  Using backward error analysis, we compute implicit training biases in
multitask and continual learning settings for neural networks trained with
stochastic gradient descent. In particular, we derive modified losses that are
implicitly minimized during training. They have three terms: the original loss,
accounting for convergence, an implicit flatness regularization term
proportional to the learning rate, and a last term, the conflict term, which
can theoretically be detrimental to both convergence and implicit
regularization. In multitask, the conflict term is a well-known quantity,
measuring the gradient alignment between the tasks, while in continual learning
the conflict term is a new quantity in deep learning optimization, although a
basic tool in differential geometry: The Lie bracket between the task
gradients.
