Laplacian regularized stratified models (LRSM) are models that utilize the explicit or implicit network structure of the sub-problems as defined by the categorical features called strata (e.g., age, region, time, forecast horizon, etc.), and draw upon data from neighboring strata to enhance the parameter learning of each sub-problem. They have been widely applied in machine learning and signal processing problems, including but not limited to time series forecasting, representation learning, graph clustering, max-margin classification, and general few-shot learning. Nevertheless, existing works on LRSM have either assumed a known graph or are restricted to specific applications. In this paper, we start by showing the importance and sensitivity of graph weights in LRSM, and provably show that the sensitivity can be arbitrarily large when the parameter scales and sample sizes are heavily imbalanced across nodes. We then propose a generic approach to jointly learn the graph while fitting the model parameters by solving a single optimization problem. We interpret the proposed formulation from both a graph connectivity viewpoint and an end-to-end Bayesian perspective, and propose an efficient algorithm to solve the problem. Convergence guarantees of the proposed optimization algorithm is also provided despite the lack of global strongly smoothness of the Laplacian regularization term typically required in the existing literature, which may be of independent interest. Finally, we illustrate the efficiency of our approach compared to existing methods by various real-world numerical examples.

翻译：拉普拉斯正则化分层模型（LRSM）是一类利用由分类特征（称为层，如年龄、区域、时间、预测范围等）定义的子问题显式或隐式网络结构，并通过借用相邻层数据来增强每个子问题参数学习的模型。该类模型已广泛应用于机器学习与信号处理问题，包括但不限于时间序列预测、表示学习、图聚类、最大间隔分类以及通用小样本学习。然而，现有关于LRSM的研究要么假设图结构已知，要么局限于特定应用场景。本文首先论证图权重在LRSM中的重要性与敏感性，并通过严格证明表明，当节点间参数尺度与样本量严重不均衡时，该敏感性可能任意大。进而提出一种通用方法，通过求解单一优化问题实现图结构与模型参数的联合学习。我们从图连通性视角和端到端贝叶斯视角两个维度对提出的公式进行解释，并设计高效算法求解该问题。尽管拉普拉斯正则化项缺乏现有文献通常要求的全局强光滑性，我们仍给出了所提优化算法的收敛性保证，该分析本身可能具有独立价值。最后，通过多个真实世界数值案例，与现有方法相比验证了本方法的有效性。