  The question of how and why the phenomenon of mode connectivity occurs in
training deep neural networks has gained remarkable attention in the research
community. From a theoretical perspective, two possible explanations have been
proposed: (i) the loss function has connected sublevel sets, and (ii) the
solutions found by stochastic gradient descent are dropout stable. While these
explanations provide insights into the phenomenon, their assumptions are not
always satisfied in practice. In particular, the first approach requires the
network to have one layer with order of $N$ neurons ($N$ being the number of
training samples), while the second one requires the loss to be almost
invariant after removing half of the neurons at each layer (up to some
rescaling of the remaining ones). In this work, we improve both conditions by
exploiting the quality of the features at every intermediate layer together
with a milder over-parameterization condition. More specifically, we show that:
(i) under generic assumptions on the features of intermediate layers, it
suffices that the last two hidden layers have order of $\sqrt{N}$ neurons, and
(ii) if subsets of features at each layer are linearly separable, then no
over-parameterization is needed to show the connectivity. Our experiments
confirm that the proposed condition ensures the connectivity of solutions found
by stochastic gradient descent, even in settings where the previous
requirements do not hold.
