Clustered Task-Aware Meta-Learning by Learning from Learning Paths

1 INTRODUCTION The astonishing performance of deep learning relies on large amounts of data , which are not always available . Humans , on the other hand , are able to learn new tasks much more quickly , leveraging prior experience to relate knowledge among tasks . Inspired by this property of human intelligence , meta-learning ( also known as learning to learn ) ( Thrun & Pratt , 1998 ) acquires transferable commonalities from existing tasks for more efficient learning of new tasks . Although recent advances in meta-learning have demonstrated success in fields like few-shot image classification and cold-start recommendation ( Vinyals et al. , 2016 ; Snell et al. , 2017 ; Ravi & Larochelle , 2016 ; Munkhdalai & Yu , 2017 ) , most of them typically assume that all the tasks are drawn from a single distribution and face the challenge of handling tasks that come from different underlying distributions , a problem known as task heterogeneity ( Devos & Dandi , 2020 ; Yao et al. , 2020 ; Suo et al. , 2020 ) . To overcome this challenge , many recently developed works leverage task-specific information to customize the global meta-learner ( Vuorio et al. , 2019 ; Oreshkin et al. , 2018 ; Requeima et al. , 2019 ; Finn et al. , 2018 ; Yoon et al. , 2018 ; Li et al. , 2019 ; Rusu et al. , 2018 ) . To further consider generalization among related tasks , methods that perform various types of task clustering are proposed ( Yao et al. , 2019 ; 2020 ; Suo et al. , 2020 ; Lu et al. , 2020 ; Dong et al. , 2020 ; Lin et al. , 2021 ) . Despite their effectiveness in improving over the globally shared meta-learning algorithms , they learn task representations only based on the input data distribution in ( original or projected ) feature space , while the interaction between data and the base-learner is often neglected . The amount of information contained in the task-specific data about a network that is responsible for performing the task can be seen as a good representation of the task itself . This can be manifested as the gradients of the network parameters with respect to the task-specific loss , or Fisher Information Matrix ( FIM ) which indicates the importance of different network parameters in solving the task . Achille et al . ( 2019 ) introduce a task embedding method based on FIM to assist the meta task of selecting a pre-trained feature extractor . To alleviate the conflict issue of the global initialization methods , Baik et al . ( 2020 ) utilize task gradients at the initialization as task representation to produce attenuation . However , these methods represent task based on gradients at only a single point in parameter space ( e.g. , at the initialization ) , while the potential of exploiting a wider range on the task manifold remains unexplored . For the parameter initialization approach , the key of success lies in that the task adaptation process is accounted for when training the meta-learner . This task-specific learning may involve multiple gradient descent steps , and thus constitute a learning trajectory on the loss surface ( Flennerhag et al. , 2018 ; Zenke et al. , 2017 ) . To better characterize the task optimization behaviors , it is more beneficial to look at the complete learning trajectory as opposed to only the first gradient step at the initialization , as it is likely that tasks with similar gradients at first will have their learning paths diverged as the update proceeds . With that in mind , we are motivated to leverage the entire learning path for better task representation , which will then be used to condition the global initialization . In particular , we propose a Clustered Task-aware Meta-Learning ( CTML ) framework , building upon the well-known global initialization method Model-Agnostic Meta-Learning ( MAML ) ( Finn et al. , 2017 ) . To address the problem of task heterogeneity with a good balance between customization and generalization , we modulate the common initialization based on task representation learned from both its local task-specific information and global clustering results . In addition to using the input features to represent task , we further leverage the task learning path with respect to the baselearner to characterize task from the perspective of optimization . To facilitate clustering , we devise a GRU-based meta path learner which automatically abstracts path representations from the step-wise geometric quantities . We realize that it may be too costly to rehearse the entire learning process for task representation . Hence , we further propose a shortcut tunnel to bypass the rehearsed task learning during meta-testing and predict path cluster assignment directly from the feature cluster assignment . We carefully study the effectiveness of CTML in two real-world application domains : few-shot image classification and cold-start recommendation , and show that our method is able to outperform the baselines with comparable inference time . 2 RELATED WORK To address the problem of task heterogeneity , recent developments often tailor the shared metalearner to different tasks using task-specific information . Finn et al . ( 2018 ) and Yoon et al . ( 2018 ) model the uncertainty exists in task distribution with probabilistic framework . Oreshkin et al . ( 2018 ) , Li et al . ( 2019 ) , Requeima et al . ( 2019 ) and Bateni et al . ( 2020 ) condition the base network on taskspecific data by designing a meta adaptation network . To enable more robust training , Rusu et al . ( 2018 ) propose to learn a lower-dimensional latent space specific to each task to generate the base network parameters . Relevant to our work , Vuorio et al . ( 2019 ) build upon MAML and modulate the global initialization based on the task representation learned from an encoder network . However , customizing the common knowledge to individual task without considering the relations among similar tasks may lead to poor generalization . In regard to this , Dong et al . ( 2020 ) and Lin et al . ( 2021 ) apply K-means clustering on users ( treated as tasks ) based on their profile information to address the cold-start problem in recommender systems . Yao et al . ( 2019 ) employ a hierarchical structure to model the nested relationships inherent in domains with clear taxonomy , such as image classification . Yao et al . ( 2020 ) further develop an automatic relational graph method by constructing a meta-knowledge graph to preserve and propagate the global structural information . Suo et al . ( 2020 ) take advantages of an external knowledge base to facilitate task clustering . Despite their effectiveness in generalizing across tasks , they rely solely on input features for task representation , while the interaction between task data and the base-learner ( e.g. , gradients ) is often neglected . The idea of using gradients for task representation is commonly adopted in continual learning ( Kirkpatrick et al. , 2017 ; Zenke et al. , 2017 ; Aljundi et al. , 2018 ) . In the context of task-aware metalearning , Achille et al . ( 2019 ) propose to use the FIM of a “ probe ” network to represent task , and Baik et al . ( 2020 ) leverage gradients of the globally initialized parameters to generate task-specific attenuation . However , these methods utilize gradients only at a single point in parameter space , whereas meta-learning often incorporates the entire task learning trajectory while learning the metalearner . Garcia et al . ( 2021 ) perform task clustering at each adaptation step and aggregate the gradients for more stable task adaptation . Flennerhag et al . ( 2018 ) highlight the benefits of leveraging the learning paths for deriving the common knowledge , but the proposed algorithm only relies on one geometric quantity – the length of the path , which can be limited in characterizing the learning paths , and the initialization is not tailored to specific task . Our work serves to address the above-mentioned limitations , whereby higher-order behaviours of the learning path are also taken into account to learn a better task representation for modulating the global initialization . 3 PRELIMINARIES AND PROBLEM STATEMENT In task-heterogeneous setting , tasks { T1 , T2 , ... , TN } are sampled from a mixture of task distributions { p1 ( T ) , p2 ( T ) , ... } , where the number of underlying distributions is usually unknown . The goal of meta-learning is to learn the sharable knowledge by training the meta-learner on a set of metatraining tasks Ttr = { Ti } N tr i=1 , and test it on a set of meta-testing tasks Tte = { Ti } Ni=Ntr+1 . For each task Ti ∈ Ttr ∪Tte , its samples are further divided into a training set ( also termed support set ) DtrTi = { ( xi , j , yi , j ) } ntrTi j=1 and a test set ( also termed query set ) DteTi = { ( xi , j , yi , j ) } nTi j=ntrTi +1 , which together form an “ episode ” ( Vinyals et al. , 2016 ) . This episodic scheme allows us to train tasks to learn fast during meta-training , and test the learning performance of new tasks in the same way during meta-testing . For few-shot learning , the size of the training set ntrTi is usually small . Our work builds upon Model-Agnostic Meta-Learning ( MAML ) ( Finn et al. , 2017 ) . It implements the meta-learner as an initialization of parameters θ0 ∈ RD of the base-learner fθ responsible for the prediction task . During meta-training , the global initialization θ0 is first adapted to each metatraining task Ti ∈ Ttr by learning on the respective training set DtrTi , which yields the task-specific parameters θTi . After that , loss is computed on the test set DteTi based on θTi and backward propagated to update θ0 . Taking one step adaptation as an example , the meta-optimization is as follows : θ∗0 = argmin θ0 ∑ Ti∈Ttr L ( fθTi , D te Ti ) = argmin θ0 ∑ Ti∈Ttr L ( fθ0−α▽θL ( fθ0 , DtrTi ) , D te Ti ) , ( 1 ) where α is the adaptation rate , L ( fθ , D ) can be mean square error loss for regression problem ( i.e. , 1|D| ∑ ( x , y ) ∈D ( y − fθ ( x ) ) 2 ) , or cross-entropy loss for classification problem ( i.e. , − 1|D| ∑ ( x , y ) ∈D y log fθ ( x ) ) . During meta-testing , the learned initialization θ ∗ 0 is adapted to each meta-testing task Ti ∈ Tte using DtrTi , and the learning performance is evaluated on D te Ti . However , with a globally shared initialization , MAML is not capable of handling task heterogeneity . Though task-specic methods have been developed to tailor the global initialization , they lack explicit modeling of the global clustering structure . Hence , a framework considering both task-specific information and global structure is desired . 4 METHODOLOGY Grounded on MAML , our CTML framework modulates the common initialization based on task representation learned from two different perspectives enhanced with the clustering information . Specifically , given the training set of a specific task , we first conduct rehearsed learning of the task from the common initialization and compute a set of step-wise quantities along the learning path . This set of values will be inputted into the meta path learner to generate a task-specific path embedding . On the other hand , input features of the given task will also be extracted to form the feature embedding . Both feature and path embeddings will then undergo soft K-means clustering in their respective latent space . Finally , the two embeddings enriched with cluster information will be aggregated to produce the task-aware modulation to be applied on the global initialization . From this modulated intialization , standard MAML follows , which conducts the actual task learning on the training set and perform meta-update across tasks using the test sets . To further improve the inference efficiency during meta-testing , a shortcut tunnel is used to bypass the rehearsed learning process and generate the path cluster assignment directly from the feature cluster assignment . An overview of CTML framework is shown in Figure 1 . In the following sections , we first elaborate on task representation learning based on learning path and features respectively , and then introduce the task-aware modulation and the shortcut tunnel . 4.1 TASK REPRESENTATION BASED ON LEARNING PATH 4.1.1 PATH CONSTRUCTION To obtain representation of task Ti based on learning path , we first conduct a τ -step rehearsed learning from the global initialization θ0 on training set DtrTi . Applying the same gradient descent update as in MAML , we obtain the updated parameters at each step t ∈ { 1 , 2 , ... , τ } by : θ̃tTi = θ̃ t−1 Ti − α▽θL ( fθ̃t−1Ti , D tr Ti ) , where θ̃ 0 Ti = θ0 . The overhead ·̃ is used to differentiate between the parameters updated from the rehearsed learning and from the actual learning . Joining the discrete points { ( θ̃tTi , L̃ t Ti ) } τ t=0 constitutes the rehearsed learning path of Ti ( we write L ( fθ̃tTi , DtrTi ) as L̃ t Ti for brevity ) on the unique task manifold MTi ∈ R D+1 , characterized by the base-learner function and the task-specific data distribution . We collect the coordinates of point ( θ̃tTi , L̃ t Ti ) at each update step t ∈ { 0 , 1 , 2 , ... , τ } to indicate the exact learning trajectory . To account for higher-order behaviours of the learning path , we further incorporate the gradients ▽θL̃tTi and the Fisher Information Matrix ( FIM ) F̃ tTi at each step , which specify the direction and curvature respectively . Note that by definition , the FIM is equivalent to the Hessian matrix in expectation for cross-entropy loss : F̃ tTi = Ex , y∼Ti [ ▽θL ( fθ̃tTi , x , y ) ▽θL ( fθ̃tTi , x , y ) ⊤ ] = Ex , y∼Ti [ ▽2θL ( fθ̃tTi , x , y ) ] . It provides a measure of the amount of information that task Ti contains about the parameters θ ( Gianfelici & Battistelli , 2009 ) . Following Achille et al . ( 2019 ) , we assume negligible correlation among different parameters and consider only the diagonal entries of F̃ tTi , denoted by F̃ t Ti . As a result , we obtain a set of step-wise quantities { ( θ̃tTi , L̃ t Ti , ▽θL̃ t Ti , F̃ t Ti ) } τ t=0 that adequately describes the rehearsed learning path . As opposed to looking at only a single point in the parameter space ( e.g. , use gradients at the initialization to represent task ( Baik et al. , 2020 ) ) , it is benificial to take into account the entire learning path traversed , which gives a more complete picture of the optimization process . Take for instance , two tasks Ta and Tb may have gradients forming a small angle ψ0a , b at the initialization , i.e. , cos ( ψ0a , b ) > 0 . However , it is likely that their learning paths will diverge as the gradient update proceeds , i.e. , the accumulated angle ∑τ t=0 cos ( ψ t a , b ) < 0 ( see dotted paths of T2 and T3 in Figure 2 ) . Conversely , the tasks may have gradients pointing at different directions at first , but eventually converge towards the same direction as the learning proceeds ( see dotted paths of T1 and T2 ) . Considering a single step only can be restrictive for task representation from the optimization perspective . Looking further down the paths allows better characterization of the learning behaviors and even the flexibility to determine the “ important ” steps ( as shown in experiments later ) , producing more informative task representations . Furthermore , leveraging path representations serves to provide a stronger inductive bias to condition the initialization in favorable direction , as the rehearsed learning path can be considered as encoding the quality of the initialization with respect to the task ( i.e. , tasks with longer and more zigzag paths imply that the initialization is not a good one for them ) . 4.1.2 PATH REPRESENTATION Having collected the step-wise quantities { ( θ̃tTi , L̃ t Ti , ▽θL̃ t Ti , F̃ t Ti ) } τ t=0 along the rehearsed learning path , the problem now is how to evaluate “ similarity ” among different task learning paths for clustering . Previous works with single-step gradients usually compute dot product or cosine similarity between gradients of two tasks ( Riemer et al. , 2018 ; Katoch et al. , 2019 ; Yu et al. , 2020 ; Baik et al. , 2020 ) . With multiple steps , a straightforward adaptation will be to simply sum up the similarity scores at all steps . However , this human-defined rule may not be the best way of measuring path similarity . Instead , we propose to employ a meta path learner to automatically learn a path embedding from the step-wise quantities , and then measure similarity based on these vector representations . In other words , the meta path learner induces a latent space on which the distance metric best characterizes what is considered as “ similar ” ( or “ disimilar ” ) among task learning paths . Before we delve into the design of the path learner , we first elaborate how we construct the functional input from the step-wise quantities . Recall that at each step t , we have θ̃tTi , ▽θL̃ t Ti , F̃ t Ti ∈ R D and L̃tTi ∈ R. To obtain a regularly shaped input , we duplicate L̃ t Ti to form a D-dimensional vector and stack it together with the other 3 components to form a matrix PtTi ∈ R 4×D . Further stacking the τ + 1 steps gives the overall 3-D matrix PTi ∈ R ( τ+1 ) ×4×D . To avoid high model complexity , we apply the meta path learner coordinate-wise on the base-learner parameters . That is , the same path learner is used to process the matrix PdTi ∈ R ( τ+1 ) ×4 independently for all d ∈ { 1 , 2 , ... , D } . For the path learner design , we propose to leverage the Gated Recurrent Units ( GRUs ) to model the sequential dependencies among steps1 . Specifically , the hidden state hd , ti at step t is obtained by inputting the t-th sliced vector Pdt , : ∈ R4 of PdTi and the hidden state h d , t−1 i at step t − 1 into the GRU cell : rd , ti = σ ( Wr · [ h d , t−1 i , P d t , : ] ) , z d , t i = σ ( Wz · [ h d , t−1 i , P d t , : ] ) , h̃d , ti = tanh ( Wh̃ · [ r d , t i ⊙ h d , t−1 i , P d t , : ] ) , h d , t i = ( 1− z d , t i ) ⊙ h d , t−1 i + z d , t i ⊙ h̃ d , t i , ( 2 ) where [ · , · ] denotes concatenation , σ ( · ) is the sigmoid activation , rd , ti , z d , t i ∈ ( 0 , 1 ) are gates that control how much of past and present information to be retained , and Wr , Wz , Wh̃ are learnable weights shared across all steps . For step t = 0 , we use a zero-initialized input hidden state . The path representation hdi at the d-th dimension is obtained from the final step hidden state h d , τ i via a linear transformation : hdi = Wo · h d , τ i + bo . Concatenating the path representations at all D dimensions and passing it through a fully-connected layer gives the final path embedding epathTi ∈ R de for task Ti : epathTi = FC ( [ h 1 i , ... , h D i ] ⊤ ) . Inspired by ( Suo et al. , 2020 ; Dong et al. , 2020 ; Lin et al. , 2021 ) , we handle task heterogeneity without jeopardizing generalization among similar tasks by employing a simple yet effective soft K-means clustering on the path embeddings2 . Specifically , we maintain kpath learnable cluster centroids { apathj |∀j ∈ [ 1 , kpath ] } ∈ Rkpath×de for path embeddings . The cluster-enhanced path embedding ẽpathTi ∈ R de is simply the weighted sum of the cluster centroids : ẽpathTi = ∑kpath j=1 qpathij a path j , ( 3 ) where qpathij = ( 1+||epathTi −a path j || 2 ) −1 ∑ j′ ( 1+||e path Ti −apath j′ || 2 ) −1 is the probability of assigning epathTi to cluster j , computed using the Student ’ s t -distribution as a kernel , following Xie et al . ( 2016 ) . 4.2 TASK REPRESENTATION BASED ON FEATURES Task representation based on learning path can be interpreted as encoding the conditional distribution p ( y|x ) ( as the learning path is informed by the labels ) , whereas task is best described by the joint distribution p ( x , y ) = p ( y|x ) p ( x ) . Therefore , we further incorporate the marginal distribution of input features by learning another representation solely based on features , as what have been done in most of the existing task-aware meta-learning methods ( Vuorio et al. , 2019 ; Yao et al. , 2019 ; 2020 ; Lin et al. , 2021 ) . Another significance of including feature-based representation is that , we can create a mapping between the path and feature representations to bypass the rehearsed learning during the meta-testing phase ( details will be elaborated in Section 4.4 ) . Generally , the design of the feature extractor may vary for different application domains . Let E ( · ) denote an arbitrary feature extractor , the feature embedding efeatTi ∈ R de for task Ti is obtained by aggregating the extracted features of all samples in the training set : efeatTi = 1 ntrTi ∑ntrTi j=1 ( E ( xi , j ) ) . Similar to the path embeddings , we also employ a soft K-means clustering to promote generalization among related feature embeddings . Specifically , given kfeat cluster centroids { afeatj |∀j ∈ 1Other network designs are possible for the meta path learner . We compare their efficacy in ablation study . 2Other clustering algorithms are possible . We leave it to future work . [ 1 , kfeat ] } ∈ Rkfeat×de , the cluster-enhanced feature embedding ẽfeatTi ∈ R de is obtained by : ẽfeatTi = ∑kfeat j=1 qfeatij a feat j , ( 4 ) where qfeatij = ( 1+||efeatTi −a feat j || 2 ) −1 ∑ j′ ( 1+||e feat Ti −afeat j′ || 2 ) −1 is the probability of assigning efeatTi to cluster j . 4.3 TASK-AWARE MODULATION We aggregate the path and feature embeddings via a learnable weight vector λ to generate the final task representation . This task representation will then be used to produce a modulation to tailor the global initialization θ0 to specific task . The modulated initialization θ0i for task Ti is obtained by : θ0i = σ ( W · ( λẽpathTi ⊕ ( 1− λ ) ẽ feat Ti ) + b ) ⊙ θ0 , ( 5 ) where W , b , λ are learnable parameters , σ ( · ) is the sigmoid function , ⊕ and ⊙ denote element-wise addition and element-wise multiplication respectively . From this modulated initialization , task Ti will undergo regular task adaptation learning for τ steps . After that , the meta-learner φ will be updated by optimizing loss across all the test sets , the same procedure as in MAML . The meta-optimization objective is : min φ ∑ Ti∈Ttr L ( fθ0i−α∑τ−1t=0 ▽θL ( fθtTi , D tr Ti ) , D te Ti ) , ( 6 ) where φ includes the global initialization θ0 , the learnable parameters used to generate the clusterenhanced path and feature embeddings , and also the final modulation . 4.4 IMPROVING META-TESTING EFFICIENCY VIA SHORTCUT TUNNEL The need to conduct rehearsed learning before the actual learning of each task inevitably leads to twice the inference time compared to the original MAML . Though it is not possible to bypass the rehearsed learning during meta-training as we need to optimize the meta path learner and the downstream cluster centroids , it is possible to improve the inference efficiency during meta-testing with the well-trained cluster centroids . Note that the cluster-enhanced path embedding ẽpathTi used to generate modulation is obtained based on the path cluster centroids and the soft assignment ( see equation 3 ) . If we can estimate the path cluster assignment without actually going through the rehearsed learning process to obtain epathTi , we will be able to reduce the inference time by half . Inspired by this , we devise a shortcut tunnel to predict the path cluster assignment directly from the feature cluster assignment of the same task . The assumption behind is that there exists a one-toone mapping between the feature cluster assignments and the path cluster assignments , which can be linear or non-linear . For better expressivity , we employ a two-layer fully connected network to approximate the mapping and use it to reconstruct the path cluster assignment of Ti from its feature cluster assignment : q̂pathi = softmax ( FCs ( q feat i ) ) , ( 7 ) where qfeati = [ q feat i1 , ... , q feat ikfeat ] ⊤ ∈ Rkfeat and q̂pathi = [ q̂ path i1 , ... , q̂ path ikpath ] ⊤ ∈ Rkpath . The mapping handles cases where kpath ∕= kfeat and allows for different softness of cluster assignments . To align the reconstructed and the actual assignment distributions , we use the Jensen-Shannon ( JS ) divergence ( a symmetrized version of Kullback-Leibler ( KL ) divergence ) as the reconstruction loss : Lr ( DtrTi ) = JS ( q̂ path i ||q path i ) = 1 2 KL ( q̂pathi ||pi ) + 1 2 KL ( qpathi ||pi ) , ( 8 ) where pi = 12 ( q̂ path i + q path i ) , and KL ( q||p ) = ∑ j qj log qj pj is the KL divergence . During metatraining , the reconstruction loss is optimized together with the loss in equation 6 , resulting in the following overall objective : min φ ∑ Ti∈Ttr L ( fθ0i−α∑τ−1t=0 ▽θL ( fθtTi , D tr Ti ) , D te Ti ) + ζLr ( D tr Ti ) , ( 9 ) where ζ controls the weight of Lr ( DtrTi ) , and φ now further includes the shortcut tunnel parameters . To apply the shortcut tunnel during meta-testing , we simply replace qpathi with q̂ path i in equation 3 to obtain the cluster-enhanced path embedding ẽpath , i.e. , ẽpathTi = ∑kpath j=1 q̂ path ij a path j . The overall meta-training and meta-testing procedures of CTML are summarized in Algorithm 1 and 2 . Algorithm 1 : Meta-Training of CTML Required : Meta-training tasks Ttr = { Ti } N tr i=1 ; Number of steps τ ; Adaptation rate α ; Meta-update rate β ; Number of clusters kpath and kfeat 1 Randomly initialize all learnable parameters φ 2 while not converged do 3 Sample a batch of tasks B from Ttr 4 for Ti ∈ B do 5 Extract efeatTi and cluster to obtain ẽ feat Ti 6 Extract epathTi from the τ -step rehearsed learning and cluster to obtain ẽpathTi 7 Aggregate ẽfeatTi and ẽ path Ti to generate θ0i 8 Evaluate L ( fθTi , D te Ti ) and Lr ( D tr Ti ) 9 Meta-update φ ← φ− β▽φ ∑ Ti∈B [ L ( fθTi , D te Ti ) + ζLr ( D tr Ti ) ] Algorithm 2 : Meta-Testing of CTML Required : Meta-testing tasks Tte = { Ti } Ni=NtrTi+1 ; Number of steps τ ; Adaptation rate α ; Number of clusters kpath and kfeat ; Meta-trained φ 1 for Ti ∈ Tte do 2 Extract efeatTi and cluster to obtain ẽ feat Ti 3 if use shortcut approximation then 4 Generate path assignment via the shortcut tunnel to obtain ẽpathTi 5 else 6 Extract epathTi from the τ -step rehearsed learning and cluster to obtain ẽpathTi 7 Aggregate ẽfeatTi and ẽ path Ti to generate θ0i 8 Evaluate L ( fθTi , D te Ti ) 5 EXPERIMENTS To test the effectiveness of the proposed CTML framework , we conduct experiments3 on two realworld application domains : few-shot image classification and cold-start recommendation . 5.1 FEW-SHOT IMAGE CLASSIFICATION Datasets and Settings In few-shot image classification , each task is defined as assigning images to N classes after training with K samples ( i.e. , N -way K-shot ) ( Vinyals et al. , 2016 ) . To simulate task heterogeneity , we construct the Mixture-Of-Datasets consisting of 6 widely used benchmark datasets : CUB-200-2011 ( Bird ) ( Wah et al. , 2011 ) , FGVC-Aircraft ( Aircraft ) ( Maji et al. , 2013 ) , FGVCx-Fungi ( Fungi ) ( Schroeder & Cui , 2018 ) , VGG Flower ( Flower ) ( Nilsback & Zisserman , 2008 ) , Describable Textures ( Texture ) ( Cimpoi et al. , 2014 ) , and GTSRB Traffic Signs ( Traffic Sign ) ( Stallkamp et al. , 2012 ) . Following Yao et al . ( 2019 ) , we create each task by sampling N classes from one of the 6 sub-datasets . This step ensures that tasks are drawn from different underlying distributions . Apart from the Mixture-Of-Datasets , we also conduct experiments on the MiniImagenet ( Ravi & Larochelle , 2016 ) , which can be considered as a task-homogeneous setting as tasks are sampled from a single dataset ( results on MiniImagenet are discussed in Appendix D.2 ) . More details on datasets and pre-processing can be found in Appendix A.1 . Baselines and Our Method We compare the performance of our method against 5 baselines : ( 1 ) global initialization method : MAML ( Finn et al. , 2017 ) ; ( 2 ) task-specific initialization methods : MMAML ( feature-based customization ) ( Vuorio et al. , 2019 ) and L2F ( gradient-based customization ) ( Baik et al. , 2020 ) ; ( 3 ) cluster-enhanced initialization methods : HSML ( feature-based hierarchical clustering ) ( Yao et al. , 2019 ) and ARML ( feature-based relational graph ) ( Yao et al. , 2020 ) . For our proposed CTML , we report both the original meta-testing performance ( CTML ) and the one applying shortcut approximation ( CTML ( approx. ) ) . We further implement 2 variants to compare with the baselines : CTML-feat which only depends on feature representation and CTML-path which only depends on path representation . Following Finn et al . ( 2017 ) , we adopt both the feature extractor E ( · ) and the base-learner fθ as a four-layer 3 × 3 convolutions with 32 filters for all the compared methods ( experiments on deeper network for base-learner can be found in Appendix D.1 ) . Further details on hyper-parameters settings can be found in Appendix B.1 . Baseline Comparison Table 1a presents the 5-way 1-shot & 5-shot classification performance on Mixture-Of-Datasets ( disentangled performance on individual sub-datasets are included in Appendix D.3 ) . First of all , we see that all the task-aware methods outperform the globally initialized MAML in task-heterogeneous setting . L2F with gradient-based conditioning on the initialization performs better than the feature-based MMAML . After considering generalization across similar tasks with some clustering techniques , HSML and ARML further improve over MMAML . Under our proposed CTML framework , CTML-feat with K-means clustering on features exhibits comparable performance as HSML and ARML . For CTML-path , the improvements over baselines are more 3All experiments are conducted using NVIDIA Tesla P100 GPU with 16GB memory . significant , especially for the 5-shot scenario where the learning paths are more reliable with more training data . The final CTML combining the advantages of the two representations achieves further improvements and outperforms the strongest baseline by a significant margin . Nevertheless , we can see that the meta-testing time of CTML is about twice that of the baselines due to the additional rehearsed learning process . After applying the shortcut approximation , we are able to cut down the inference time by half with just a small compromise on performance . Ablation Study We further conduct ablation study to test the efficacy of various CTML designs , including the step-wise input components , the path learner design , and the effects of clustering . From the results in Table 1b , we see that the lower-order quantities seem more important . Regarding the path learner design , our proposed GRU-based model achieves the best result due to its ability to capture the sequential dependencies among the steps . Attention model is also a competitive alternative , as the positional encodings and the causal mask serve to constrain the sequential order ( detailed descriptions of the different path learner designs can be found in Appendix C ) . As for clustering , we see that it indeed helps to generalize better as the global structure is explicitly considered . Understanding the Path Learning Mechanism To understand what has actually been learned by the GRU path learner , we randomly sample 6 tasks ( 5-way 1-shot ) from Aircraft , Flower and Traffic Sign sub-datasets ( Figure 3a ) and visualize their path learning results . In Figure 3b , we plot the 5- step rehearsed learning paths ( upper ) and the corresponding path embeddings generated by the GRU ( lower ) on the first two PCA components . We can see that the learning paths of tasks from the same sub-dataset generally move along the same direction , and the corresponding path embeddings are able to encode the path geometric information and positioned closely for similar tasks in the latent space . In Figure 3c , we further visulize the z gates and r gates of the GRU at each step t , which control how much to retain for the current step input and previous step memory respectively . We can see that the GRU path learner is able to identify the “ important ” steps ( darker-blue blocks ) and absorbs a larger portion of the inputs at these steps into the final path embedding . More examples on path visualization can be found in Appendix D.5 5.2 COLD-START RECOMMENDATION Datasets and Settings In the field of recommender systems , meta-learning has been applied to solve the cold-start problem , which refers to making recommendations related to new items or new users unseen before ( Vartak et al. , 2017 ; Volkovs et al. , 2017 ; Lee et al. , 2019 ) . Here , we focus on the user cold-start problem , where recommending for each user is treated as a task . This setting is task-heterogeneous in nature as users may belong to different preference groups . We conduct experiments on 3 public datasets : MovieLens-1M ( Harper & Konstan , 2015 ) , Yelp ( Dataset , 2019 ) , and Amazon-CDs ( McAuley et al. , 2015 ) , consisting of user ’ s ratings on movies , business services and CD products respectively . We split each dataset into meta-train/val/test by ratio 7/1/2 according to the timestamps of ratings . For each user , we use the first 10 samples as training set , and the rest as test set . Further details about datasets and pre-processing can be found in Appendix A.2 . Baselines and Our Method We choose 4 baselines that specifically tackle the user cold-start problem in recommender systems developed based on MAML . MeLU ( Lee et al. , 2019 ) is the first to adopt MAML in recommender systems with locally adapted decision layers ; MetaHIN ( Lu et al. , 2020 ) further incorporates Heterogeneous Information Networks ( HIN ) for data augmentation and multi-faceted adaptations ; MAMO ( Dong et al. , 2020 ) involves a global memory module to group users based on profile information and personalizes the initial bias ; PAML ( Yu et al. , 2021 ) personalizes the adaptation learning rate to allow for better fitting of the minor users . Following the baselines , we implement the feature extractor E ( · ) as several embedding lookup matrices for different features , and the base-learner fθ as a general recommender system model consists of embedding matrices followed by multiple fully-connected layers ( Lee et al. , 2019 ) . Detailed hyper-parameters settings can be found in Appendix B.2 . Baseline Comparison We evaluate the rating prediction performance in terms of mean absolute error ( MAE ) on 3 datasets , reported in Table 2 ( more detailed results can be found in Appendix D.4 ) . From the results , we can see that PAML and MAMO with personalized adaptation rate and initialization achieve better performance than MELU . Our CTML surpasses all the baselines , including MetaHIN which leverages HIN for data augmentation . Furthermore , we notice that for MovieLens , CTML-feat is slightly better than CTML-path , while for the other two datasets , CTML-path significantly outperforms CTML-feat . Given that the MovieLens dataset contains more complete user profile , this result implies the potential of representing users based on learning paths when the side information is seriously lacking or inaccurate , which is often the case in real-world applications . Number of Clusters For recommendation problem , there is no ground-truth grouping of users . In Figure 4a , we investigate the effect of varying the number of clusters kpath and kfeat for MovieLens dataset . We see that the best performance occurs at smaller kfeat values and larger kpath values , which implies greater complexity for clustering the learning paths . Our design of the shortcut tunnel allows flexibility to set different kpath and kfeat . In Figure 4b , we visualize the shortcut approximation of path assignment from feature assignment for a random user of MovieLens . We can see that the shortcut tunnel is rather reliable as the reconstructed path assignment highly resembles the actual path assignment . More examples on shortcut approximation are provided in Appendix D.6 . 6 CONCLUSION In this work , we introduce a CTML framework which leverages both features and learning paths for task representation . We employ a GRU-based meta path learner to process the step-wise geometric quantities , and introduce a shortcut tunnel to bypass the rehearsed learning during meta-testing . Experiments on two real-world application domains demonstrate the effectiveness of our framework . 7 REPRODUCIBILITY STATEMENT Our code can be accessed at : https : //github.com/didiya0825/ctml_code For data pre-processing and hyper-parameters settings , please see Appendix A and B . 8 ETHICS STATEMENT Since the proposed method is a general meta-learning framework , we do not foresee any potential ethic issue to the best our knowledge . REFERENCES Alessandro Achille , Michael Lam , Rahul Tewari , Avinash Ravichandran , Subhransu Maji , Charless C Fowlkes , Stefano Soatto , and Pietro Perona . Task2vec : Task embedding for meta-learning . In Proceedings of the IEEE/CVF International Conference on Computer Vision , pp . 6430–6439 , 2019 . 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The open-source datasets are licensed under CC0 : Public Domain . For recommendation datasets , all records are anonymized and no personal identifiable information is involved . A.1 IMAGE DATASETS In our experiments , Mixture-Of-Datasets consist of six widely used benchmark datasets . For the CUB-200-2011 ( Bird ) , FGVC-Aircraft ( Aircraft ) , FGVCx-Fungi ( Fungi ) , and DTD ( Texture ) datasets , we directly acquire the pre-processed version released by the authors of HSML4 ( Yao et al. , 2019 ) . Pre-processing details of these four sub-datasets can be found in the original paper of HSML . We have obtained the authors ’ consent to use this asset for our research . For the VGG Flower ( Flower ) 5 and GTSRB ( Traffic Sign ) 6 datasets , we download the raw datasets from the respective official websites . We follow the pre-processing protocols of Finn et al . ( 2017 ) to resize all the images to 84×84×3 . We also try to maintain the same meta-train/validation/test ratio as Yao et al . ( 2019 ) for class splits . For VGG Flower , there are in total 102 flower classes , and each class consists of 40 to 258 images . For GTSRB , we use the official training data which contains 43 traffic sign classes , and each class has 30 images . For MiniImagenet , we acquire the dataset as well as the meta-train/validation/test splits from Ravi & Larochelle ( 2016 ) . Table 3 summarizes the meta-train/validation/test splits for each of the subdatasets as well as MiniImagenet . A.2 RECOMMENDATION DATASETS A.2.1 MOVIELENS-1M MovieLens-1M7 dataset ( Harper & Konstan , 2015 ) contains 1,000,209 anonymous ratings on around 3,900 movies made by 6,040 users . We acquire the additional side information released by the authors of MeLU8 ( Lee et al. , 2019 ) , which contains user profile information of gender , age group , occupation , zipcode , and movie profile information of rate , genre , director and actor . The dataset is split into meta-train/validation/test set according to the timestamps of ratings by ratio 7/1/2 . As a result , the start time is 2000-04-26 , the meta-train cut-off time is 2000-11-22 , the metavalidation cut-off time is 2000-12-02 , and the end time is 2003-03-01 . Users with less than 30 samples in each split are removed . In the end , the average sample size for each user is 163 . The statistics of the pre-processed MovieLens-1M dataset and the number of meta-train/validation/test users are recorded in Table 4 and 5 . Note that it is possible for meta-train users to appear in metavalidation and meta-test set , and these users are termed warm users . Conversely , meta-validation or meta-test users that have never appeared in the meta-train set are termed cold users . 4https : //github.com/huaxiuyao/HSML 5https : //www.robots.ox.ac.uk/˜vgg/data/flowers/ 6https : //benchmark.ini.rub.de/ 7https : //grouplens.org/datasets/movielens/ 8https : //github.com/hoyeoplee/MeLU A.2.2 YELP Yelp9 dataset ( Dataset , 2019 ) contains 8,635,403 reviews for 160,585 business services made by 2,189,457 users . We extract the star ratings from the reviews for our experiments . For each business service , we collect the city it resides and the categories it belongs to from the business metadata . However , for the user , the personal information is lacking , and we only have access to the user ID . The dataset is split into meta-train/validation/test set according to the timestamps of ratings by ratio 7/1/2 . As a result , the start time is 2004-10-12 , the meta-train cut-off time is 2018-02-07 , the metavalidation cut-off time is 2018-09-04 , and the end time is 2019-12-13 . Users with less than 30 samples in each split are removed . In the end , the average sample size for each user is 89 . The statistics of the pre-processed Yelp dataset and the number of meta-train/validation/test users are recorded in Table 4 and 5 . A.2.3 AMAZON-CDS Amazon-CDs10 dataset ( McAuley et al. , 2015 ) contains 3,749,004 ratings for 544,442 CD and Vinyl products made by 1,578,597 users . For each CD , we collect its brand and category , while for the user , the personal profile information is lacking and we only have access to the user ID . The dataset is split into meta-train/validation/test set according to the timestamps of ratings by ratio 7/1/2 . As a result , the start time is 1997-09-13 , the meta-train cut-off time is 2012-01-13 , the metavalidation cut-off time is 2013-03-05 , and the end time is 2014-07-23 . Users with less than 30 samples in each split are removed . In the end , the average sample size for each user is 82 . The statistics of the pre-processed Amazon-CDs dataset and the number of meta-train/validation/test users are recorded in Table 4 and 5 . B HYPER-PARAMETERS SETTINGS B.1 FEW-SHOT IMAGE CLASSIFICATION For fair comparison , we adopt the same feature extractor and base-learner for all the compared methods . Following Finn et al . ( 2017 ) , the feature extractor is a four-layer 3 × 3 convolutions with 32 filters , followed by a one-layer fully-connected layer that maps the extracted features into the task embedding dimension de . The base-learner is also a four-layer 3 × 3 convolutions with 32 filters , followed by a one-layer fully-connected layer that projects the output to prediction scores for 5 classes . Each convolution layer is accompanied with batch normalization , ReLU nonlinearity , and 2× 2 max-pooling . For all the compared methods , we follow the hyper-parameters settings for meta-training in MAML ( Finn et al. , 2017 ) , which sets the adaptation learning rate α as 0.01 , the meta-update learning rate β as 0.001 , the number of adaptation steps τ as 5 , the meta batch size |B| as 4 , the size of test set |DteTi | as 15 , and the number of meta-update iterations as 60,000 . For all the task-aware modulation methods ( i.e. , MMAML ( Vuorio et al. , 2019 ) , L2F ( Baik et al. , 2020 ) , HSML ( Yao et al. , 2019 ) , ARML ( Yao et al. , 2020 ) ) , we set the dimension of task embedding de as 128 ( following Yao et al . ( 2019 ) ) . Additionally , for HSML ( Yao et al. , 2019 ) , we set the number of clusters in the first and second layer of hierarchical tree as 4 and 2 respectively . For ARML ( Yao et al. , 2020 ) , we set the number of vertices in the meta-knowledge graph as 4 . For our proposed CTML , we set the number of clusters for both feature and path representations as 6 , i.e. , kpath = kfeat = 6 . B.2 COLD-START RECOMMENDATION Different from the image dataset , recommendation dataset typically consists of user-item interactions and profile features for both sides . The feature extractor is implemented as several embedding lookup tables , each corresponds to one categorical feature . We set the embedding dimension as 8 for all features . The base-learner is implemented as a general deep learning-based recommender system model , composed of embedding matrices followed by a two-layer fully connected layer that models the interaction between different features . The embedding dimension for base-learner is set to 8 , and the hidden sizes for the fully-connected layers is set to [ 32 , 16 ] . For all the compared methods , we follow the hyper-parameters settings for meta-training in MeLU ( Lee et al. , 2019 ) , which sets the adaptation learning rate α as 5e-3 , the meta-update learning rate β as 5e-5 , the number of adaptation steps τ as 5 . For other hyper-parameters that are not explicitly specified by Lee et al . ( 2019 ) , we set the meta batch size |B| as 1 , the size of training set |DtrTi | for each user as 10 ( the remaining are used as test set DteTi for that user , whose size varies for different users ) , and the number of meta-training epochs as 20 . For MAMO ( Dong et al. , 2020 ) which also employs K-means clustering , we apply the same hyper-parameters tuning procedure as our CTML . That is , we tune both kpath and kfeat in { 4 , 8 , 12 , 16 , 20 } , and tune the dimension of task embedding de in { 64 , 128 , 256 , 512 , 1024 } . Other hyper-parameters settings specific to the baseline methods are directly adopted from the source codes released by the authors . C OTHER PATH LEARNER DESIGNS In addition to the GRU-based meta path learner design described in equation 2 , we also experiment on 3 other network designs to test their effectiveness in modeling the step-wise geometric quantities . Recall that the meta path learner is applied coordinate-wise on the base-learner parameters . That is , the same meta path learner is used to process the sequence of step-wise tuples PdTi ∈ R ( τ+1 ) ×4 independently for all parameters at different dimensions d ∈ { 1 , 2 , ... , D } . The first design linearly transforms the vectorized PdTi with learnable weight and bias : Linear : hdi = W · vec ( PdTi ) + b . ( 10 ) The second one employs multiple fully-connected layers with nonlinear activation σ ( · ) : FC : u0 = vec ( PdTi ) , u1 = σ ( W1 · u0 + b1 ) , ... uL = σ ( WL · uL−1 + bL ) , hdi = u L. ( 11 ) The third design applies multiple self-attention layers on the sequence with scaled dot-product similarity following Vaswani et al . ( 2017 ) . To inform about the sequential order , we apply sinusoid positional encodings at the input level , and also apply causal mask to prevent interaction with future steps ( i.e. , set the upper triangular entries of the similarity matrix to −∞ before inputting into softmax ) . The final representation hdi is given by the ( τ + 1 ) -th row ( i.e. , the last row ) of the final-layer output : Attention : U0 = PdTi + positional encodings , U1 = softmax ( Q1 · ( K1 ) ⊤√ da ) ·V1 , ... UL = softmax ( QL · ( KL ) ⊤√ da ) ·VL , hdi = U L τ+1 , : , where Ql = σ ( Ul−1 ·WlQ + blQ ) ∈ R ( τ+1 ) ×da , Kl = σ ( Ul−1 ·WlK + blK ) ∈ R ( τ+1 ) ×da , Vl = σ ( Ul−1 ·WlV + blV ) ∈ R ( τ+1 ) ×da . ( 12 ) Concatenating the path representations at all D dimensions and passing it through a fully-connected layer gives the final path embedding epathTi ∈ R de for task Ti : epathTi = FC ( [ h 1 i , ... , h D i ] ⊤ ) . ( 13 ) D ADDITIONAL EXPERIMENTAL RESULTS D.1 PERFORMANCE ON RESNET-12 We have conducted additional experiments to test the effectiveness of our method using a deeper network ResNet-12 as the backbone ( i.e. , the base-learner ) . We follow the implementation of Sun et al . ( 2019 ) which adopts ( 64-128-256-512 ) for the number of filters at the four blocks . It was shown recently that , with a deeper backbone , simple training paradigm like pre-training and fine-tuning ( Tian et al. , 2020 ; Chen et al. , 2018 ) can achieve very satisfactory performance . We therefore further include the following 3 baselines for comparison on ResNet-12 : • Finetune : This baseline simply pretrains a global feature extractor on the entire training dataset and then finetunes the classifier for individual tasks . • Finetune-distance ( Chen et al. , 2018 ) : This method modifies Finetune by computing cosine distance between the input feature vector and the class vectors for the classifier . • Finetune-distill ( Tian et al. , 2020 ) : This method modifies Finetune by employing a sequential self-distillation technique to pretrain the feature extractor . Table 6 below presents the 5-way 1-shot performance on the Mixture-Of-Datasets with 2 different backbones : Conv-4 and ResNet-12 . We can see that the gaps between Finetune methods and MAML are narrower for deeper backbone , which can be attributed to that the deeper models have larger capacity to accommodate the transferable knowledge obtained from the pre-training ( in line with the findings of Chen et al . ( 2018 ) ) . However , under this task-heterogeneous setting , the taskconditioned methods still yield better performance than MAML and the Finetune baselines for ResNet-12 , demonstrating the effectiveness of task-conditioning even for deeper backbone . And our method with path representation yields the best result . D.2 PERFORMANCE ON MINIIMAGENET Table 7 presents the results on MiniImagenet benchmark ( Ravi & Larochelle , 2016 ) , where tasks are constructed from a single dataset ( i.e. , there is only one underlying distribution ) and hence can be considered as a task-homogeneous setting . Since in this case , task-aware methods do not have advantages over the global initialization methods , we see that all the baselines have comparable performance as the original MAML . Nevertheless , the results show that though the task-aware modulation is rendered less meaningful in task-homogeneous setting , CTML will yield performance at least comparable to the global initialization method in this scenario . D.3 PERFORMANCE ON SUB-DATASETS Table 8 presents the few-shot classification performance of CTML and the baselines on each subdataset . We see that CTML consistently outperforms all the baselines for all the sub-datasets . Other trends are similar to that of the overall results in Table 1a . D.4 PERFORMANCE ON RECOMMENDATION DATASETS Table 9 presents the rating prediction performance on 3 datasets evaluated in two metrics : mean absolute error ( MAE ) and normalized discounted cumulative gain at top 20 ( NDCG @ 20 ) . The meta-testing users are further divided into warm and cold users , where warm users are those that have appeared in the meta-training set , and cold users are those that have not11 . From the results , we see that CTML consistently outperforms the baselines . 11Note that for few-shot image classification , all the meta-testing tasks are unseen during meta-training , while for recommendation , it is possible that the meta-testing users also appeared in meta-training set . D.5 ADDITIONAL LEARNING PATH VISUALIZATION D.5.1 FEW-SHOT IMAGE CLASSIFICATION Here we provide more examples for visualizing the path learning results for the few-shot image classification experiments . Figure 5 , 6 and 7 each shows the path learning results of 6 tasks randomly sampled from 3 of the 6 sub-datasets . We see that in this application scenario , the overall direction of the learning path plays an important role in determining the task representation . Looking at the GRU gates at each step , it seems that the first 2 to 3 steps are often the most important/informative steps , after which the learning begins to converge and the subsequent steps no longer provide useful information about the learning direction . D.5.2 COLD-START RECOMMENDATION We also provide examples for visualizing the learning paths for the cold-start recommendation experiments . Figure 8 and 9 each shows the path learning results of 6 users randomly selected from the MovieLens-1M dataset , and Figure 10 and 11 each shows the results of 6 users sampled from the Amazon-CDs dataset . In this application scenario , we notice that apart from the overall direction of the path , the length of the path also plays an important role in determining the task representation . For instance , in Figure 11 , the blue , orange , green and brown paths ( upper plot ) are clustered closely in the representation space ( lower plot ) due to similar lengths , although they are pointing at very different directions . Similar observation can also be found in Figure 8 . Although blue and orange paths are pointing at the same directions , their embeddings are positioned far away from each other due to the big difference in path lengths . D.6 ADDITIONAL SHORTCUT APPROXIMATION VISUALIZATION In this section , we give more examples to verify the effectiveness of the shortcut approximation . Figure 12 shows the approximation results of 2 users sampled from the MovieLens-1M dataset , and Figure 13 shows the results of 3 tasks sampled from 3 different image sub-datasets . We can see that the shortcut tunnel is able to learn and accommodate different mappings , provide guarantee on the performance of the approximated model .

Summary: The paper proposes to apply the task-aware-modulation on global Meta-learning for learning on tasks sampled from heterogeneous distributions. This paper builds on prior work on the feature-based task characterization to integrate the rehearsed task gradient descent trajectory into task representation. As the rehearsal task is computationally expensive, the authors learn a different network to estimate the rehearsed task-trajectory characterization from the feature representation. The proposed framework is tested on few-shot image classification (meta dataset and miniimagenet) and cold-start recommendation tasks.