Stratified models are models that depend in an arbitrary way on a set of selected categorical features, and depend linearly on the other features. In a basic and traditional formulation a separate model is fit for each value of the categorical feature, using only the data that has the specific categorical value. To this formulation we add Laplacian regularization, which encourages the model parameters for neighboring categorical values to be similar. Laplacian regularization allows us to specify one or more weighted graphs on the stratification feature values. For example, stratifying over the days of the week, we can specify that the Sunday model parameter should be close to the Saturday and Monday model parameters. The regularization improves the performance of the model over the traditional stratified model, since the model for each value of the categorical `borrows strength' from its neighbors. In particular, it produces a model even for categorical values that did not appear in the training data set. We propose an efficient distributed method for fitting stratified models, based on the alternating direction method of multipliers (ADMM). When the fitting loss functions are convex, the stratified model fitting problem is convex, and our method computes the global minimizer of the loss plus regularization; in other cases it computes a local minimizer. The method is very efficient, and naturally scales to large data sets or numbers of stratified feature values. We illustrate our method with a variety of examples.

Recent work has shown how to embed differentiable optimization problems (that is, problems whose solutions can be backpropagated through) as layers within deep learning architectures. This method provides a useful inductive bias for certain problems, but existing software for differentiable optimization layers is rigid and difficult to apply to new settings. In this paper, we propose an approach to differentiating through disciplined convex programs, a subclass of convex optimization problems used by domain-specific languages (DSLs) for convex optimization. We introduce disciplined parametrized programming, a subset of disciplined convex programming, and we show that every disciplined parametrized program can be represented as the composition of an affine map from parameters to problem data, a solver, and an affine map from the solver’s solution to a solution of the original problem (a new form we refer to as affine-solver-affine form). We then demonstrate how to efficiently differentiate through each of these components, allowing for end-to-end analytical differentiation through the entire convex program. We implement our methodology in version 1.1 of CVXPY, a popular Python-embedded DSL for convex optimization, and additionally implement differentiable layers for disciplined convex programs in PyTorch and TensorFlow 2.0. Our implementation significantly lowers the barrier to using convex optimization problems in differentiable programs. We present applications in linear machine learning models and in stochastic control, and we show that our layer is competitive (in execution time) compared to specialized differentiable solvers from past work.

Models for predicting aircraft motion are an important component of modern aeronautical systems. These models help aircraft plan collision avoidance maneuvers and help conduct off-line performance and safety analyses. In this paper, we develop a method for learning a probabilistic generative model of aircraft motion in terminal airspace, the controlled airspace surrounding a given airport. The method fits the model based on a historical dataset of radar-based position measurements of aircraft landings and takeoffs at that airport. We find that the model generates realistic trajectories, provides accurate predictions, and captures the statistical properties of the aircraft trajectories. Furthermore, the model trains quickly, is compact, and allows for efficient real-time inference.

Recent work [1], [2], [3] has shown promising results in learning from demonstrations for the manipulation of deformable objects. Their approach finds a non-rigid registration between points in the demonstration scene and points in the test scene. This registration is then extrapolated and applied to the gripper motions in the demonstration scene to obtain the gripper motions for the test scene. If more than one demonstration is available, a quality score for the non-rigid registration is used to determine the best matching training scene. For many manipulation tasks, however, the gripper's direction of approach with respect to the objects' surface normals is important in order to succeed at the task. This prior work only registers points across scenes and does not register the surface normals, often leading to warps between scenes that are inappropriate for transfer of manipulation primitives. The main contributions of this paper are (i) An algorithm for non-rigid registration that considers both points and normals, and (ii) An evaluation of this registration approach in the context of learning from demonstrations for robotic manipulation. Our experiments, which consider an insertion task in simulation and also knot-tying and towel-folding executions in a PR2, show that incorporating normals results in improved performance and qualitatively better grasps.