*Allen Wang (MIT); Xin Huang (MIT); Ashkan Jasour (MIT); Brian Williams (Massachusetts Institute of Technology)*

This paper presents fast non-sampling based methods to assess the risk of trajectories for autonomous vehicles when probabilistic predictions of other agentsâ€™ futures are generated by deep neural networks (DNNs). The presented methods address a wide range of representations for uncertain predictions including both Gaussian and non-Gaussian mixture models for predictions of both agent positions and controls. We show that the problem of risk assessment when Gaussian mixture models (GMMs) of agent positions are learned can be solved rapidly to arbitrary levels of accuracy with existing numerical methods. To address the problem of risk assessment for non-Gaussian mixture models of agent position, we propose finding upper bounds on risk using Chebyshevâ€™s Inequality and sums-of-squares (SOS) programming; they are both of interest as the former is much faster while the latter can be arbitrarily tight. These approaches only require statistical moments of agent positions to determine upper bounds on risk. To perform risk assessment when models are learned for agent controls as opposed to positions, we develop TreeRing, an algorithm analogous to tree search over the ring of polynomials that can be used to exactly propagate moments of control distributions into position distributions through nonlinear dynamics. The presented methods are demonstrated on realistic predictions from DNNs trained on the Argoverse and CARLA datasets and are shown to be effective for rapidly assessing the probability of low probability events.

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07/16 15:00 UTC | 07/16 17:00 UTC |

The core contribution of this paper is a method for rapidly making predictions for whether or not a particular self-driving car (SDC) trajectory will collide with another vehicle or a pedestrian, under a Gaussian or non-Gaussian assumption. Second, they develop TreeRing, a tree-search-like algorithm for computing probabilities of rate events, so that they can use non-Gaussian models of probability. Finally, they apply to deep neural network models trained on the Argoverse and CARLA datasets. I think these are strong contributions that could be useful in a variety of real-world robot applications in the future. Estimates of these sorts of rare-event probabilities seem extremely important in any situation where robots will coexist with humans. ------------------- This paper summarizes methods for fast estimation of collision probabilities from either Gaussian or non-Gaussian models. A collision is considered to be any time when another agent enters an ellipse around the vehicle center. I think the paper is largely well-written, but I had a few notable issues. In particular, Sec. IV is on risk assessment, and dives into the moment-based SoS methods that this approach is based on. The primary goal is to discuss how these moments are computed, in (A) for new reference frames. Throughout this section I was a bit confused about *what* exactly these moments were, or how we compute them from our observations of other actors in the scene. IV.B seems to consist of a relatively straightforward equation for agent risk, and then a lit of references we could use to solve this expression. Fig. 1 is a nice illustration of the sorts of multi-modal predictions we want to analyze, but is a bit hard to read. Why isn't the ego vehicle trajectory directional? It's hard to understand how we expect time to be flowing or what we expect to happen. Why use red squares for agent observations? I would have apreciated the examples from Sec. V being applied earlier and carried through the paper, just to give me something more concrete to follow. I also think it's hard to keep track of the full list of assumptions the authors are making: - elliptical collision detection region - characteristic functions of controls - moments up to some order The experiments support the main thesis of the paper, but I had some reservations. They compare two methods with Monte Carlo simulations and show that they are able to compute error in less time, and with lower maximum relative error. Authors should include error bars on these means, since they're dealing with 170 scenarios. I'm also suspicious of the fact that the authors used Imhof -- one of their proposed methods -- as ground truth. Of course this method had zero error. Why can't they use observed trajectories from the dataset? I also found the TreeRing results hard to interpret. All in all, I think this paper has a lot of interesting ideas but its clarity and experiments could be improved. Side note, please compile the LaTeX in your supplement. It's very hard to read, and I'm not sure what I'm supposed to be getting out of it. You mention the appendix but not the supplement in the text. Minor: pg. 3: "drivers high level" --> "driver's high-level" pg. 3: "across a $n$ node trajectory" --> "... an $n$ node..." pg. 7: "outcomes becomes is" --> "outcomes are"