  We investigate a sequential optimization procedure to minimize the empirical
risk functional $f_{\hat\theta}(x) = \frac{1}{2}\|G_{\hat\theta}(x) - y\|^2$
for certain families of deep networks $G_{\theta}(x)$. The approach is to
optimize a sequence of objective functions that use network parameters obtained
during different stages of the training process. When initialized with random
parameters $\theta_0$, we show that the objective $f_{\theta_0}(x)$ is "nice''
and easy to optimize with gradient descent. As learning is carried out, we
obtain a sequence of generative networks $x \mapsto G_{\theta_t}(x)$ and
associated risk functions $f_{\theta_t}(x)$, where $t$ indicates a stage of
stochastic gradient descent during training. Since the parameters of the
network do not change by very much in each step, the surface evolves slowly and
can be incrementally optimized. The algorithm is formalized and analyzed for a
family of expansive networks. We call the procedure {\it surfing} since it
rides along the peak of the evolving (negative) empirical risk function,
starting from a smooth surface at the beginning of learning and ending with a
wavy nonconvex surface after learning is complete. Experiments show how surfing
can be used to find the global optimum and for compressed sensing even when
direct gradient descent on the final learned network fails.
