Task-similarity Aware Meta-learning through Nonparametric Kernel Regression

1 INTRODUCTION . Meta-learning seeks to abstract a general learning rule applicable to a class of different learning problems or tasks , given the knowledge of a set of training tasks from the class ( Finn & Levine , 2018 ; Denevi et al. , 2018 ; Hospedales et al. , 2020 ; Grant et al. , 2018 ; Yoon et al. , 2018 ) . The setting is such that the data available for solving each task is often severely limited , resulting in a poor performance when the tasks are solved independently from each other . This also sets meta-learning apart from the transfer learning paradigm where the focus is to transfer a well-trained network from existing domain to another ( Pan & Yang , 2010 ) . While existing meta-learning approaches implicitly assume the tasks as being similar , it is generally unclear how this task-similarity could be quantified and used in the learning . As a result , most popular meta-learning approaches do not actively use the similarity/dissimilarity between the tasks , but rely on availability of huge number of tasks for their working . In many practical applications , the number of tasks could be limited and the tasks may not always be very similar . There might even be ‘ outlier tasks ’ or ‘ out-of-the-distribution tasks ’ that are less similar or dissimilar from the rest of the tasks . Our conjecture is that the explicit incorporation or awareness of task-similarity helps improve meta-learning performance in such task-limited and adverse settings . The goal of this paper is to test this hypothesis by developing a task-similarity aware meta-learning algorithm using nonparametric kernel regression . Specifically , our contribution is a novel metalearning algorithm called the Task-similarity Aware Nonparametric Meta-Learning ( TANML ) that : • Explicitly employs similarity across the tasks to fast adapt the meta-information to a given task , by using nonparametric kernel regression . • Models the parameters of a task as belonging to a reproducing kernel Hilbert space ( RKHS ) , obtained by viewing the popular meta-learning of MAML and Meta-SGD approaches through the lens of linear/kernel regression . • Uses task-descriptors through a kernel function to quantify task-similarity/dissimilarity . • Offers a general framework for incorporating task-similarity in the meta-learning process . Though we pursued the algorithm with a specific choice of the task-descriptors , the proposed RKHS task-similarity aware framework can be extended to other formulations . We wish to emphasize that our goal is not to propose another meta-learning algorithm that outperforms the state-of-the-art , but rather to investigate if task-similarity can be explicitly incorporated and used to advantage in a meaningful manner . We show how this is achieved as the consequence of viewing the popular MAML and Meta-SGD formulations through the lens of nonparametric kernel regression . In order to keep the comparison fair on an apple-to-apple level , we compare the performance of TANML with that of MAML and Meta-SGD algorithms . 1.1 MATHEMATICAL OVERVIEW OF THE PROPOSED TASK-SIMILARITY AWARE FRAMEWORK . Given pairs of data ( xk , yk ) ∈ Rnx ×Rny where k ∈ { 1 , 2 , · · · , K } generated by an unknown data source , we are interested in learning a predictor Rnx × RD 3 ( x , θ ) 7→ f ( x , θ ) ∈ Rny from the given data . For example , f ( x , θ ) could be a function defined by an artificial neural network ( ANN ) . We collect pairs of data in X = ( x1 , x2 , · · · , xK ) and Y = ( y1 , y2 , · · · , yK ) and define the loss function RKnx×RKny×RD 3 ( X , Y , θ ) 7→ L ( X , Y , θ ) ∈ R which we then minimize with respect to θ . This constitutes the training of the predictor . In the case of a ANN , L ( X , Y , θ ) ∈ R could be the mean-square loss or the cross-entropy function . The data X , Y used for training is referred to as the training data . Let θ̂ denote the optimal value of θ obtained from training . Given a new x ∈ Rnx , we use ŷ = f ( x , θ̂ ) to predict y . The goodness of θ̂ is evaluated using y − ŷ on a sequence of pairs of new data called the test data X̄ , Ȳ , defined similarly as X and Y , but with K̄ number of data pairs . The training of the predictor for the given data source is defined as a task . Now , consider that we are interested in carrying out several such tasks for data coming from different but similar sources . Let Xi , Yi , X̄i , Ȳi , i = 1 , · · · , Ttr denote the data from Ttr different datasources , and defined similarly as X , Y , X̄ , Ȳ above . We refer to the training of the predictor for data Xi , Yi , X̄i , Ȳi as the ith training task , and θi is referred to as the parameter for the task . Metalearning captures similarities across the tasks by learning a common θ̂ ( which we denote by θ0 ) from the data of these Ttr tasks ( called the meta-training data ) , such that θ0 can be quickly adapted to train a predictor for data from new and different but similar data-sources . Depending on how θ is obtained from θ0 , various meta-learning algorithms exist ( Denevi et al. , 2018 ; Finn & Levine , 2018 ; Allen et al. , 2019 ) . The performance of the meta-learning algorithm is evaluated on previously unseen data from several other similar sources X vi , Yvi , X̄ vi , Ȳvi , i = 1 , · · · , Tv ( called the meta-test data ) defined similarly to X , Y , X̄ , Ȳ − this constitutes the meta-test phase . The training of the predictor for test data X vi , Yvi , X̄ vi , Ȳvi is referred to as the ith test task , θvi denotes the parameter for the ith test task . In the existing meta-learning approaches , both θi and θvi are obtained by adapting θ0 using the gradient of L ( Xi , Yi , θ ) and L ( X vi , Yvi , θ ) , respectively , evaluated at θ0 . In our work , we propose a meta-learning algorithm where θi explicitly uses a similarity between the ith training task and all the training tasks . Similarly , the parameters θvi for the test tasks also use explicitly a similarity between the ith test task and all the training tasks . As specified later , we define this task-similarity between two tasks through kernel regression , and our algorithm learns the kernel regression coefficients Ψ as meta-parameters in addition to θ0 . A motivating example Let us now consider a specific loss function given by L ( X , Y , θ ) =∑K k=1 ‖yk − f ( xk , θ ) ‖22 . Training for tasks independently with limited training data will typically result in a predictor that overfits to X , Y , and generalizes poorly to X̄ , Ȳ . MAML-type metalearning approaches ( Finn et al. , 2017 ) solve this by inferring the information across tasks in the form of a good initialization θ0– specialized/adapted to a new task using the adaptation function RD × RKnx × RKny 3 ( θ0 , X , Y ) 7→ gMAML ( θ0 , X , Y ) ∈ RD defined as : gMAML ( θ0 , X , Y ) , θ0 − α∇θ0L ( X , Y , θ0 ) The parameters for the training and test tasks as obtained through adaptation of θ0 as θi = gMAML ( θ0 , Xi , Yi ) , i = 1 , · · · , Ttr , and θvi = gMAML ( θ0 , X vi , Yvi ) , i = 1 , · · · , Tv . The meta-parameter θ0 is learnt by iteratively taking a gradient descent with respect to the test loss on the training tasks given by ∑Ttr i=1 L ( X̄i , Ȳi , gMAML ( θ0 , Xi , Yi ) ) . The parameters for a task are obtained directly from θ0 and does not make use of any information from the other training tasks . As a result , the common θ0 learnt during the meta-training treats all tasks equally − the algorithm implicitly assumes similarity of all tasks , but is not able to discern or quantify the degree of similarity or dissimilarity among the tasks . In contrast , our algorithm involves an adaptation function gTANML ( to be defined later ) that explicitly uses a notion of similarity between the tasks to predict parameters for a task . As a result , we expect that our approach helps train predictors even when the data-sources that are not very similar to each other . In our numerical experiments in Section 4 , we see that this is indeed the case the sinusoidal function as the data source . 1.2 RELATED WORK . The structural characterization of tasks and use of task-dependent knowledge has gained interest in meta-learning recently . Edwards & Storkey ( 2017 ) proposed a variational autoencoder based approach to generate task/dataset statistics used to measure similarity . Ruder & Plank ( 2017 ) considered domain similarity and diversity measures in the context of transfer learning ( Ruder & Plank , 2017 ) . The study of how task properties affect the catastrophic forgetting in continual learning was pursued by Nguyen et al . ( 2019 ) . Lee et al . ( 2020 ) proposed a task-adaptive meta-learning approach for classification that adaptively balances meta-learning and task-specific learning differently for every task and class . Bayesian approaches have been proposed to capture the similarity across tasks in the form of task hyperpriors ( Yoon et al. , 2018 ; Finn et al. , 2018 ; Grant et al. , 2018 ; Rothfuss et al. , 2020 ) . Task-similarity defined through effective sample size has been used to develop a new offpolicy algorithm for meta-reinforcement learning ( Fakoor et al. , 2020 ) . It was shown by Oreshkin et al . ( 2018 ) that the performance few-shot learning shows significant improvements with the use of task-dependent metrics . While the use of kernels or similarity metrics is not new in meta-learning , they are typically seen in the context of defining relations between the classes or samples within a given task ( Qiao et al. , 2018 ; Rusu et al. , 2019 ; Vinyals et al. , 2016 ; Snell et al. , 2017 ; Oreshkin et al. , 2018 ; Fortuin & Rätsch , 2019 ; Goo & Niekum , 2020 ) . Qiao et al . ( 2018 ) use similarity metrics in the activation space to predict parameters for novel categories in few-shot learning with zero-training . Information-theoretic ideas have also been used in the study of the topology and the geometry of task spaces by Nguyen et al . ( 2019 ) ; Achille et al . ( 2018 ) . Achille et al . ( 2019 ) construct vector representations for tasks using partially trained probe networks , based on which task-similarity metrics are developed . Task descriptors have been of interest specially in vision related tasks in the context of transfer learning ( Zamir et al. , 2018 ; Achille et al. , 2019 ; Tran et al. , 2019 ) . Recently , neural tangent kernels were been proposed for asymptotic analysis of meta-learning for infinitely wide neural networks by considering gradient based kernels across tasks by Wang et al . ( 2020 ) . The work of Wang et al . ( 2020 ) is the closest in spirit to our work in that they consider kernels across meta-learning tasks . However , the premise of their work is very different from ours . The aim of their work is to show how global convergence behaviour of popular MAML type task-specific adaptation can be assymptotically described using specific kernels , when every task involves training deep neural networks of asymptotically infinite or very large widths . Our premise on the other hand is entirely different − we consider a task-specific adaptation that actively employs similarity of tasks in the form of valid kernel functions , in order to improve meta-learning performance . Our work does not make assumptions on the nature of the kernel , the structure of the learner , or its dimensions .

Summary: The paper introduced a meta-learning framework in which a kernel describing similarity between the tasks is used to construct an RKHS which is used to perform kernel regression. The framework is instantiated in a form of an algorithm: TANML which can be viewed as an extension to a popular Meta-SGD algorithm. The experiments on two regression tasks are presented to analyse the efficacy of the proposed method.