Robert Dyro

Robert Dyro, Edward Schmerling, Nikos Arechiga et al.

In this work we derive a second-order approach to bilevel optimization, a type of mathematical programming in which the solution to a parameterized optimization problem (the "lower" problem) is itself to be optimized (in the "upper" problem) as a function of the parameters. Many existing approaches to bilevel optimization employ first-order sensitivity analysis, based on the implicit function theorem (IFT), for the lower problem to derive a gradient of the lower problem solution with respect to its parameters; this IFT gradient is then used in a first-order optimization method for the upper problem. This paper extends this sensitivity analysis to provide second-order derivative information of the lower problem (which we call the IFT Hessian), enabling the usage of faster-converging second-order optimization methods at the upper level. Our analysis shows that (i) much of the computation already used to produce the IFT gradient can be reused for the IFT Hessian, (ii) errors bounds derived for the IFT gradient readily apply to the IFT Hessian, (iii) computing IFT Hessians can significantly reduce overall computation by extracting more information from each lower level solve. We corroborate our findings and demonstrate the broad range of applications of our method by applying it to problem instances of least squares hyperparameter auto-tuning, multi-class SVM auto-tuning, and inverse optimal control.

Robert Dyro, James Harrison, Apoorva Sharma et al.

As robotic systems move from highly structured environments to open worlds, incorporating uncertainty from dynamics learning or state estimation into the control pipeline is essential for robust performance. In this paper we present a nonlinear particle model predictive control (PMPC) approach to control under uncertainty, which directly incorporates any particle-based uncertainty representation, such as those common in robotics. Our approach builds on scenario methods for MPC, but in contrast to existing approaches, which either constrain all or only the first timestep to share actions across scenarios, we investigate the impact of a \textit{partial consensus horizon}. Implementing this optimization for nonlinear dynamics by leveraging sequential convex optimization, our approach yields an efficient framework that can be tuned to the particular information gain dynamics of a system to mitigate both over-conservatism and over-optimism. We investigate our approach for two robotic systems across three problem settings: time-varying, partially observed dynamics; sensing uncertainty; and model-based reinforcement learning, and show that our approach improves performance over baselines in all settings.