  We consider the problem of engineering robust direct perception neural
networks with output being regression. Such networks take high dimensional
input image data, and they produce affordances such as the curvature of the
upcoming road segment or the distance to the front vehicle. Our proposal starts
by allowing a neural network prediction to deviate from the label with
tolerance $\Delta$. The source of tolerance can be either contractual or from
limiting factors where two entities may label the same data with slightly
different numerical values. The tolerance motivates the use of a non-standard
loss function where the loss is set to $0$ so long as the prediction-to-label
distance is less than $\Delta$. We further extend the loss function and define
a new provably robust criterion that is parametric to the allowed output
tolerance $\Delta$, the layer index $\tilde{l}$ where perturbation is
considered, and the maximum perturbation amount $\kappa$. During training, the
robust loss is computed by first propagating symbolic errors from the
$\tilde{l}$-th layer (with quantity bounded by $\kappa$) to the output layer,
followed by computing the overflow between the error bounds and the allowed
tolerance. The overall concept is experimented in engineering a direct
perception neural network for understanding the central position of the
ego-lane in pixel coordinates.
