On Episodes, Prototypical Networks, and Few-Shot Learning

1 Introduction . The problem of few-shot learning ( FSL ) – classifying examples from previously unseen classes given only a handful of training data – has considerably grown in popularity within the machine learning community in the last few years . The reason is likely twofold . First , being able to perform well on FSL problems is important for several applications , from learning new symbols [ 23 ] to drug discovery [ 2 ] . Second , since the aim of researchers interested in meta-learning is to design systems that can quickly learn novel concepts by generalising from previously encountered learning tasks , FSL benchmarks are often adopted as a practical way to empirically validate meta-learning algorithms . To the best of our knowledge , there is not a widely recognised definition of meta-learning . In a recent survey , Hospedales et al . [ 22 ] informally describe it as “ the process of improving a learning algorithm over multiple learning episodes ” . In practical terms , following the compelling rationale that “ test and train conditions should match ” [ 48 , 13 ] , several seminal meta-learning papers ( e.g . [ 48 , 32 , 14 ] ) have emphasised the importance of organising training into episodes , i.e . learning problems with a limited amount of “ training ” ( the support set ) and “ test ” examples ( the query set ) to mimic the test-time scenario presented by FSL benchmarks . However , several recent works ( e.g . [ 9 , 49 , 11 , 44 ] ) showed that simple baselines can outperform established FSL meta-learning methods by using embeddings pre-trained with the standard crossentropy loss , thus casting a doubt on the importance of episodes in FSL . Inspired by these results , we aim at understanding the practical usefulness of episodic learning in popular FSL methods relying on metric-based nonparametric classifiers such as Matching and Prototypical Networks [ 48 , 40 ] . We chose this family of methods because they do not perform any adaptation at test time . This allows us ∗Work done while research intern at FiveAI 35th Conference on Neural Information Processing Systems ( NeurIPS 2021 ) . to test the efficacy of episodic training without having to significantly change the baseline algorithms , which could potentially introduce confounding factors . In this work we perform a case study focussed on Matching Networks [ 48 ] and Prototypical Networks [ 40 ] , and we show that within this family of methods episodic learning a ) is detrimental for performance , b ) is analogous to randomly discarding examples from a batch and c ) introduces a set of superfluous hyperparameters that require careful tuning . Without episodic learning , these methods are closely related to the classic Neighbourhood Component Analysis ( NCA ) [ 19 , 35 ] on deep embeddings and achieve , without bells and whistles , an accuracy that is competitive with recent methods on multiple FSL benchmarks : miniImageNet , CIFAR-FS and tieredImageNet . PyTorch code is available at https : //github.com/fiveai/on-episodes-fsl . 2 Background and method . This section is divided as follows : Sec . 2.1 introduces episodic learning and illustrates a data efficiency issue encountered with nonparametric few-shot learners based on episodes ; Sec . 2.2 introduces the losses from Snell et al . [ 40 ] , Vinyals et al . [ 48 ] and Goldberger et al . [ 19 ] which we use throughout our experiments ; and Sec . 2.3 explains the three options we explored to perform FSL classification with previously-trained feature embeddings . 2.1 Episodic learning . A common strategy to train FSL methods is to consider a distribution Ê over possible subsets of labels that is as close as possible to the one encountered during evaluation E 2 [ 48 ] . Each episodic batch BE = { S , Q } is obtained by first sampling a subset of labels L from Ê , and then sampling images constituting both support set S and query set Q from the set of images with labels in L , where S = { ( s1 , y1 ) , . . . , ( sn , yn ) } , Q = { ( q1 , y1 ) , . . . , ( qm , ym ) } , and Sk and Qk denote the sets of images with label y = k in the support set and query set respectively . For most methods , this corresponds to training on a series of mini-batches in which each image belongs to either the support or the query set . Support and query sets are constructed such that they both contain all the classes of L , and a fixed number of images per class . Therefore , episodes are defined by three variables : the number of classes w = |L| ( the “ ways ” ) , the number of examples per class in the support set n = |Sk| ( the “ shots ” ) , and the number of examples per class in the query set m = |Qk| . During evaluation , the set { w , n , m } defines the problem setup . Instead , at training time { w , n , m } can be seen as a set of hyperparameters controlling the batch creation , and that ( as we will see in Sec . 3.2 ) requires careful tuning . 2Note that , in FSL , the sets of classes encountered during training and evaluation are disjoint . In a Maximum Likelihood Estimation framework , training on these episodes can be written as arg max θ E L∼Ê E S∼L Q∼L ∑ ( qi , yi ) ∈Q logPθ ( yi|qi , S , ρ ) . ( 1 ) For the sake of brevity , with a slight abuse of notation we omit the function fθ ( e.g . a deep neural network ) which is used to obtain a representation for the images in S and Q , and whose parameters θ are optimised during the training process . Note that the optimisation of Eq . 1 depends on an optional set of parameters ρ . This is obtained by an “ inner ” optimisation procedure , whose scope is limited to the current episode [ 22 ] . The idea is that the “ outer ” optimisation loop , by attending to a distribution of episodes , will appropriately shift the inductive bias of the algorithm located in the inner loop , thus learning how to learn [ 47 ] . In recent years , many interesting proposals have been made about what form ρ should have , and how it should be computed . For instance , in MAML [ 14 ] ρ takes the form of an update of the global parameters θ , while Ravi and Larochelle [ 32 ] learn to optimise by considering ρ as set of the hyper-parameters of the optimiser ’ s update rule . Other methods , such as Matching and Prototypical Networks [ 48 , 40 ] , avoid learning a separate set of parameters ρ altogether , and utilise a nonparametric learner ( such as nearest neighbour classifiers ) at the inner level . We chose to focus our case study on these methods not only because they have been seminal for the community , but also for ease of experimental design . Having ρ = ∅ considerably reduces the design complexity of the algorithm , thus allowing precise ablations to understand the efficacy of episodic learning without considerably changing the nature of the original algorithms . Considerations on data efficiency . The constraints imposed by episodic learning on the role each image has in a training batch has subtle but important implications , illustrated in Fig . 1 by highlighting the number of distances contributing to the loss . By dividing batches between support and query set ( S and Q ) during training , episodes have the side effect of disregarding many of the distances between labelled examples that would constitute useful training signal for nonparametric FSL methods . More specifically , for metric-based nonparametric methods , the number of training distances that are omitted in a batch because of the episodic strategy grows quadratically as O ( w2 ( m2 + n2 ) ) ( derivation shown in Appendix A ) . Table 1 breaks down this difference in terms of gradients from positives and negatives distance pairs ( which we simply refer to as positives and negatives throughout the rest of the paper ) . In a typical training batch with w = 20 , m = 15 and n = 5 [ 40 ] , ignoring the episodic constraints increases the number of both positives and negatives by more than 150 % . In the remainder of this paper , we conduct a case study to illustrate how this issue affects two of the most popular FSL algorithms relying on nonparametric approaches at the inner level : Prototypical Networks [ 40 ] and Matching Networks [ 48 ] . 2.2 Loss functions . Prototypical Networks ( PNs ) [ 40 ] are one of the most popular and effective approaches in the few-shot learning literature . They are at the core of several recently proposed FSL methods ( e.g . [ 27 , 18 , 1 , 50 , 7 ] ) , and they are used in a number of applied machine learning works ( e.g . EEG scan analysis for autism [ 36 ] and glaucoma grading [ 17 ] ) . During training , episodes consisting of a support set S and a query set Q are sampled as described in Sec . 2.1 . Then , a prototype for each class k is computed as the mean embedding of the samples from the support set belonging to that class : ck = ( 1/|Sk| ) · ∑ ( si , yk ) ∈Sk fθ ( si ) , where fθ is a deep neural network with parameters θ learned via Eq . 1 . Let C = { ( c1 , y1 ) , . . . , ( ck , yk ) } be the set of prototypes and corresponding labels . The loss can be written as follows : LPNs = −1 |Q| ∑ ( qi , yi ) ∈Q log ( exp−‖fθ ( qi ) − cyi‖2∑ k′ exp−‖fθ ( qi ) − ck′‖2 ) , where k′ is an index that goes over all classes . Matching Networks ( MNs ) [ 48 ] are closely related to PNs in the multi-shot case and equivalent in the 1-shot case . Rather than aggregating the embeddings of the same class into prototypes , this loss directly computes a softmax over individual embeddings of the support set , as : LMNs = −1 |Q| ∑ ( qi , y ) ∈Q log ∑ sj ∈Sy exp−‖fθ ( qi ) − fθ ( sj ) ‖2 ∑ sk ∈S exp−‖fθ ( qi ) − fθ ( sk ) ‖2 . In their work , Vinyals et al . [ 48 ] use the cosine rather than the Euclidean distance . However , ( as [ 40 ] ) we observed that the Euclidean distance is a better choice for FSL problems , and thus we use it in all the losses of our experiments . Note that Vinyals et al . [ 48 ] also suggest a variant to LMNs ( MNs with “ Full Context Embeddings ” ) , where an LSTM ( with an extra set of parameters ) is used to condition the way the inputs are embedded in the current support set . In our experiments , we did not consider this variant as it falls in the category of adaptive episodic learning approaches ( ρ 6= ∅ , see Sec . 2.1 ) . Neighbourhood Component Analysis ( NCA ) . LMNs and LPNs sum over the likelihoods that a query image belongs to the same class of a certain sample ( or prototype ) from the support set by computing the softmax over the distances between the query and the support samples ( or prototypes ) . This is closely related to the Neighbourhood Component Analysis approach by Goldberger et al . [ 19 ] ( and expanded to the non-linear case by Salakhutdinov et al . [ 35 ] and Frosst et al . [ 15 ] ) , except for a few important differences which we discuss at the end of this section . Let i ∈ [ 1 , b ] be the indices of the images within a batch B . The NCA loss can be written as : LNCA = −1 |B| ∑ i∈1 , ... , b log ∑ j∈1 , ... , b j 6=i yi=yj exp−‖zi − zj‖2 ∑ k∈1 , ... , b k 6=i exp−‖zi − zk‖2 , where zi = fθ ( xi ) is an image embedding and yi its corresponding label . By minimising this loss , distances between embeddings from the same class will be minimised , while distances between embeddings from different classes will be maximised . Importantly , note how the concepts of support set and query set here do not exist . More simply , the images ( and respective labels ) constituting the batch B = { ( x1 , y1 ) , . . . , ( xb , yb ) } are sampled uniformly . Given the similarity between these three losses , and considering that PNs and MNs do not perform episode-specific parameter adaptation , { w , m , n } can be simply interpreted as the set of hyperparameters controlling the sampling of mini-batches during training . More specifically , PNs , MNs and NCA differ in three aspects : I . First and foremost , due to the nature of episodic learning , PNs and MNs only consider pairwise distances between the query and the support set ( Fig . 1 left ) ; NCA instead uses all the distances within a batch and treats each example in the same way ( Fig . 1 right ) . II . Only PNs rely on the creation of prototypes . III . Because of how L , S and Q are sampled in episodic learning ( Eq . 1 ) , for PNs and MNs some images might be sampled more frequently than others ( sampling “ with replacement ” ) . NCA instead visits every image of the dataset once for each epoch ( sampling “ without replacement ” ) . To investigate the effects of these three differences , in Sec . 3 we conduct a wide range of experiments .

Summary: This paper studies the question of whether episodes (using a split of support & query sets) are necessary for non-parameteric approaches for few-shot learning (examples include Prototypical Networks & Matching Networks). The authors propose a Neighborhood Component Analysis-based method for few-shot learning, where a mini-batch consists of examples from a subset of base classes, with no support or query split. The NCA loss then involves distances computed across all examples in the mini-batch rather than only using distances computed between support and query examples (as is done in Prototypical Networks & Matching Networks). The authors show that their proposed method is able to achieve better performance for 1-shot & 5-shot classification in miniImageNet, cifar-fs, and tieredImageNet benchmarks. They also speculate that the proposed method performs better because of its use of more distance computations in the loss compared to Prototypical Networks & Matching Networks and confirm this by conducting an experiments where they randomly drop distances in their NCA loss and showing this has a negative impact on performance.