Regularization Shortcomings for Continual Learning

1 INTRODUCTION . Continual Learning is a sub-field of machine learning dealing with non-iid ( identically and independently distributed ) data French ( 1999 ) ; Lesort et al . ( 2019c ) . Its goal is to learn the global optima to an optimization problem where the data distribution changes through time . This is typically the case in databases that get regularly augmented with new data or when data is streamed to the algorithms with limited storage possibilities . Continual learning ( CL ) looks for alternative methods to the iid training to avoid the complete retraining with all data each time new data is available . CL algorithms propose different memory storage approaches to collect information from past learning experiences and learning algorithms to continue to learn with this memory and new data . In this paper , we propose to study the class-incremental setting with regularization based methods . The class-incremental setting consists of learning sets of classes incrementally . Each task is composed of new classes . As the training ends , the model should classify data from all classes correctly . Without task labels for inferences , the model needs to both learn the discrimination of intra-task classes and the trans-task classes discrimination ( i.e . distinctions between classes from different tasks ) . On the contrary , if the task label is available for inferences , only the discrimination of intra-task classes needs to be learned . The discrimination upon different tasks is given by the task label . Learning without access to task labels at test time is then much more complex since it needs to discriminate data that are not available at the same time in the data stream . In such setting , we would like to demonstrate that regularization does not help to learn the discrimination between tasks . For example , if a first task is to discriminate white cats vs black cats and the second is the same with dogs , a regularization based method does not provide the learning criteria to learn features to distinguish white dogs from white cats . We consider as regularization methods , those who aim at protecting important weights learned from past tasks without using a buffer of old data or any replay process . Those methods are widely used for continual learning Kirkpatrick et al . ( 2017 ) ; Zenke et al . ( 2017 ) ; Ritter et al . ( 2018 ) ; Schwarz et al . ( 2018 ) . In this paper , we show that in the classical setting of class-incremental tasks , this approach has theoretical limitations and can not be used alone . Indeed , it does not provide any learning criteria to distinguish classes from different tasks . Therefore , in practice , regularization algorithms need external information to make an inference in class-incremental settings . It is provided by the task label at test time . However , relying on the task label to make inferences is an important limitation for algorithms ’ autonomy , i.e . its capacity to run without external information , in most application scenarios . We believe this paper presents important results for a better understanding of CL which will help practitioners to choose the appropriate approach for practical settings . 2 RELATED WORKS . In continual learning , algorithms protect knowledge from catastrophic forgetting French ( 1999 ) by saving them into a memory . The memory should be able to incorporate new knowledge and protect existing knowledge from modification . In continual learning , we distinguish four types of memorization categories dynamic architecture Rusu et al . ( 2016 ) ; Li & Hoiem ( 2017 ) , rehearsal Chaudhry et al . ( 2019 ) ; Aljundi et al . ( 2019 ) ; Belouadah & Popescu ( 2018 ) ; Wu et al . ( 2019 ) ; Hou et al . ( 2019 ) ; Caccia et al . ( 2019 ) , generative replay Shin et al . ( 2017 ) ; Lesort et al . ( 2019a ) ; Wu et al . ( 2018 ) and regularization Kirkpatrick et al . ( 2017 ) ; Zenke et al . ( 2017 ) ; Ritter et al . ( 2018 ) ; Schwarz et al . ( 2018 ) . In this paper , we are interested in the capacity of making inferences without task labels at test time ( test task label ) . The task label t ( typically a simple integer ) is an abstract representation built to help continual algorithms to learn . It is designed to index the current task and notify if the task changes Lesort et al . ( 2019c ) . Dynamic architecture is a well-known method that needs the task label at test time for an inference . Indeed , since the inference path is different for different tasks , the task test label is needed to use the right path through the neural network Rusu et al . ( 2016 ) ; Li & Hoiem ( 2017 ) . Rehearsal and Generative Replay methods generally need the task label at training time but not for inferences Lesort et al . ( 2019a ; b ) . Finally , Regularization methods are often assumed as methods that need task labels only at training time . In this article , we show that in class-incremental settings , it is also necessary at test time . Test task labels have been used in many continual learning approaches , in particular in those referred to as “ multi-headed ” Lange et al . ( 2019 ) . However , the need for task labels for inferences makes algorithms unable to make autonomous predictions and therefore we believe that this requirement is not in the spirit of continual learning . Continual learning is about creating autonomous algorithms that can learn in dynamic environments Lesort et al . ( 2019c ) . 3 REGULARIZATION APPROACH . In this section , we present the formalism we use and we present the class-incremental learning problem with a regularization based approach . 3.1 FORMALISM . In this paper , we assume that the data stream is composed of N disjoint tasks learned sequentially one by one ( with N > = 2 ) . Task t is noted Tt and Dt is the associated dataset . The task label t is a simple integer indicating the task index . We refer to the full sequence of tasks as the continuum , noted CN . The dataset combining all data until task t is noted Ct . While learning task Tt , the algorithm has access to data from Dt only . We study a disjoint set of classification tasks where classes of each task only appear in this task and never again . We assume at least two classes per task ( otherwise a classifier can not learn ) . Let f be a function parametrized by θ that implement the neural network ’ s model . At each task t the model learn an optimal set of parameters θ∗t optimizing the task loss ` Dt ( · ) . Since we are in a continual learning setting , θ∗t should also be an optima for all tasks Tt′ , ∀t′ ∈ J0 , tK . We consider the class-incremental setting with no test task label . It means that an optima θ∗1 for T1 is a set of parameters which at test time will , for any data point x from D0 ∪ D1 , classify correctly without knowing if x comes from T0 or T1 . Therefore , in our continual learning setting , the loss to optimize when learning a given task t is augmented with a remembering loss : ` Ct ( f ( x ; θ ) , y ) = ` Dt ( f ( x ; θ ) , y ) + λΩ ( Ct−1 ) ( 1 ) where ` Ct ( . ) is the continual loss , ` Dt ( . ) is the current task loss , Ω ( Ct−1 ) is the remembering loss with Ct−1 represents past tasks , λ is the importance parameter . 3.2 PROBLEM . In continual learning , the regularization approach is to define Ω ( · ) as a regularization term to maintain knowledge fromCt−1 in the parameters θ such as while learning a new task Tt , f ( x ; θ∗t−1 ) ≈ f ( x ; θ ) , ∀x ∈ Ct−1 . In other words , it aims to keep ` Ct−1 ( f ( x ; θ ) , y ) low ∀x ∈ Ct−1 while learning Tt . The regularization term Ωt−1 act as a memory of θ∗t−1 . This memory term depends on the learned parameters θ∗t−1 , on ` Ct−1 the loss computed on Tt−1 and the current parameters θ. Ωt−1 memorizes the optimal state of the model at Tt−1 and generally the importance of each parameter with regard to the loss ` Ct−1 . We note ΩCt−1 the regularization term memorizing past tasks optimal parameters . When learning the task Tt , the loss to optimize is then : ` Ct ( f ( x ; θ ) , y ) = ` Dt ( f ( x ; θ ) , y ) + λΩCt−1 ( θ ∗ t−1 , ` Ct−1 , θ ) ( 2 ) Eq . 2 is similar to eq . 1 but in this case the function Ω ( · ) is a regularization term depending on past optimal parameters θ∗t−1 , loss on previous tasks ` Ct−1 and the vector of current model parameters θ only . It could be , for example , a matrix pondering weights importance in previous tasks Kirkpatrick et al . ( 2017 ) ; Ritter et al . ( 2018 ) ; Zenke et al . ( 2017 ) . 4 PROPOSITIONS . In this section , we present the proposition concerning the shortcomings of regularization methods in class-incremental settings . We first present definitions and lemmas to prepare for the proposition . 4.1 PRELIMINARY DEFINITION / LEMMA . Definition 1 . Linear separability Let S and S′ be two sets of points in an n-dimensional Euclidean space . S and S′ are linearly separable if there exists n+ 1 real numbers ω1 , ω2 , ... , ωn , k such that ∀x ∈ S , ∑n i=1 ωixi > k and ∀x ∈ S′ , ∑n i=1 ωixi < k where xi the ith component of x . This means that two classes are linearly separable in an embedded space if there exists a hyperplane separating both classes of data points . This property can also be written , ∀x ∈ S and ∀x′ ∈ S′ , ( q · x + q0 ) · ( q · x′ + q0 ) < 0 . With q = [ ω1 , ω2 , ... , ωn ] and q0 = −k respectively the normal vector and position vector of a hyperplane Q . In the case of learning a binary classification with linear model , the model is a hyperplane separating two dataset . As soon as this equation can be solved , then it is possible to define a function f ( x , θ ) and a loss ` ( . ) to learn a hyperplane that will separate S and S′ perfectly . Definition 2 . Interferences In machine learning , interferences are conflicts between two ( or more ) objective functions leading to prediction errors . There are interferences when optimizing one objective function degrades the optimization of , at least , another one . As such , optimizing one of the objective function increases the error on the other one . In continual learning , interferences happen often after a drift in the data distribution . The loss on previous data is increased with the optimization of the loss for the new data leading to interferences and catastrophic forgetting . Lemma 4.1 . ∀ ( S , S′ ) bounded set of discrete points in Rn and linearly separable by a hyperplane Q . For any algorithm , it is impossible to assess Q as a separation hyperplane without access to S′ set . The proof is in appendix B , but in an insightful way , for any bounded set of points S , there is a infinite number of linearly separable set of points . Thus , there exists an infinite number of potential separating hyperplanes . If the second set of points S′ is not known , then it is not possible to choose among the infinite number of potential separating hyperplane which one is a correct one . And even if one is chosen , there is no way to tell if it is better or not than another . In the context of machine learning , without an assessment criterion for a classification problem , it is not possible to learn a viable solution . Hence , we can not optimize the parameters . For binary classification , the Lemma 4.1 can be interpreted as : “ The decision boundary between two classes can not be assessed nor learned if there is no access to data from both simultaneously ” . Lemma 4.2 . ∀ ( S , S′ ) two bounded datasets not linearly separable . For any algorithm , it is impossible to assess a function g ( . ) as a projection of S and S′ into a space were they are linearly separable without access to S′ set . The proof is in appendix C , but in an insightful way , for any bounded set of points , there is an infinite number of projections of the initial set of point in a space where it could be linearly separable from another set of points . Then , If you don ’ t know the second set of points S′ you can not choose among the infinite number of potential projections which one is a good one . And if you ever choose one , you have no way to tell if it is better or not than another . In the context of binary classification , the previous lemma can be interpreted as : “ Two classes representation can not be disentangled if there is no access to data from both simultaneously ” . In those lemma , the concept of “ not having access to ” a certain dataset can both be applicable to not being able to sample data point from the distributions and to not have a model of the dataset . It can be generalized to not having access to any representative data distribution of a dataset .

Summary: This paper presents a theoretical analysis of regularization based approaches to the problem of continually learning a sequence of tasks. The point of the paper is to demonstrate shortcomings of these kinds of approaches, in the context of class-incremental learning where classes are observed once and one after another. The authors argue that these kinds of methods require task labels at test time to correctly distinguish classes from different tasks.