Minimizing Memorization in Meta-learning: A Causal Perspective

1 INTRODUCTION . Recently , there has been renewed interest in meta-learning which empowers more human-like machines that suffice to learn a wide range of tasks with minimal supervision ( Bengio et al. , 1991 ; Thrun & Pratt , 2012 ; Finn et al. , 2017 ; Raghu et al. , 2020 ) . While metric-based meta-learning algorithms ( Vinyals et al. , 2016 ; Snell et al. , 2017 ) only solve few-shot classification problems , we focus on gradient-based meta-learning algorithms in this work that are more flexible ( Finn et al. , 2017 ; Li et al. , 2017 ) . Gradient-based meta-learning algorithms formulate the meta-knowledge as the initialization for a base learner and learn the initialization by a bi-level optimization procedure during the meta-training phase . Concretely , the initialization is adapted to each meta-training task by its support set , while the performance of the adapted model on its query set in turn serves as feedback to update the initialization . This bi-level optimization scheme , though designed to learn a well-generalized initialization , runs a high risk of inducing a sufficiently expressive initialization that memorizes all meta-training tasks . This kind of overfitting is named memorization overfitting ( Yin et al. , 2020 ) , where the initialization solves the query set even without relying on the support set for adaptation . As a consequence , such an initialization poorly generalizes to meta-testing tasks . As suggested in Yin et al . ( 2020 ) , the more non-mutually exclusive meta-training tasks are and the more powerful the model initialization is , the higher risk of memorization arises . To combat the memorization overfitting , Yin et al . ( 2020 ) proposed to regularize the capacity of the initialization , and task augmentation strategies have been recently explored in ( Rajendran et al. , 2020 ; Yao et al. , 2021 ) . Despite the effectiveness of the three algorithms , understanding their benefits rigorously within a unified analytic tool remains a mystery . To bridge the gap , we develop a causal perspective on metalearning , as illustrated by the causal graph in Figure 1 . We argue that the universal label space of the base learner turns to be a confounder causing a spurious correlation between the initializations learned in different steps of meta-training . Such a spurious correlation biases the meta-knowledge that should be only updated by the performance of task-specific models . Fortunately , the causal graph in Figure 1b offers valuable insights into how to minimize memorization via deconfounder approaches . In particular , we have demonstrated the deconfounding role of both lines of existing works : 1 ) regularizer-based methods directly weaken the correlation between the initialization metatrained in the last step ( i.e. , θ′ ) and the task-specific model Φ during meta-training , though the limited flexibility of the initialization in this case still promotes spurious relations ; 2 ) augmentation-based methods take different mapping functions from task labels to the universal label space for various tasks , but the performance highly depends on the independence between mapping functions . Drawing upon the causal perspective , we put forward a new direction of deconfounder approaches by applying the causal inference principle of front-door adjustment . We propose two easy implementations of this principle , which are to sample multiple stratification of the initialization by Dropout and to predict the label as well as the bin that the label belongs to , respectively . Take the backbone of MAML ( Finn et al. , 2017 ) as an example . We name the two deconfounder approaches as MAML-Dropout and MAML-Bins , respectively . The main contributions of our paper are as follows . ( 1 ) We , for the first time , develop a causal perspective of meta-learning and shed light on the memorization overfitting with causality in Section 2.2 . ( 2 ) We place existing methods into the proposed causal framework and adequately demonstrate how they alleviate the memorization overfitting in Section 2.3 and Section 2.4 . ( 3 ) We propose a new deconfounder approach following the principle of front-door adjustment in Section 2.4 and two methods that implement the approach in Section 3 . ( 4 ) We showcase that our methods remain compatible with off-the-shelf meta-learning algorithms and consistently improve their performance . 2 PROBLEM FORMULATION . 2.1 META-LEARNING AND THE OVERFITTING PROBLEM . Meta-learning learns the model initialization θ from a series of tasks Ti sampled from a specific task distribution p ( T ) . All tasks in p ( T ) share some common features , so that starting from the initialization θ a new task sampled from the same task distribution can be quickly learned with a resulting task-specific model φ . The tasks used to learn the initialization are considered as metatraining tasks Dtrain , while novel tasks are meta-testing tasks Dtest . Each i-th task Ti consists of a support set si = { ( xsi , j , ysi , j ) } Ksi j=1 and a query set qi = { ( xqi , j , y q i , j ) } Kqi j=1 , where ( x , y ) denote the features and the label of a sample , Ksi and K q i denote the number of support and query samples . Gradient-based meta-learning formulate learning such a initialization θ as a bi-level optimization problem ( See Figure 1a ) . During inner-loop optimization , the adapted model φi for the i-th task is initialized from θ and updated by its support set si . In outer-loop optimization , the initial- ization θ is optimized according to performances of adapted models on query sets , i.e. , by losses between label yqi , j and prediction ŷ q i , j . Following ( Grant et al. , 2018 ; Gordon et al. , 2019 ; Yin et al. , 2020 ) , we formulate the objective of meta-learning as maximizing the conditional likelihood pφ ( ŷ q|xq , θ , s ) , where the inner-loop optimization learns the conditional distribution of task-specific models p ( φ|θ , s ) and the outer-loop optimizes the distribution of θ p ( θ|Dtrain ) . Consequently , the objective for inner-loop optimization ( i.e. , task objective ) is L ( φi ) = 1 Ks Ks∑ j=1 L ( fφi , θ ( xsi , j ) , ysi , j ) , and the objective for outer-loop optimization ( i.e. , meta-objective ) is L ( θ ) = 1 N N∑ i=1 Ep ( φi|θ , si ) [ 1 Kqi Kqi∑ j=1 L ( fφi , θ ( x q i , j ) , y q i , j ) ] . Take the algorithm of MAML ( Finn et al. , 2017 ) as a concrete example . During the outer-loop update , MAML optimizes the delta function p ( θ|Dtrain ) on meta-training tasks to learn the initialization θ of a base learner f ; in the inner-loop update , p ( φ|θ , s ) is also a point estimation by gradient optimization , i.e. , φi = θ − α∇θ 1Ks ∑Ks j=1 L ( fθ ( xsi , j ) , ysi , j ) . Finally , we predict for a query sample by the adapted model , i.e. , p ( ŷqi , j |x q i , j , φi ) = fφi ( x q i , j ) ( Grant et al. , 2018 ; Yin et al. , 2020 ) . In meta-learning , there are two types of overfitting problems : 1 ) memorization overfitting , which happens when the meta-knowledge memorizes all query sets in meta-training tasks even without adapting on the support sets , and 2 ) learner overfitting , which happens when meta-knowledge is only effective on meta-training tasks and fails to generalize to meta-testing tasks ( Yin et al. , 2020 ; Rajendran et al. , 2020 ) . In this paper , we focus on the former . Regularizer-based and augmentation-based methods have been proposed to combat the memorization overfitting , but how to systematically understand the benefits of these methods within a unified analytic tool is still a mystery . 2.2 A CAUSAL VIEW OF META-LEARNING . Firstly , we introduce causality and causal graph , which are main theories supporting our work . Then , we show the causal graph for gradient-based meta-learning and formulate the memorization overfitting ( Yin et al. , 2020 ; Rajendran et al. , 2020 ) in a causal view . Besides , we explain the reason why existing methods alleviate the memorization in various degrees via our causal graph . Lastly , we propose a deconfounding principle with frontdoor adjustment . Causation and Causal Graph . Causation describes causal relationship among variables instead of correlation . Causal graph ( Pearl et al. , 2016 ) addresses causality problems with a directed acyclic graph G = < V , E > , where a node Vi ∈ V denotes a variable and a directed edge Vi → Vj ∈ E denotes that variable Vi is a direct cause of Vj . Revisit of meta-learning : a causal view . During the meta-training phase , given N meta-training tasks , we define si and qi to be the support set and the query set of task i , respectively . Then , in meta-training data Dtrain , we have all support sets S as { si } Ni=1 and all query sets Q as { qi } Ni=1 . Support sets S and query sets Q consist of randomly drawn examples that are i.i.d . Thus , given meta-training tasks , S andQ are independent . Given query sets of the training set , the input variables XQ = { xqi } Ni=1 is determined , so Q→ XQ . It is obvious that query sets Q has a causal effect on labels of query sets ( i.e. , Q→ Y ) . According to the workflow of meta-learning shown in Figure 1a , we can easily find the causal links of inner-loop optimization S → Φ and outer-loop optimization Φ→ ŶQ→ θ← Y as shown in Figure 1b . In Figure 1b , let θ′ denote the meta-knowledge learned from last step . It is trained in the same way as θ ( the meta-knowledge learned from current step ) ; therefore , there is also a causal link from Y to θ′ ( i.e. , Y → θ′ ) . We omit the connection from the predictions in the last step since the causal effect can be merged into the effect from Φ→ ŶQ ( See Appendix A.1 ) . The connection θ′ → Φ denotes that meta-knowledge θ′ obtained in last update has a causal effect on task-specific models Φ since Φ is always trained by leveraging meta-knowledge θ′ as initialization . Finally , we can obtain the causal graph of meta-learning as shown in Figure 1b . In Figure 1b , the key idea of meta-learning is that by utilizing the past meta-knowledge θ′ as initialization , one can optimize the task-specific model Φ with new support sets S for a more general meta-knowledge θ . But it ignores the confounder Y ( in causal path ŶQ ← Φ ← θ′ ← Y → θ ) which influences both the meta-knowledge in the past step and the current step , leading to spurious correlation between θ′ and θ . This spurious correlation biased by the confounder Y makes the taskspecific model especially challenging to be sufficiently adapted by the support set S , thereby putting the initialization at high risk of memorization . We would highlight the difference of labels Y in meta-learning from those in conventional machine learning . In meta-learning , despite the universal label space , the same label varies from task to task in semantic meanings . For example , the label of 0 may indicate “ dog ” in one task and represent “ cat ” in another . Thus , Y is not only affected by query sets Q , but also by a hidden variable ( i.e. , how to map a task label to the universal label space for various tasks ) . The hidden variable can be denoted as a unobserved exogenous variable and omitted in Figure 1b . Deconfounded meta-learning . In meta-learning , the meta-knowledge θ is learned by p ( θ|θ′ , S , Q ) in each step , although the correlation between θ′ and θ would be spurious since the causal path ŶQ ← Φ← θ′ ← Y → θ shown in Figure 1b demonstrates Y is a confounder of path θ′ → · · · → θ ( see proof in A.2 ) . Given Dtrain = ( S , Q ) and omitting the intermediate nodes , we simplify the causal graph with three nodes { Y , θ′ , θ } as shown in Figure 2a . Y opens the backdoor path from θ and θ′ . However , the backdoor adjustment is not applicable to to cut the causal relationship between Y and θ′ because the edges Y → θ and Y → θ′ in the causal graph would be changed simultaneously since they perform exactly the same roles in meta-learning . Despite this , we propose two kinds of deconfounded methods applying to MAML ( Finn et al. , 2017 ) —one is inspired by some recent works ( Rajendran et al. , 2020 ; Yin et al. , 2020 ; Yao et al. , 2021 ; Tseng et al. , 2020 ) ; the other is based on front-door criterion . These methods are introduced in Section 2.3 and Section 2.4 .

Summary: The paper addresses the undesirable memorization problem in the context of gradient-based meta-learning, where the meta-learning knowledge obtained as a result of memorizing all meta-training tasks yields poor generalization and is detrimental to task specific adaptation. While previous work has focused on regularization-based and augmentation-based solutions to prevent the memorization of all meta-training tasks, the authors argue these methods and their benefits are still not well understood and propose a causal perspective of meta-learning in a unified causal framework. In this context the authors identify the universal label space of the base model as the confounding factor of memorization causing spurious correlations between initializations learnt in different meta-training steps. Informed by this analysis, the authors demonstrate why existing meta-learning overfitting solutions work, and propose two deconfounder approaches to address the issue of memorization, namely i) sampling multiple versions of the meta-knowledge via dropout (MAML-Dropout), and ii) grouping the meta-knowledge into bins (MAML-Bins).