  Drawing inspiration from gradient-based meta-learning methods with infinitely
small gradient steps, we introduce Continuous-Time Meta-Learning (COMLN), a
meta-learning algorithm where adaptation follows the dynamics of a gradient
vector field. Specifically, representations of the inputs are meta-learned such
that a task-specific linear classifier is obtained as a solution of an ordinary
differential equation (ODE). Treating the learning process as an ODE offers the
notable advantage that the length of the trajectory is now continuous, as
opposed to a fixed and discrete number of gradient steps. As a consequence, we
can optimize the amount of adaptation necessary to solve a new task using
stochastic gradient descent, in addition to learning the initial conditions as
is standard practice in gradient-based meta-learning. Importantly, in order to
compute the exact meta-gradients required for the outer-loop updates, we devise
an efficient algorithm based on forward mode differentiation, whose memory
requirements do not scale with the length of the learning trajectory, thus
allowing longer adaptation in constant memory. We provide analytical guarantees
for the stability of COMLN, we show empirically its efficiency in terms of
runtime and memory usage, and we illustrate its effectiveness on a range of
few-shot image classification problems.
