Joseph A. Vincent

Joseph A. Vincent, Mac Schwager

Neural networks are increasingly used in robotics as policies, state transition models, state estimation models, or all of the above. With these components being learned from data, it is important to be able to analyze what behaviors were learned and how this affects closed-loop performance. In this paper we take steps toward this goal by developing methods for computing control invariant sets and regions of attraction (ROAs) of dynamical systems represented as neural networks. We focus our attention on feedforward neural networks with the rectified linear unit (ReLU) activation, which are known to implement continuous piecewise-affine (PWA) functions. We describe the Reachable Polyhedral Marching (RPM) algorithm for enumerating the affine pieces of a neural network through an incremental connected walk. We then use this algorithm to compute exact forward and backward reachable sets, from which we provide methods for computing control invariant sets and ROAs. Our approach is unique in that we find these sets incrementally, without Lyapunov-based tools. In our examples we demonstrate the ability of our approach to find non-convex control invariant sets and ROAs on tasks with learned van der Pol oscillator and pendulum models. Further, we provide an accelerated algorithm for computing ROAs that leverages the incremental and connected enumeration of affine regions that RPM provides. We show this acceleration to lead to a 15x speedup in our examples. Finally, we apply our methods to find a set of states that are stabilized by an image-based controller for an aircraft runway control problem.

Joseph A. Vincent, Haruki Nishimura, Masha Itkina et al.

With the rise of stochastic generative models in robot policy learning, end-to-end visuomotor policies are increasingly successful at solving complex tasks by learning from human demonstrations. Nevertheless, since real-world evaluation costs afford users only a small number of policy rollouts, it remains a challenge to accurately gauge the performance of such policies. This is exacerbated by distribution shifts causing unpredictable changes in performance during deployment. To rigorously evaluate behavior cloning policies, we present a framework that provides a tight lower-bound on robot performance in an arbitrary environment, using a minimal number of experimental policy rollouts. Notably, by applying the standard stochastic ordering to robot performance distributions, we provide a worst-case bound on the entire distribution of performance (via bounds on the cumulative distribution function) for a given task. We build upon established statistical results to ensure that the bounds hold with a user-specified confidence level and tightness, and are constructed from as few policy rollouts as possible. In experiments we evaluate policies for visuomotor manipulation in both simulation and hardware. Specifically, we (i) empirically validate the guarantees of the bounds in simulated manipulation settings, (ii) find the degree to which a learned policy deployed on hardware generalizes to new real-world environments, and (iii) rigorously compare two policies tested in out-of-distribution settings. Our experimental data, code, and implementation of confidence bounds are open-source.

Joseph A. Vincent, Mac Schwager

We present a method for computing exact reachable sets for deep neural networks with rectified linear unit (ReLU) activation. Our method is well-suited for use in rigorous safety analysis of robotic perception and control systems with deep neural network components. Our algorithm can compute both forward and backward reachable sets for a ReLU network iterated over multiple time steps, as would be found in a perception-action loop in a robotic system. Our algorithm is unique in that it builds the reachable sets by incrementally enumerating polyhedral cells in the input space, rather than iterating layer-by-layer through the network as in other methods. If an unsafe cell is found, our algorithm can return this result without completing the full reachability computation, thus giving an anytime property that accelerates safety verification. In addition, our method requires less memory during execution compared to existing methods where memory can be a limiting factor. We demonstrate our algorithm on safety verification of the ACAS Xu aircraft advisory system. We find unsafe actions many times faster than the fastest existing method and certify no unsafe actions exist in about twice the time of the existing method. We also compute forward and backward reachable sets for a learned model of pendulum dynamics over a 50 time step horizon in 87s on a laptop computer. Algorithm source code: https://github.com/StanfordMSL/Neural-Network-Reach.

Javier Yu, Joseph A. Vincent, Mac Schwager

We present a distributed algorithm that enables a group of robots to collaboratively optimize the parameters of a deep neural network model while communicating over a mesh network. Each robot only has access to its own data and maintains its own version of the neural network, but eventually learns a model that is as good as if it had been trained on all the data centrally. No robot sends raw data over the wireless network, preserving data privacy and ensuring efficient use of wireless bandwidth. At each iteration, each robot approximately optimizes an augmented Lagrangian function, then communicates the resulting weights to its neighbors, updates dual variables, and repeats. Eventually, all robots' local network weights reach a consensus. For convex objective functions, we prove this consensus is a global optimum. We compare our algorithm to two existing distributed deep neural network training algorithms in (i) an MNIST image classification task, (ii) a multi-robot implicit mapping task, and (iii) a multi-robot reinforcement learning task. In all of our experiments our method out performed baselines, and was able to achieve validation loss equivalent to centrally trained models. See \href{https://msl.stanford.edu/projects/dist_nn_train}{https://msl.stanford.edu/projects/dist\_nn\_train} for videos and a link to our GitHub repository.