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Title: Meta-Learning and Universality: Deep Representations and Gradient Descent can Approximate any Learning Algorithm

Paper content:
Learning to learn is a powerful paradigm for enabling models to learn from data more effectively and efficiently.
A popular approach to meta-learning is to train a recurrent model to read in a training dataset as input and output the parameters of a learned model, or output predictions for new test inputs.
Alternatively, a more recent approach to meta-learning aims to acquire deep representations that can be effectively fine-tuned, via standard gradient descent, to new tasks.
In this paper, we consider the meta-learning problem from the perspective of universality, formalizing the notion of learning algorithm approximation and comparing the expressive power of the aforementioned recurrent models to the more recent approaches that embed gradient descent into the meta-learner.
In particular, we seek to answer the following question: does deep representation combined with standard gradient descent have sufficient capacity to approximate any learning algorithm?
We find that this is indeed true, and further find, in our experiments, that gradient-based meta-learning consistently leads to learning strategies that generalize more widely compared to those represented by recurrent models.
Deep neural networks that optimize for effective representations have enjoyed tremendous success over human-engineered representations.
Meta-learning takes this one step further by optimizing for a learning algorithm that can effectively acquire representations.
A common approach to metalearning is to train a recurrent or memory-augmented model such as a recurrent neural network to take a training dataset as input and then output the parameters of a learner model (Schmidhuber, 1987; Bengio et al., 1992; Li & Malik, 2017a; BID0 .
Alternatively, some approaches pass the dataset and test input into the model, which then outputs a corresponding prediction for the test example (Santoro et al., 2016; Duan et al., 2016; Wang et al., 2016; Mishra et al., 2018) .
Such recurrent models are universal learning procedure approximators, in that they have the capacity to approximately represent any mapping from dataset and test datapoint to label.
However, depending on the form of the model, it may lack statistical efficiency.In contrast to the aforementioned approaches, more recent work has proposed methods that include the structure of optimization problems into the meta-learner (Ravi & Larochelle, 2017; Finn et al., 2017a; Husken & Goerick, 2000) .
In particular, model-agnostic meta-learning (MAML) optimizes only for the initial parameters of the learner model, using standard gradient descent as the learner's update rule (Finn et al., 2017a) .
Then, at meta-test time, the learner is trained via gradient descent.
By incorporating prior knowledge about gradient-based learning, MAML improves on the statistical efficiency of black-box meta-learners and has successfully been applied to a range of meta-learning problems (Finn et al., 2017a; b; Li et al., 2017) .
But, does it do so at a cost?
A natural question that arises with purely gradient-based meta-learners such as MAML is whether it is indeed sufficient to only learn an initialization, or whether representational power is in fact lost from not learning the update rule.
Intuitively, we might surmise that learning an update rule is more expressive than simply learning an initialization for gradient descent.
In this paper, we seek to answer the following question: does simply learning the initial parameters of a deep neural network have the same representational power as arbitrarily expressive meta-learners that directly ingest the training data at meta-test time?
Or, more concisely, does representation combined with standard gradient descent have sufficient capacity to constitute any learning algorithm?We
analyze this question from the standpoint of the universal function approximation theorem. We
compare the theoretical representational capacity of the two meta-learning approaches: a deep network updated with one gradient step, and a meta-learner that directly ingests a training set and test input and outputs predictions for that test input (e.g. using a recurrent neural network). In
studying the universality of MAML, we find that, for a sufficiently deep learner model, MAML has the same theoretical representational power as recurrent meta-learners. We
therefore conclude that, when using deep, expressive function approximators, there is no theoretical disadvantage in terms of representational power to using MAML over a black-box meta-learner represented, for example, by a recurrent network.Since MAML has the same representational power as any other universal meta-learner, the next question we might ask is: what is the benefit of using MAML over any other approach? We
study this question by analyzing the effect of continuing optimization on MAML performance. Although
MAML optimizes a network's parameters for maximal performance after a fixed small number of gradient steps, we analyze the effect of taking substantially more gradient steps at meta-test time. We find
that initializations learned by MAML are extremely resilient to overfitting to tiny datasets, in stark contrast to more conventional network initialization, even when taking many more gradient steps than were used during meta-training. We also
find that the MAML initialization is substantially better suited for extrapolation beyond the distribution of tasks seen at meta-training time, when compared to meta-learning methods based on networks that ingest the entire training set. We analyze
this setting empirically and provide some intuition to explain this effect.
In this paper, we show that there exists a form of deep neural network such that the initial weights combined with gradient descent can approximate any learning algorithm.
Our findings suggest that, from the standpoint of expressivity, there is no theoretical disadvantage to embedding gradient descent into the meta-learning process.
In fact, in all of our experiments, we found that the learning strategies acquired with MAML are more successful when faced with out-of-domain tasks compared to recurrent learners.
Furthermore, we show that the representations acquired with MAML are highly resilient to overfitting.
These results suggest that gradient-based meta-learning has a num-ber of practical benefits, and no theoretical downsides in terms of expressivity when compared to alternative meta-learning models.
Independent of the type of meta-learning algorithm, we formalize what it means for a meta-learner to be able to approximate any learning algorithm in terms of its ability to represent functions of the dataset and test inputs.
This formalism provides a new perspective on the learning-to-learn problem, which we hope will lead to further discussion and research on the goals and methodology surrounding meta-learning.
While there are likely a number of ways to prove Lemma 4.1 (copied below for convenience), here we provide a simple, though inefficient, proof of Lemma 4.1.Lemma 4.1 Let us assume that e(y) can be chosen to be any linear (but not affine) function of y.
Then, we can choose θ ft , θ h , {A i ; i > 1}, {B i ; i < N } such that the function DISPLAYFORM0 can approximate any continuous function of (x, y, x ) on compact subsets of R dim(y) .

assistant: text: Deep representations combined with gradient descent can approximate any learning algorithm.