We consider joint trajectory generation and tracking control for under-actuated robotic systems. A common solution is to use a layered control architecture, where the top layer uses a simplified model of system dynamics for trajectory generation, and the low layer ensures approximate tracking of this trajectory via feedback control. While such layered control architectures are standard and work well in practice, selecting the simplified model used for trajectory generation typically relies on engineering intuition and experience. In this paper, we propose an alternative data-driven approach to dynamics-aware trajectory generation. We show that a suitable augmented Lagrangian reformulation of a global nonlinear optimal control problem results in a layered decomposition of the overall problem into trajectory planning and feedback control layers. Crucially, the resulting trajectory optimization is dynamics-aware, in that, it is modified with a tracking penalty regularizer encoding the dynamic feasibility of the generated trajectory. We show that this tracking penalty regularizer can be learned from system rollouts for independently-designed low layer feedback control policies, and instantiate our framework in the context of a unicycle and a quadrotor control problem in simulation. Further, we show that our approach handles the sim-to-real gap through experiments on the quadrotor hardware platform without any additional training. For both the synthetic unicycle example and the quadrotor system, our framework shows significant improvements in both computation time and dynamic feasibility in simulation and hardware experiments.

We consider the problem of certifying the robustness of deep neural networks against real-world distribution shifts. To do so, we bridge the gap between hand-crafted specifications and realistic deployment settings by proposing a novel neural-symbolic verification framework, in which we train a generative model to learn perturbations from data and define specifications with respect to the output of the learned model. A unique challenge arising from this setting is that existing verifiers cannot tightly approximate sigmoid activations, which are fundamental to many state-of-the-art generative models. To address this challenge, we propose a general meta-algorithm for handling sigmoid activations which leverages classical notions of counter-example-guided abstraction refinement. The key idea is to "lazily" refine the abstraction of sigmoid functions to exclude spurious counter-examples found in the previous abstraction, thus guaranteeing progress in the verification process while keeping the state-space small. Experiments on the MNIST and CIFAR-10 datasets show that our framework significantly outperforms existing methods on a range of challenging distribution shifts.

We exploit player symmetry to formulate the representation of large normal-form games as a regression task. This formulation allows arbitrary regression methods to be employed in in estimating utility functions from a small subset of the game's outcomes. We demonstrate the applicability both neural networks and Gaussian process regression, but focus on the latter. Once utility functions are learned, computing Nash equilibria requires estimating expected payoffs of pure-strategy deviations from mixed-strategy profiles. Computing these expectations exactly requires an infeasible sum over the full payoff matrix, so we propose and test several approximation methods. Three of these are simple and generic, applicable to any regression method and games with any number of player roles. However, the best performance is achieved by a continuous integral that approximates the summation, which we formulate for the specific case of fully-symmetric games learned by Gaussian process regression with a radial basis function kernel. We demonstrate experimentally that the combination of learned utility functions and expected payoff estimation allows us to efficiently identify approximate equilibria of large games using sparse payoff data.