  This paper presents fast non-sampling based methods to assess the risk for
trajectories of 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 to predict both agent
positions and control inputs conditioned on the scene contexts. 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 nonlinear 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 higher order
statistical moments of agent positions to determine upper bounds on risk. To
perform risk assessment when models are learned for agent control inputs as
opposed to positions, we propagate the moments of uncertain control inputs
through the nonlinear motion dynamics to obtain the exact moments of uncertain
position over the planning horizon. To this end, we construct deterministic
linear dynamical systems that govern the exact time evolution of the moments of
uncertain position in the presence of uncertain control inputs. 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.
