  We show that learning can be improved by using loss functions that evolve
cyclically during training to emphasize one class at a time. In
underparameterized networks, such dynamical loss functions can lead to
successful training for networks that fail to find a deep minima of the
standard cross-entropy loss. In overparameterized networks, dynamical loss
functions can lead to better generalization. Improvement arises from the
interplay of the changing loss landscape with the dynamics of the system as it
evolves to minimize the loss. In particular, as the loss function oscillates,
instabilities develop in the form of bifurcation cascades, which we study using
the Hessian and Neural Tangent Kernel. Valleys in the landscape widen and
deepen, and then narrow and rise as the loss landscape changes during a cycle.
As the landscape narrows, the learning rate becomes too large and the network
becomes unstable and bounces around the valley. This process ultimately pushes
the system into deeper and wider regions of the loss landscape and is
characterized by decreasing eigenvalues of the Hessian. This results in better
regularized models with improved generalization performance.
