In this paper, we propose Domain Agnostic Meta Score-based Learning (DAMSL), a novel, versatile and highly effective solution that delivers significant out-performance over state-of-the-art methods for cross-domain few-shot learning. We identify key problems in previous meta-learning methods over-fitting to the source domain, and previous transfer-learning methods under-utilizing the structure of the support set. The core idea behind our method is that instead of directly using the scores from a fine-tuned feature encoder, we use these scores to create input coordinates for a domain agnostic metric space. A graph neural network is applied to learn an embedding and relation function over these coordinates to process all information contained in the score distribution of the support set. We test our model on both established CD-FSL benchmarks and new domains and show that our method overcomes the limitations of previous meta-learning and transfer-learning methods to deliver substantial improvements in accuracy across both smaller and larger domain shifts.
University of Massachusetts Amherst（马萨诸塞大学阿默斯特分校）；IBM Research（IBM研究院）；Michigan State University（密歇根州立大学）；MIT-IBM Watson AI Lab（麻省理工学院-IBM沃森人工智能实验室）
We propose a new computationally-efficient first-order algorithm for Model-Agnostic Meta-Learning (MAML). The key enabling technique is to interpret MAML as a bilevel optimization (BLO) problem and leverage the sign-based SGD(signSGD) as a lower-level optimizer of BLO. We show that MAML, through the lens of signSGD-oriented BLO, naturally yields an alternating optimization scheme that just requires first-order gradients of a learned meta-model. We term the resulting MAML algorithm Sign-MAML. Compared to the conventional first-order MAML (FO-MAML) algorithm, Sign-MAML is theoretically-grounded …
Capturing scenes with a high dynamic range is crucial to reproducing images that appear similar to those seen by the human visual system. Despite progress in developing data-driven deep learning approaches for converting low dynamic range images to high dynamic range images, existing approaches are limited by the assumption that all conversions are governed by the same nonlinear mapping. To address this problem, we propose "Model-Agnostic Meta-Learning for HDR Image Reconstruction" (MetaHDR), which applies meta-learning to the LDR-to-HDR conversion problem using existing HDR datasets. Our key …
Meta learning aims at learning a model that can quickly adapt to unseen tasks. Widely used meta learning methods include model agnostic meta learning (MAML), implicit MAML, Bayesian MAML. Thanks to its ability of modeling uncertainty, Bayesian MAML often has advantageous empirical performance. However, the theoretical understanding of Bayesian MAML is still limited, especially on questions such as if and when Bayesian MAML has provably better performance than MAML. In this paper, we aim to provide theoretical justifications for Bayesian MAML's advantageous performance by comparing the meta tes…
Rensselaer Polytechnic Institute（伦斯勒理工学院）；University of Michigan（密歇根大学）；MIT-IBM Watson AI Lab（麻省理工学院-IBM沃森人工智能实验室）；IBM Research（IBM研究院）；Northeastern University（东北大学）；Michigan State University（密歇根州立大学）
Model-agnostic meta-learning (MAML) has emerged as one of the most successful meta-learning techniques in few-shot learning. It enables us to learn a meta-initialization} of model parameters (that we call meta-model) to rapidly adapt to new tasks using a small amount of labeled training data. Despite the generalization power of the meta-model, it remains elusive that how adversarial robustness can be maintained by MAML in few-shot learning. In addition to generalization, robustness is also desired for a meta-model to defend adversarial examples (attacks). Toward promoting adversarial robustnes…
Model-agnostic meta-learning (MAML) is a meta-learning technique to train a model on a multitude of learning tasks in a way that primes the model for few-shot learning of new tasks. The MAML algorithm performs well on few-shot learning problems in classification, regression, and fine-tuning of policy gradients in reinforcement learning, but comes with the need for costly hyperparameter tuning for training stability. We address this shortcoming by introducing an extension to MAML, called Alpha MAML, to incorporate an online hyperparameter adaptation scheme that eliminates the need to tune meta-…
Huazhong University of Science and Technology（华中科技大学）
Model-Agnostic Meta-Learning (MAML) is a famous few-shot learning method that has inspired many follow-up efforts, such as ANIL and BOIL. However, as an inductive method, MAML is unable to fully utilize the information of query set, limiting its potential of gaining higher generality. To address this issue, we propose a simple yet effective method that generates psuedo-labels adaptively and could boost the performance of the MAML family. The proposed methods, dubbed Generative Pseudo-label based MAML (GP-MAML), GP-ANIL and GP-BOIL, leverage statistics of the query set to improve the performanc…
University of California, Berkeley（加州大学伯克利分校）；OpenAI
We propose an algorithm for meta-learning that is model-agnostic, in the sense that it is compatible with any model trained with gradient descent and applicable to a variety of different learning problems, including classification, regression, and reinforcement learning. The goal of meta-learning is to train a model on a variety of learning tasks, such that it can solve new learning tasks using only a small number of training samples. In our approach, the parameters of the model are explicitly trained such that a small number of gradient steps with a small amount of training data from a new ta…
Many meta-learning algorithms can be formulated into an interleaved process, in the sense that task-specific predictors are learned during inner-task adaptation and meta-parameters are updated during meta-update. The normal meta-training strategy needs to differentiate through the inner-task adaptation procedure to optimize the meta-parameters. This leads to a constraint that the inner-task algorithms should be solved analytically. Under this constraint, only simple algorithms with analytical solutions can be applied as the inner-task algorithms, limiting the model expressiveness. To lift the …
Architecture, Dataset and Model-Scale Agnostic Data-free Meta-Learning
Tsinghua Shenzhen International Graduate School（清华大学深圳国际研究生院）；JD Explore Academy（京东探索研究院）；State University of New York at Buffalo（纽约州立大学布法罗分校）；The University of Sydney（悉尼大学）
Element AI（Element AI公司）；Kakao Brain（Kakao Brain公司）；MILA Université de Montréal（蒙特利尔大学MILA研究所）；Rutgers University（罗格斯大学）
Learning to infer Bayesian posterior from a few-shot dataset is an important step towards robust meta-learning due to the model uncertainty inherent in the problem. In this paper, we propose a novel Bayesian model-agnostic meta-learning method. The proposed method combines scalable gradient-based meta-learning with nonparametric variational inference in a principled probabilistic framework. During fast adaptation, the method is capable of learning complex uncertainty structure beyond a point estimate or a simple Gaussian approximation. In addition, a robust Bayesian meta-update mechanism with …
Current deep learning based text classification methods are limited by their ability to achieve fast learning and generalization when the data is scarce. We address this problem by integrating a meta-learning procedure that uses the knowledge learned across many tasks as an inductive bias towards better natural language understanding. Based on the Model-Agnostic Meta-Learning framework (MAML), we introduce the Attentive Task-Agnostic Meta-Learning (ATAML) algorithm for text classification. The essential difference between MAML and ATAML is in the separation of task-agnostic representation lear…
Meta-learning for few-shot learning entails acquiring a prior over previous tasks and experiences, such that new tasks be learned from small amounts of data. However, a critical challenge in few-shot learning is task ambiguity: even when a powerful prior can be meta-learned from a large number of prior tasks, a small dataset for a new task can simply be too ambiguous to acquire a single model (e.g., a classifier) for that task that is accurate. In this paper, we propose a probabilistic meta-learning algorithm that can sample models for a new task from a model distribution. Our approach extends…
Few-shot classification (FSC) requires training models using a few (typically one to five) data points per class. Meta learning has proven to be able to learn a parametrized model for FSC by training on various other classification tasks. In this work, we propose PLATINUM (semi-suPervised modeL Agnostic meTa-learnIng usiNg sUbmodular Mutual information), a novel semi-supervised model agnostic meta-learning framework that uses the submodular mutual information (SMI) functions to boost the performance of FSC. PLATINUM leverages unlabeled data in the inner and outer loop using SMI functions durin…
SK T-Brain；University of Southern California（南加州大学）
Gradient-based meta-learners such as MAML are able to learn a meta-prior from similar tasks to adapt to novel tasks from the same distribution with few gradient updates. One important limitation of such frameworks is that they seek a common initialization shared across the entire task distribution, substantially limiting the diversity of the task distributions that they are able to learn from. In this paper, we augment MAML with the capability to identify tasks sampled from a multimodal task distribution and adapt quickly through gradient updates. Specifically, we propose a multimodal MAML alg…
Natural language understanding(NLU) is challenging for finance due to the lack of annotated data and the specialized language in that domain. As a result, researchers have proposed to use pre-trained language model and multi-task learning to learn robust representations. However, aggressive fine-tuning often causes over-fitting and multi-task learning may favor tasks with significantly larger amounts data, etc. To address these problems, in this paper, we investigate model-agnostic meta-learning algorithm(MAML) in low-resource financial NLU tasks. Our contribution includes: 1. we explore the p…
For machine learning applications in medical imaging, the availability of training data is often limited, which hampers the design of radiological classifiers for subtle conditions such as autism spectrum disorder (ASD). Transfer learning is one method to counter this problem of low training data regimes. Here we explore the use of meta-learning for very low data regimes in the context of having prior data from multiple sites - an approach we term site-agnostic meta-learning. Inspired by the effectiveness of meta-learning for optimizing a model across multiple tasks, here we propose a framewor…
University of Southern California（南加州大学）；University of Michigan（密歇根大学）
Model-agnostic meta-learners aim to acquire meta-learned parameters from similar tasks to adapt to novel tasks from the same distribution with few gradient updates. With the flexibility in the choice of models, those frameworks demonstrate appealing performance on a variety of domains such as few-shot image classification and reinforcement learning. However, one important limitation of such frameworks is that they seek a common initialization shared across the entire task distribution, substantially limiting the diversity of the task distributions that they are able to learn from. In this pape…
Inception Institute of Artificial Intelligence（人工智能初始研究院）；University of Central Florida（中佛罗里达大学）
Humans can continuously learn new knowledge as their experience grows. In contrast, previous learning in deep neural networks can quickly fade out when they are trained on a new task. In this paper, we hypothesize this problem can be avoided by learning a set of generalized parameters, that are neither specific to old nor new tasks. In this pursuit, we introduce a novel meta-learning approach that seeks to maintain an equilibrium between all the encountered tasks. This is ensured by a new meta-update rule which avoids catastrophic forgetting. In comparison to previous meta-learning techniques,…
Technische Universität Berlin（柏林工业大学）；Bernstein Center for Computational Neuroscience Berlin（柏林伯恩斯坦计算神经科学中心）
In past years model-agnostic meta-learning (MAML) has been one of the most promising approaches in meta-learning. It can be applied to different kinds of problems, e.g., reinforcement learning, but also shows good results on few-shot learning tasks. Besides their tremendous success in these tasks, it has still not been fully revealed yet, why it works so well. Recent work proposes that MAML rather reuses features than rapidly learns. In this paper, we want to inspire a deeper understanding of this question by analyzing MAML's representation. We apply representation similarity analysis (RSA), a…
Institute for Natural Language Processing（自然语言处理研究所）；University of Stuttgart（斯图加特大学）
While neural text-to-speech systems perform remarkably well in high-resource scenarios, they cannot be applied to the majority of the over 6,000 spoken languages in the world due to a lack of appropriate training data. In this work, we use embeddings derived from articulatory vectors rather than embeddings derived from phoneme identities to learn phoneme representations that hold across languages. In conjunction with language agnostic meta learning, this enables us to fine-tune a high-quality text-to-speech model on just 30 minutes of data in a previously unseen language spoken by a previously…
Northeastern University（东北大学）；IBM Research（IBM研究院）；Michigan State University（密歇根州立大学）
Adversarial perturbations are critical for certifying the robustness of deep learning models. A universal adversarial perturbation (UAP) can simultaneously attack multiple images, and thus offers a more unified threat model, obviating an image-wise attack algorithm. However, the existing UAP generator is underdeveloped when images are drawn from different image sources (e.g., with different image resolutions). Towards an authentic universality across image sources, we take a novel view of UAP generation as a customized instance of few-shot learning, which leverages bilevel optimization and lea…
Center for Research in Computer Vision（计算机视觉研究中心）；University of Central Florida（中佛罗里达大学）
Meta-learning approaches have been proposed to tackle the few-shot learning problem.Typically, a meta-learner is trained on a variety of tasks in the hopes of being generalizable to new tasks. However, the generalizability on new tasks of a meta-learner could be fragile when it is over-trained on existing tasks during meta-training phase. In other words, the initial model of a meta-learner could be too biased towards existing tasks to adapt to new tasks, especially when only very few examples are available to update the model. To avoid a biased meta-learner and improve its generalizability, we…
Janelia Research Campus, HHMI（霍华德·休斯医学研究所珍妮莉亚研究园区）；Harvard University（哈佛大学）；Columbia University（哥伦比亚大学）
Meta-learning has emerged as an important framework for learning new tasks from just a few examples. The success of any meta-learning model depends on (i) its fast adaptation to new tasks, as well as (ii) having a shared representation across similar tasks. Here we extend the model-agnostic meta-learning (MAML) framework introduced by Finn et al. (2017) to achieve improved performance by analyzing the temporal dynamics of the optimization procedure via the Runge-Kutta method. This method enables us to gain fine-grained control over the optimization and helps us achieve both the adaptation and …
Korea Advanced Institute of Science and Technology (KAIST)（韩国科学技术院）
Model agnostic meta-learning (MAML) is a popular state-of-the-art meta-learning algorithm that provides good weight initialization of a model given a variety of learning tasks. The model initialized by provided weight can be fine-tuned to an unseen task despite only using a small amount of samples and within a few adaptation steps. MAML is simple and versatile but requires costly learning rate tuning and careful design of the task distribution which affects its scalability and generalization. This paper proposes a more robust MAML based on an adaptive learning scheme and a prioritization task …
The capacity of meta-learning algorithms to quickly adapt to a variety of tasks, including ones they did not experience during meta-training, has been a key factor in the recent success of these methods on few-shot learning problems. This particular advantage of using meta-learning over standard supervised or reinforcement learning is only well founded under the assumption that the adaptation phase does improve the performance of our model on the task of interest. However, in the classical framework of meta-learning, this constraint is only mildly enforced, if not at all, and we only see an im…
Model-agnostic meta-learning (MAML) formulates meta-learning as a bilevel optimization problem, where the inner level solves each subtask based on a shared prior, while the outer level searches for the optimal shared prior by optimizing its aggregated performance over all the subtasks. Despite its empirical success, MAML remains less understood in theory, especially in terms of its global optimality, due to the nonconvexity of the meta-objective (the outer-level objective). To bridge such a gap between theory and practice, we characterize the optimality gap of the stationary points attained by…
We introduce here the idea of Meta-Learning for training EEG BCI decoders. Meta-Learning is a way of training machine learning systems so they learn to learn. We apply here meta-learning to a simple Deep Learning BCI architecture and compare it to transfer learning on the same architecture. Our Meta-learning strategy operates by finding optimal parameters for the BCI decoder so that it can quickly generalise between different users and recording sessions -- thereby also generalising to new users or new sessions quickly. We tested our algorithm on the Physionet EEG motor imagery dataset. Our ap…
Although few-shot learning research has advanced rapidly with the help of meta-learning, its practical usefulness is still limited because most of them assumed that all meta-training and meta-testing examples came from a single domain. We propose a simple but effective way for few-shot classification in which a task distribution spans multiple domains including ones never seen during meta-training. The key idea is to build a pool of models to cover this wide task distribution and learn to select the best one for a particular task through cross-domain meta-learning. All models in the pool share…
Dynamic scheduling is an important problem in applications from queuing to wireless networks. It addresses how to choose an item among multiple scheduling items in each timestep to achieve a long-term goal. Conventional approaches for dynamic scheduling find the optimal policy for a given specific system so that the policy from these approaches is usable only for the corresponding system characteristics. Hence, it is hard to use such approaches for a practical system in which system characteristics dynamically change. This paper proposes a novel policy structure for MDP-based dynamic schedulin…
This work addresses meta-learning (ML) by considering deep networks with stochastic local winner-takes-all (LWTA) activations. This type of network units results in sparse representations from each model layer, as the units are organized into blocks where only one unit generates a non-zero output. The main operating principle of the introduced units rely on stochastic principles, as the network performs posterior sampling over competing units to select the winner. Therefore, the proposed networks are explicitly designed to extract input data representations of sparse stochastic nature, as oppo…
University of Adelaide（阿德莱德大学）；University of Liverpool（利物浦大学）
We introduce a new, rigorously-formulated Bayesian meta-learning algorithm that learns a probability distribution of model parameter prior for few-shot learning. The proposed algorithm employs a gradient-based variational inference to infer the posterior of model parameters to a new task. Our algorithm can be applied to any model architecture and can be implemented in various machine learning paradigms, including regression and classification. We show that the models trained with our proposed meta-learning algorithm are well calibrated and accurate, with state-of-the-art calibration and classi…
Speaker adaptive training (SAT) of neural network acoustic models learns models in a way that makes them more suitable for adaptation to test conditions. Conventionally, model-based speaker adaptive training is performed by having a set of speaker dependent parameters that are jointly optimised with speaker independent parameters in order to remove speaker variation. However, this does not scale well if all neural network weights are to be adapted to the speaker. In this paper we formulate speaker adaptive training as a meta-learning task, in which an adaptation process using gradient descent …
As a popular meta-learning approach, the model-agnostic meta-learning (MAML) algorithm has been widely used due to its simplicity and effectiveness. However, the convergence of the general multi-step MAML still remains unexplored. In this paper, we develop a new theoretical framework to provide such convergence guarantee for two types of objective functions that are of interest in practice: (a) resampling case (e.g., reinforcement learning), where loss functions take the form in expectation and new data are sampled as the algorithm runs; and (b) finite-sum case (e.g., supervised learning), whe…
Meta-learning methods have shown an impressive ability to train models that rapidly learn new tasks. However, these methods only aim to perform well in expectation over tasks coming from some particular distribution that is typically equivalent across meta-training and meta-testing, rather than considering worst-case task performance. In this work we introduce the notion of "task-robustness" by reformulating the popular Model-Agnostic Meta-Learning (MAML) objective [Finn et al. 2017] such that the goal is to minimize the maximum loss over the observed meta-training tasks. The solution to this …
Massachusetts Institute of Technology（麻省理工学院）；The University of Texas at Austin（德克萨斯大学奥斯汀分校）
In this paper, we study the generalization properties of Model-Agnostic Meta-Learning (MAML) algorithms for supervised learning problems. We focus on the setting in which we train the MAML model over $m$ tasks, each with $n$ data points, and characterize its generalization error from two points of view: First, we assume the new task at test time is one of the training tasks, and we show that, for strongly convex objective functions, the expected excess population loss is bounded by ${\mathcal{O}}(1/mn)$. Second, we consider the MAML algorithm's generalization to an unseen task and show that th…
Few-shot learning aims to generalize to novel classes with only a few samples with class labels. Research in few-shot learning has borrowed techniques from transfer learning, metric learning, meta-learning, and Bayesian methods. These methods also aim to train models from limited training samples, and while encouraging performance has been achieved, they often fail to generalize to novel domains. Many of the existing meta-learning methods rely on training data for which the base classes are sampled from the same domain as the novel classes used for meta-testing. However, in many applications i…
While neural networks are powerful function approximators, they suffer from catastrophic forgetting when the data distribution is not stationary. One particular formalism that studies learning under non-stationary distribution is provided by continual learning, where the non-stationarity is imposed by a sequence of distinct tasks. Most methods in this space assume, however, the knowledge of task boundaries, and focus on alleviating catastrophic forgetting. In this work, we depart from this view and move the focus towards faster remembering -- i.e measuring how quickly the network recovers perf…
Rensselaer Polytechnic Institute（伦斯勒理工学院）；IBM Thomas J. Watson Research Center（IBM托马斯·J·沃森研究中心）
Model-agnostic meta learning (MAML) is currently one of the dominating approaches for few-shot meta-learning. Albeit its effectiveness, the optimization of MAML can be challenging due to the innate bilevel problem structure. Specifically, the loss landscape of MAML is much more complex with possibly more saddle points and local minimizers than its empirical risk minimization counterpart. To address this challenge, we leverage the recently invented sharpness-aware minimization and develop a sharpness-aware MAML approach that we term Sharp-MAML. We empirically demonstrate that Sharp-MAML and its…
Fuzhou University（福州大学）；Hikvision Research Institute（海康威视研究院）
Domain generalization (DG) is a fundamental yet very challenging research topic in machine learning. The existing arts mainly focus on learning domain-invariant features with limited source domains in a static model. Unfortunately, there is a lack of training-free mechanism to adjust the model when generalized to the agnostic target domains. To tackle this problem, we develop a brand-new DG variant, namely Dynamic Domain Generalization (DDG), in which the model learns to twist the network parameters to adapt the data from different domains. Specifically, we leverage a meta-adjuster to twist th…
Korea Advanced Institute of Science and Technology (KAIST)（韩国科学技术院）
We tackle a novel few-shot learning challenge, which we call few-shot semantic edge detection, aiming to localize crisp boundaries of novel categories using only a few labeled samples. We also present a Class-Agnostic Few-shot Edge detection Network (CAFENet) based on meta-learning strategy. CAFENet employs a semantic segmentation module in small-scale to compensate for lack of semantic information in edge labels. The predicted segmentation mask is used to generate an attention map to highlight the target object region, and make the decoder module concentrate on that region. We also propose a …
Georgia Institute of Technology（佐治亚理工学院）；University of California, Berkeley（加州大学伯克利分校）；Mitsubishi Electric Research Laboratories（三菱电机研究实验室）
The rapid development of artificial intelligence and deep learning has provided many opportunities to further enhance the safety, stability, and accuracy of industrial Cyber-Physical Systems (CPS). As indispensable components to many mission-critical CPS assets and equipment, mechanical bearings need to be monitored to identify any trace of abnormal conditions. Most of the data-driven approaches applied to bearing fault diagnosis up-to-date are trained using a large amount of fault data collected a priori. In many practical applications, however, it can be unsafe and time-consuming to collect …
SnT, University of Luxembourg（卢森堡大学SnT）；Baidu Inc.（百度公司）；CITADEL, Université Virtuelle du Burkina Faso（布基纳法索虚拟大学CITADEL）
Representation learning of source code is essential for applying machine learning to software engineering tasks. Learning code representation from a multilingual source code dataset has been shown to be more effective than learning from single-language datasets separately, since more training data from multilingual dataset improves the model's ability to extract language-agnostic information from source code. However, existing multilingual training overlooks the language-specific information which is crucial for modeling source code across different programming languages, while only focusing o…
Few-shot or one-shot learning of classifiers requires a significant inductive bias towards the type of task to be learned. One way to acquire this is by meta-learning on tasks similar to the target task. In this paper, we propose UMTRA, an algorithm that performs unsupervised, model-agnostic meta-learning for classification tasks. The meta-learning step of UMTRA is performed on a flat collection of unlabeled images. While we assume that these images can be grouped into a diverse set of classes and are relevant to the target task, no explicit information about the classes or any labels are need…
Massachusetts Institute of Technology（麻省理工学院）；The University of Texas at Austin（德克萨斯大学奥斯汀分校）
We study the convergence of a class of gradient-based Model-Agnostic Meta-Learning (MAML) methods and characterize their overall complexity as well as their best achievable accuracy in terms of gradient norm for nonconvex loss functions. We start with the MAML method and its first-order approximation (FO-MAML) and highlight the challenges that emerge in their analysis. By overcoming these challenges not only we provide the first theoretical guarantees for MAML and FO-MAML in nonconvex settings, but also we answer some of the unanswered questions for the implementation of these algorithms inclu…
University of Oxford（牛津大学）；University College London（伦敦大学学院）
When experience is scarce, models may have insufficient information to adapt to a new task. In this case, auxiliary information - such as a textual description of the task - can enable improved task inference and adaptation. In this work, we propose an extension to the Model-Agnostic Meta-Learning algorithm (MAML), which allows the model to adapt using auxiliary information as well as task experience. Our method, Fusion by Meta-Initialization (FuMI), conditions the model initialization on auxiliary information using a hypernetwork, rather than learning a single, task-agnostic initialization. F…
End-to-end Speech Translation (ST) models have several advantages such as lower latency, smaller model size, and less error compounding over conventional pipelines that combine Automatic Speech Recognition (ASR) and text Machine Translation (MT) models. However, collecting large amounts of parallel data for ST task is more difficult compared to the ASR and MT tasks. Previous studies have proposed the use of transfer learning approaches to overcome the above difficulty. These approaches benefit from weakly supervised training data, such as ASR speech-to-transcript or MT text-to-text translation…
Graph Neural Networks (GNNs) are a framework for graph representation learning, where a model learns to generate low dimensional node embeddings that encapsulate structural and feature-related information. GNNs are usually trained in an end-to-end fashion, leading to highly specialized node embeddings. However, generating node embeddings that can be used to perform multiple tasks (with performance comparable to single-task models) is an open problem. We propose a novel meta-learning strategy capable of producing multi-task node embeddings. Our method avoids the difficulties arising when learni…
