  Existing depth separation results for constant-depth networks essentially
show that certain radial functions in $\mathbb{R}^d$, which can be easily
approximated with depth $3$ networks, cannot be approximated by depth $2$
networks, even up to constant accuracy, unless their size is exponential in
$d$. However, the functions used to demonstrate this are rapidly oscillating,
with a Lipschitz parameter scaling polynomially with the dimension $d$ (or
equivalently, by scaling the function, the hardness result applies to
$\mathcal{O}(1)$-Lipschitz functions only when the target accuracy $\epsilon$
is at most $\text{poly}(1/d)$). In this paper, we study whether such depth
separations might still hold in the natural setting of
$\mathcal{O}(1)$-Lipschitz radial functions, when $\epsilon$ does not scale
with $d$. Perhaps surprisingly, we show that the answer is negative: In
contrast to the intuition suggested by previous work, it \emph{is} possible to
approximate $\mathcal{O}(1)$-Lipschitz radial functions with depth $2$, size
$\text{poly}(d)$ networks, for every constant $\epsilon$. We complement it by
showing that approximating such functions is also possible with depth $2$, size
$\text{poly}(1/\epsilon)$ networks, for every constant $d$. Finally, we show
that it is not possible to have polynomial dependence in both $d,1/\epsilon$
simultaneously. Overall, our results indicate that in order to show depth
separations for expressing $\mathcal{O}(1)$-Lipschitz functions with constant
accuracy -- if at all possible -- one would need fundamentally different
techniques than existing ones in the literature.
