Title: Representing smooth functions as compositions of near-identity functions with implications for deep network optimization,
Abstract: We show that any smooth bi-Lipschitz $h$ can be represented exactly as a
composition $h_m \circ ... \circ h_1$ of functions $h_1,...,h_m$ that are close
to the identity in the sense that each $\left(h_i-\mathrm{Id}\right)$ is
Lipschitz, and the Lipschitz constant decreases inversely with the number $m$
of functions composed. This implies that $h$ can be represented to any accuracy
by a deep residual network whose nonlinear layers compute functions with a
small Lipschitz constant. Next, we consider nonlinear regression with a
composition of near-identity nonlinear maps. We show that, regarding Fréchet
derivatives with respect to the $h_1,...,h_m$, any critical point of a
quadratic criterion in this near-identity region must be a global minimizer. In
contrast, if we consider derivatives with respect to parameters of a fixed-size
residual network with sigmoid activation functions, we show that there are
near-identity critical points that are suboptimal, even in the realizable case.
Informally, this means that functional gradient methods for residual networks
cannot get stuck at suboptimal critical points corresponding to near-identity
layers, whereas parametric gradient methods for sigmoidal residual networks
suffer from suboptimal critical points in the near-identity region.