The supplementary file is organized as follows: Section 1 restates the assumptions and main theorems on the convergence of parameter iterates and the full gradient; Section 2 is devoted to the proofs of the two main theorems, while Section 3 includes the proofs of supporting lemmas; Section 4 includes additional figures from the numerical study. Under Assumptions 1.1 to 1.3, when m > C for some constant C > 0, we have the following results under two corresponding conditions on sl(m): First we present the following lemma, showing that the loss function has a property similar from strong convexity. For the first case discussed in Lemma 2.1, define eg(θ(k)) = (g(θ(k)))2, and for the second case define eg(θ(k)) = g(θ(k)). Therefore, combining Lemma 2.1, Lemma 2.2 and (7) leads to the following conclusion. Apply(15)inLemma 2.3 with = 12, then for any 0<α<1, with probability at least 1 2m α, we have A11 1 Under this case, we can still apply (15) in Lemma 2.3.

In particular, using stochastic gradients in MAML update steps is crucial for RL problems since computation of exact gradients requires access to a large number of possible trajectories.

A.1 ImplementationDetails Network Architecture: Inspired by [33], we utilize a pre-trained ResNet-50 [20] as the feature extractor for object recognition tasks (i.e., Office-31 [22], Office-Caltech [18] and Office-Home [46]). Theoverallframeworkis trained under an end-to-end manner via back-propagation. The stochastic gradient descent with momentum value as 0.9 is employed as the network optimizer. The initial learning rates for feature extractor and bottleneck layer are respectively set as 10 3 and 10 2, while the parameters of classifier are frozen. It is exponentially decayed as the training process.