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Title: Recasting Gradient-Based Meta-Learning as Hierarchical Bayes

Paper content:
Meta-learning allows an intelligent agent to leverage prior learning episodes as a basis for quickly improving performance on a novel task.
Bayesian hierarchical modeling provides a theoretical framework for formalizing meta-learning as inference for a set of parameters that are shared across tasks.
Here, we reformulate the model-agnostic meta-learning algorithm (MAML) of Finn et al. (2017) as a method for probabilistic inference in a hierarchical Bayesian model.
In contrast to prior methods for meta-learning via hierarchical Bayes, MAML is naturally applicable to complex function approximators through its use of a scalable gradient descent procedure for posterior inference.
Furthermore, the identification of MAML as hierarchical Bayes provides a way to understand the algorithm’s operation as a meta-learning procedure, as well as an opportunity to make use of computational strategies for efficient inference.
We use this opportunity to propose an improvement to the MAML algorithm that makes use of techniques from approximate inference and curvature estimation.
A remarkable aspect of human intelligence is the ability to quickly solve a novel problem and to be able to do so even in the face of limited experience in a novel domain.
Such fast adaptation is made possible by leveraging prior learning experience in order to improve the efficiency of later learning.
This capacity for meta-learning also has the potential to enable an artificially intelligent agent to learn more efficiently in situations with little available data or limited computational resources BID45 BID4 BID37 .In
machine learning, meta-learning is formulated as the extraction of domain-general information that can act as an inductive bias to improve learning efficiency in novel tasks (Caruana, 1998; BID52 . This
inductive bias has been implemented in various ways: as learned hyperparameters in a hierarchical Bayesian model that regularize task-specific parameters BID18 , as a learned metric space in which to group neighbors BID7 , as a trained recurrent neural network that allows encoding and retrieval of episodic information BID43 , or as an optimization algorithm with learned parameters BID45 BID3 .The model-agnostic
meta-learning (MAML) of BID12 is an instance of a learned optimization procedure that directly optimizes the standard gradient descent rule. The algorithm estimates
an initial parameter set to be shared among the task-specific models; the intuition is that gradient descent from the learned initialization provides a favorable inductive bias for fast adaptation. However, this inductive
bias has been evaluated only empirically in prior work BID12 .In this work, we present
a novel derivation of and a novel extension to MAML, illustrating that this algorithm can be understood as inference for the parameters of a prior distribution in a hierarchical Bayesian model. The learned prior allows
for quick adaptation to unseen tasks on the basis of an implicit predictive density over task-specific parameters. The reinterpretation as
hierarchical Bayes gives a principled statistical motivation for MAML as a meta-learning algorithm, and sheds light on the reasons for its favorable performance even among methods with significantly more parameters.More importantly, by casting gradient-based meta-learning within a Bayesian framework, we are able to improve MAML by taking insights from Bayesian posterior estimation as novel augmentations to the gradient-based meta-learning procedure. We experimentally demonstrate
that this enables better performance on a few-shot learning benchmark.
We have shown that model-agnostic meta-learning (MAML) estimates the parameters of a prior in a hierarchical Bayesian model.
By casting gradient-based meta-learning within a Bayesian framework, our analysis opens the door to novel improvements inspired by probabilistic machinery.As a step in this direction, we propose an extension to MAML that employs a Laplace approximation to the posterior distribution over task-specific parameters.
This technique provides a more accurate estimate of the integral that, in the original MAML algorithm, is approximated via a point estimate.
We show how to estimate the quantity required by the Laplace approximation using Kroneckerfactored approximate curvature (K-FAC), a method recently proposed to approximate the quadratic curvature of a neural network objective for the purpose of a second-order gradient descent technique.Our contribution illuminates the road to exploring further connections between gradient-based metalearning methods and hierarchical Bayesian modeling.
For instance, in this work we assume that the predictive distribution over new data-points is narrow and well-approximated by a point estimate.
We may instead employ methods that make use of the variance of the distribution over task-specific parameters in order to model the predictive density over examples from a novel task.Furthermore, it is known that the Laplace approximation is inaccurate in cases where the integral is highly skewed, or is not unimodal and thus is not amenable to approximation by a single Gaussian mode.
This could be solved by using a finite mixture of Gaussians, which can approximate many density functions arbitrarily well BID49 BID0 .
The exploration of additional improvements such as this is an exciting line of future work.

assistant: text: A specific gradient-based meta-learning algorithm, MAML, is equivalent to an inference procedure in a hierarchical Bayesian model. We use this connection to improve MAML via methods from approximate inference and curvature estimation.