  Model Agnostic Meta Learning (MAML) is widely used to find a good
initialization for a family of tasks. Despite its success, a critical challenge
in MAML is to calculate the gradient w.r.t. the initialization of a long
training trajectory for the sampled tasks, because the computation graph can
rapidly explode and the computational cost is very expensive. To address this
problem, we propose Adjoint MAML (A-MAML). We view gradient descent in the
inner optimization as the evolution of an Ordinary Differential Equation (ODE).
To efficiently compute the gradient of the validation loss w.r.t. the
initialization, we use the adjoint method to construct a companion, backward
ODE. To obtain the gradient w.r.t. the initialization, we only need to run the
standard ODE solver twice -- one is forward in time that evolves a long
trajectory of gradient flow for the sampled task; the other is backward and
solves the adjoint ODE. We need not create or expand any intermediate
computational graphs, adopt aggressive approximations, or impose proximal
regularizers in the training loss. Our approach is cheap, accurate, and
adaptable to different trajectory lengths. We demonstrate the advantage of our
approach in both synthetic and real-world meta-learning tasks.
