in expectation, and we also derive high-probability estimates that imply almost sure convergence. Our analysis introduces a general technique for obtaining high-probability guarantees in infinite-dimensional settings. We illustrate the practical scope of our framework with applications to structured prediction and parametric PDEs, providing examples that reflect how the approach can be applied in practice.
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