Given two Deep Neural Network (DNN) classifiers with the same input and output domains, our goal is to quantify the robustness of the two networks in relation to each other. Towards this, we introduce the notion of Relative Safety Margins (RSMs). Intuitively, given two classes and a common input, RSM of one classifier with respect to another reflects the relative margins with which decisions are made. The proposed notion is relevant in the context of several applications domains, including to compare a trained network and its corresponding compact network (e.g., pruned, quantized, distilled network). Not only can RSMs establish whether decisions are preserved, but they can also quantify their qualities. We also propose a framework to establish safe bounds on RSM gains or losses given an input and a family of perturbations. We evaluate our approach using the MNIST, CIFAR10, and two real-world medical datasets, to show the relevance of our results.

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.