Learning-to-learn or meta-learning leverages data-driven inductive bias to increase the efficiency of learning on a novel task. This approach encounters difficulty when transfer is not mutually beneficial, for instance, when tasks are sufficiently dissimilar or change over time. Here, we use the connection between gradient-based meta-learning and hierarchical Bayes to propose a mixture of hierarchical Bayesian models over the parameters of an arbitrary function approximator such as a neural network. Generalizing the model-agnostic meta-learning (MAML) algorithm, we present a stochastic expectation maximization procedure to jointly estimate parameter initializations for gradient descent as well as a latent assignment of tasks to initializations. This approach better captures the diversity of training tasks as opposed to consolidating inductive biases into a single set of hyperparameters. Our experiments demonstrate better generalization on the standard miniImageNet benchmark for 1-shot classification. We further derive a novel and scalable non-parametric variant of our method that captures the evolution of a task distribution over time as demonstrated on a set of few-shot regression tasks. Meta-learning algorithms aim to increase the efficiency of learning by treating task-specific learning episodes as examples from which to generalize BID39 . The central assumption of a meta-learning algorithm is that some tasks are inherently related and so inductive transfer can improve generalization and sample efficiency BID4 BID2 . Recent metalearning algorithms have encoded this assumption by learning global hyperparameters that provide a task-general inductive bias. In learning a single set of hyperparameters that parameterize, for example, a metric space BID46 or an optimizer for gradient descent BID31 BID8 , these meta-learning algorithms make the assumption that tasks are equally related and therefore mutual transfer is appropriate. This assumption has been cemented in recent few-shot learning benchmarks, which consist of a set of tasks generated in a systematic manner (e.g., BID8 BID46 .However , the real world often presents scenarios in which an agent must decide what degree of transfer is appropriate. In the case of positive transfer, a subset of tasks may be more strongly related to each other and so non-uniform transfer poses a strategic advantage. Negative transfer in the presence of dissimilar or outlier tasks worsens generalization performance BID34 . Moreover , when the underlying task distribution is non-stationary, inductive transfer to initial tasks should exhibit graceful degradation to address the catastrophic forgetting problem BID16 . However , the consolidation of all inductive biases into a single set of hyperparameters cannot flexibly account for variability in the task distribution. In contrast , in order to deal with this degree of task heterogeneity, extensive task-switching literature reveals that people detect and readily adapt even in the face of significantly novel contexts (see BID5 , for a review).In this work , we learn a mixture of hierarchical models that allows the meta-learner to adaptively select over a set of learned parameter initializations for gradient-based fast adaptation BID8 to a new task. The method is equivalent to clustering task-specific parameters in the hierarchical model induced by recasting gradient-based meta-learning as hierarchical Bayes BID13 and generalizes the model-agnostic meta-learning (MAML) algorithm introduced in BID8 .By treating the assignment of task-specific parameters to clusters as latent variables in a probabilistic model, we can directly detect similarities between tasks on the basis of the task-specific likelihood, which may be parameterized by a black-box model such as a neural network. Our approach therefore alleviates the need for explicit geometric or probabilistic modelling assumptions about the weights of a parametric model and provides a scalable method to regulate information transfer between episodes.We extend our latent variable model to the non-parametric setting and leverage stochastic point estimation for scalable inference in a Dirichlet process mixture model (DPMM) BID30 . To the best of our knowledge, no previous work has considered a scalable stochastic point estimation in a non-parametric mixture model. Furthermore , we are not aware of prior work applying nonparametric mixture modelling techniques to high-dimensional parameter spaces such as those of deep neural networks. The non-parametric extension allows the complexity of a meta-learner to evolve by introducing or removing clusters in alignment with the changing composition of the dataset and preserves performance on previously encountered tasks better than a parametric counterpart. Meta-learning is a source of learned inductive bias. Occasionally, the inductive bias is harmful because the experience gained from solving one task does not transfer well to another. On the other hand, if tasks are closely related, they can benefit from a greater amount of inductive transfer. Here, we present an approach that allows a gradient-based meta-learner to explicitly modulate the amount of transfer between tasks, as well as to adapt its parameter dimensionality when the underlying task distribution evolves. We formulate this as probabilistic inference in a mixture model that defines a clustering of task-specific parameters. To ensure scalability, we make use of the recent connection between gradient-based meta-learning and hierarchical Bayes BID13 to perform approximate maximum a posteriori (MAP) inference in both a finite and an infinite mixture model. This approach admits non-conjugate likelihoods parameterised with a black-box function approximator such as a deep neural network, and therefore learns to identify underlying genres of tasks using the standard gradient descent learning rule. We demonstrate that this approach allows the model complexity to grow along with the evolving complexity of the observed tasks in both a few-shot regression and a few-shot classification problem. BID31 43.44 ± 0.77 60.60 ± 0.71 SNAIL BID20 b 45.1 ± --55.2 ± --prototypical networks BID42 c 46.61 ± 0.78 65.77 ± 0.70 MAML BID8 48.70 ± 1.84 63.11 ± 0.92 LLAMA BID13 49.40 BID45 49.82 ± 0.78 63.70 ± 0.67 KNN + GNN embedding BID10 49.44 ± 0.28 64.02 ± 0.51 GNN BID10 50.33 ± 0.36 66.41 ± 0.63 fwCNN (Hebb) BID22 50.21 ± 0.37 64.75 ± 0.49 fwResNet (Hebb) BID22 56.84 ± 0.52 71.00 ± 0.34 SNAIL BID20 55. BID23 57.10 ± 0.70 70.04 ± 0.63 MAML BID8 Figure 7: An evolving dataset of miniImageNet few-shot classification tasks where for the first 20k iterations we train on the standard dataset, then switch to a "pencil" effect set of tasks for 10k iterations before finally switching to a "blurred" effect set of tasks until 40k. Responsibilities γ ( ) for each cluster are plotted over time. Note the change in responsibilities as the dataset changes at iterations 20k and 30k.
We use the connection between gradient-based meta-learning and hierarchical Bayes to learn a mixture of meta-learners that is appropriate for a heterogeneous and evolving task distribution.
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