  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.
