  This paper focuses on training implicit models of infinite layers.
Specifically, previous works employ implicit differentiation and solve the
exact gradient for the backward propagation. However, is it necessary to
compute such an exact but expensive gradient for training? In this work, we
propose a novel gradient estimate for implicit models, named phantom gradient,
that 1) forgoes the costly computation of the exact gradient; and 2) provides
an update direction empirically preferable to the implicit model training. We
theoretically analyze the condition under which an ascent direction of the loss
landscape could be found, and provide two specific instantiations of the
phantom gradient based on the damped unrolling and Neumann series. Experiments
on large-scale tasks demonstrate that these lightweight phantom gradients
significantly accelerate the backward passes in training implicit models by
roughly 1.7 times, and even boost the performance over approaches based on the
exact gradient on ImageNet.
