Roland Andrews

Roland Andrews, Justin Carpentier, Ajay Sathya

Imitation learning (IL) is an effective approach to train complex robotics policies. Recent works have introduced hard constraints into imitation-learning optimization problems to ensure safety, stability, and robustness of the learned policy. However, we argue that these constraints are sometimes infeasible, which can lead to unstable or difficult training dynamics. We study a simple remedy for such situations based on recent theoretical results on the augmented Lagrangian method in infeasible settings. We show that our approach drives the learned policy toward the solution of a closest-feasible constrained IL problem with desirable properties. The method is illustrated on a toy driving example with a total-acceleration constraint and pedestrian-safety constraints, a setting in which infeasibility can naturally arise while still allowing a safe learned policy.

Roland Andrews, Justin Carpentier, Adrien Taylor

This work investigates the convergence behavior of augmented Lagrangian methods (ALMs) when applied to convex optimization problems that may be infeasible. ALMs are a popular class of algorithms for solving constrained optimization problems. We demonstrate that, under mild assumptions, the sequences of iterates generated by ALMs converge to solutions of the ``closest feasible problem''. We establish progressively stronger convergence results, ranging from basic sequence convergence to more precise convergence rates, under a hierarchy of assumptions. This study leverages the classical relationship between ALMs and the proximal-point algorithm applied to the dual problem. A key technical contribution is a set of concise results on the behavior of the proximal-point algorithm when applied to functions that may lack minimizers. These results pertain to its convergence in terms of its subgradients and of the values of the convex conjugate.