| Summary: | This paper presents fast non-sampling based methods to assess the risk of
trajectories for autonomous vehicles when probabilistic predictions of other
agents' 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 Chebyshev'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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