Bayesian Online Meta-Learning

Neural networks are known to suffer from catastrophic forgetting when trained on sequential datasets . While there have been numerous attempts to solve this problem for large-scale supervised classification , little has been done to overcome catastrophic forgetting for few-shot classification problems . Few-shot metalearning algorithms often require all few-shot tasks to be readily available in a batch for training . The popular gradient-based model-agnostic meta-learning algorithm ( MAML ) is a typical algorithm that suffers from these limitations . This work introduces a Bayesian online meta-learning framework to tackle the catastrophic forgetting and the sequential few-shot tasks problems . Our framework incorporates MAML into a Bayesian online learning algorithm with Laplace approximation or variational inference . This framework enables few-shot classification on a range of sequentially arriving datasets with a single meta-learned model and training on sequentially arriving few-shot tasks . The experimental evaluations demonstrate that our framework can effectively prevent catastrophic forgetting and is capable of online meta-learning in various few-shot classification settings . 1 INTRODUCTION . Image classification models and algorithms often require an enormous amount of labelled examples for training to achieve state-of-the-art performance . Labelled examples can be expensive and time-consuming to acquire . Human visual systems , on the other hand , are able to recognise new classes after being shown a few labelled examples . Few-shot classification ( Miller et al. , 2000 ; Li et al. , 2004 ; 2006 ; Lake et al. , 2011 ) tackles this issue by learning to adapt to unseen classes ( known as novel classes ) with very few labelled examples from each class . Recent works show that metalearning provides promising approaches to few-shot classification problems ( Santoro et al. , 2016 ; Finn et al. , 2017 ; Li et al. , 2017 ; Ravi & Larochelle , 2017 ) . Meta-learning or learning-to-learn ( Schmidhuber , 1987 ; Thrun & Pratt , 1998 ) takes the learning process a level deeper – instead of learning from the labelled examples in the training classes ( known as base classes ) , meta-learning learns the example-learning process . The training process in meta-learning that utilises the base classes is called the meta-training stage , and the evaluation process that reports the few-shot performance on the novel classes is known as the meta-evaluation stage . Despite being a promising solution to few-shot classification problems , meta-learning methods suffer from several limitations : 1 . Unable to continually learn from sequential few-shot tasks : It is mandatory to have all base classes readily available for meta-training . Such meta-learning algorithms often require sampling a number of few-shot tasks in every iteration for optimisation . 2 . Unable to retain few-shot classification ability on sequential datasets that have evident distributional shift : A meta-learned model is restricted to perform few-shot classification on a specific dataset , in the sense that the base and novel classes have to originate from the same dataset distribution . A meta-learned model loses its few-shot classification ability on previous datasets as new ones arrive subsequently for meta-training . We emphasise that the task mentioned in this paper refers to the few-shot task for meta-learning . This paper considers meta-learning a single model for few-shot classification in the sequential datasets and sequential few-shot tasks settings respectively . We introduce a Bayesian online meta-learning framework that can train a few-shot learning model under the sequential few-shot tasks setting and train a model that is applicable to a broader scope of few-shot classification datasets by overcoming catastrophic forgetting . We extend the Bayesian online learning ( BOL ) framework ( Opper , 1998 ) to a Bayesian online meta-learning framework using the model-agnostic meta-learning ( MAML ) algorithm ( Finn et al. , 2017 ) . MAML finds a good model parameter initialisation ( called meta-parameters ) that can quickly adapt to novel classes using very few labelled examples , while BOL provides a principled framework for finding the posterior of the model parameters . Our framework aims to combine both BOL and MAML to find the posterior of the meta-parameters . Our work builds on Ritter et al . ( 2018a ) which combines the BOL framework and Laplace approximation with block-diagonal Kronecker-factored Fisher approximation , and Nguyen et al . ( 2018 ) which uses variational inference with BOL to overcome catastrophic forgetting in large-scale supervised classification . An important reason to implement Bayesian inference over non-Bayesian methods for an online @ @ @ Newsetting is that BOL provides a grounded framework that suggests using the previous posterior as the prior recursively . Bayesian inference inherits an advantage for robust meta-learning ( Yoon et al. , 2018 ) to overcome training instability problems addressed by Antoniou et al . ( 2019 ) . BOL implicitly keeps a memory on previous knowledge via the posterior , in contrast to recent online meta-learning methods that explicitly accumulate previous data in a task buffer ( Finn et al. , 2019 ; Zhuang et al. , 2019 ) . Explicitly keeping a memory on previous data often triggers an important question : how should the carried-forward data be processed in future task rounds , in order to accumulate knowledge ? Finn et al . ( 2019 ) update the meta-parameters at each iteration using previous few-shot tasks in the task buffer . This defeats the purpose of online learning , which by definition means to update the parameters each round using only the new data encountered . Having to re-train on previous data to avoid forgetting also increases the training time as the data accumulate ( Finn et al. , 2019 ; He et al. , 2019 ) . Certainly one can clamp the amount of data at some maximal limit and sample from the buffer , but the final performance of such an algorithm would be dependent on the samples being informative and of good quality which may vary across different seed runs . In contrast to memorising the datasets , having an implicit memory via the posterior automatically deals with the question on how to process carried-forward data and allows a better carry forward in previous experiences . Below are the contributions we make in this paper : • We develop the Bayesian online meta-learning ( BOML ) framework for sequential few-shot classification problems . Under this framework we introduce the algorithms Bayesian online meta-learning with Laplace approximation ( BOMLA ) and Bayesian online meta-learning with variational inference ( BOMVI ) . • We propose a simple approximation to the Fisher corresponding to the BOMLA algorithm that carries over the desirable block-diagonal Kronecker-factored structure from the Fisher approximation in the non-meta-learning setting . • We demonstrate that BOML can overcome catastrophic forgetting in the sequential few-shot datasets setting with apparent distributional shift in the datasets . • We demonstrate that BOML can continually learn to few-shot classify the novel classes in the sequential meta-training few-shot tasks setting . 2 META-LEARNING . Most meta-learning algorithms comprise an inner loop for example-learning and an outer loop that learns the example-learning process . Such algorithms often require sampling a meta-batch of tasks at each iteration , where a task is formed by sampling a subset of classes from the pool of base classes or novel classes during meta-training or meta-evaluation respectively . The N -way K-shot task , for instance , refers to sampling N classes and using K examples per class for few-shot quick adaptation . An offline meta-learning algorithm learns a few-shot classification model only for a specific dataset Dt+1 where all base classes of Dt+1 have to be readily available for meta-training . For notational convenience , we drop the t + 1 subscript in this section , as there is only one dataset involved in offline meta-learning . The datasetDt+1 is divided into the set of base classes D̃ and novel classes D̂ for meta-training and meta-evaluation respectively . Upon completing meta-training on the base class set D̃ , the goal of few-shot classification is to perform well on an unseen task D̂∗ sampled from the novel class set D̂ after a quick adaptation on a small subset D̂∗ , S ( known as the support set ) of D̂∗ . The performance of this unseen task is evaluated on the query set D̂∗ , Q , where D̂∗ , Q = D̂∗\D̂∗ , S . Since D̂ is not accessible during meta-training , this support-query split is mimicked on the base class set D̃ for meta-training . Model-agnostic meta-learning We are interested in the well-known meta-learning algorithm MAML ( Finn et al. , 2017 ) . Each updating step of MAML aims to improve the ability of the metaparameters to act as a good model initialisation for a quick adaptation on unseen tasks . Each iteration of the MAML algorithm samples M tasks from the base class set D̃ and runs a few steps of stochastic gradient descent ( SGD ) for an inner loop task-specific learning . The number of tasks sampled per iteration is known as the meta-batch size . For task m , the inner loop outputs the task-specific parameters θ̃m from a k-step SGD quick adaptation on the objective L ( θ , D̃m , S ) with the support set D̃m , S and initialised at θ : θ̃m = SGDk ( L ( θ , D̃m , S ) ) , ( 1 ) where m = 1 , . . . , M . The outer loop gathers all task-specific adaptations to update the metaparameters θ using the loss L ( θ̃m , D̃m , Q ) on the query set D̃m , Q . The overall MAML optimisation objective is arg min θ 1 M M∑ m=1 L ( SGDk ( L ( θ , D̃m , S ) ) , D̃m , Q ) . ( 2 ) Like most offline meta-learning algorithms , MAML requires all base classes to be readily available for tasks-sampling at each iteration . We aim to overcome this limitation by meta-learning a model that can few-shot classify unseen tasks from the novel classes , while the tasks from the base classes arrive sequentially for meta-training . MAML also assumes a stationary task distribution during meta-training and meta-evaluation . Under this assumption , a meta-learned model is only applicable to a specific dataset distribution . When the model encounters a sequence of datasets with apparent distributional shift , it loses the few-shot classification ability on previous datasets as new ones arrive for meta-training . Our work also aims to meta-learn a single model for few-shot classification on multiple datasets that arrive sequentially for meta-training . We achieve these two goals by incorporating MAML into the BOL framework to give the Bayesian online meta-learning ( BOML ) framework that finds the posterior of the meta-parameters . 3 OVERVIEW OF OUR BAYESIAN ONLINE META-LEARNING APPROACH . Our central contribution is to extend the benefits of meta-learning to the Bayesian online scenario , thereby training models that can generalise across tasks whilst dealing with parameter uncertainty in the setting of sequential tasks or sequential datasets . Sequential datasets setting In this setting , online meta-training occurs sequentially on the @ @ @ Newdatasets D1 , . . . , DT . Each dataset Di can be seen as a knowledge domain with an associated underlying task distribution p ( Ti ) . A newly-arrived Dt+1 is separated into the base class set D̃t+1 and novel class set D̂t+1 for meta-training and meta-evaluation respectively , where the tasks in these two stages are drawn from the task distribution p ( Tt+1 ) . Sequential tasks setting The sequential tasks setting only involves one dataset D with an associ- @ @ @ Newated underlying task distribution p ( T ) , where D is separated into the base and novel class sets . In this setting , D̃1 , . . . , D̃t+1 denote the non-overlapping tasks formed from the base class set and they arrive sequentially for meta-training . These tasks D̃1 , . . . , D̃t+1 and the meta-evaluation tasks are drawn from the task distribution p ( T ) . Notationally , for both sequential tasks and sequential datasets settings , let D̃St+1 and D̃ Q t+1 denote the collection of support sets and query sets respectively from D̃t+1 , so that D̃t+1 = D̃St+1 ∪ D̃ Q t+1 . We are interested in a MAP estimate θ∗ = arg maxθ p ( θ|D̃1 : t+1 ) . Using Bayes ’ rule on the posterior gives the recursive formula p ( θ|D̃1 : t+1 ) ∝ p ( D̃St+1 , D̃ Q t+1|θ ) p ( θ|D̃1 : t ) ( 3 ) = p ( D̃Qt+1|θ , D̃St+1 ) p ( D̃St+1|θ ) p ( θ|D̃1 : t ) ( 4 ) = { ∫ p ( D̃Qt+1|θ̃ ) p ( θ̃|θ , D̃St+1 ) dθ̃ } p ( D̃St+1|θ ) p ( θ|D̃1 : t ) ( 5 ) where Eq . ( 3 ) follows from the assumption that each dataset is independent given θ . From the meta-learning perspective , the parameters θ̃ introduced in Eq . ( 5 ) can be viewed as the task-specific parameters in MAML . There are various choices for the distribution p ( θ̃|θ , D̃St+1 ) in Eq . ( 5 ) . In particular if we choose to set it as the deterministic function of taking several steps of SGD on loss L with the support set collection D̃St+1 and initialised at θ , we have p ( θ̃|θ , D̃St+1 ) = 1 { θ̃ = SGDk ( L ( θ , D̃St+1 ) ) } . ( 6 ) and this recovers the MAML inner loop with SGD quick adaptation in Eq . ( 1 ) . The recursion given by Eq . ( 5 ) forms the basis of our approach and the remainder of this paper explains how we implement this . In order to do so we give a mini tutorial in Appendix A on Bayesian online learning , Laplace approximation and variational continual learning .

Summary: This work proposes a Bayesian approach to meta-learning from sequential data. Two algorithms are proposed. The first is based on the Laplace approximation to the model posterior which is made tractable by using K-FAC approximation of the Hessian. The second approaches uses a variational approximation for the posterior, where meta-learning corresponds to learning the variational prior. The experiments present results of the proposed method in sequential Omniglot and a pentathlon task involving different datasets.