Learning curves for continual learning in neural networks: Self-knowledge transfer and forgetting

1 INTRODUCTION . As deep learning develops for a single task , it enables us to work on more complicated learning frameworks where the model is sequentially trained on multiple tasks , e.g. , transfer learning , curriculum learning , and continual learning . Continual learning deals with the situation in which the learning machine can not access previous data due to memory constraints or privacy reasons . It has attracted much attention due to the demand on applications , and fundamental understanding and algorithms are being explored ( Hadsell et al. , 2020 ) . One well-known phenomenon is catastrophic forgetting ; when a network is trained between different tasks , naive training can not maintain performance on the previous task ( McCloskey & Cohen , 1989 ; Kirkpatrick et al. , 2017 ) . It remains unclear in most cases under what conditions a trained model ’ s performance improves or deteriorates in continual learning . Understanding its generalization performance is still limited ( Pentina & Lampert , 2014 ; Bennani et al. , 2020 ; Lee et al. , 2021 ) . For single-task training , however , many empirical and theoretical studies have succeeded in characterizing generalization performance in over-parameterized neural networks and given quantitative evaluation on sample-size dependencies , e.g. , double descent ( Nakkiran et al. , 2020 ) . For further development , it will be helpful to extend the analyses on single-task training to sequential training on multiple tasks and give theoretical backing . To deepen our understanding of continual learning , this study shows a theoretical analysis of its generalization error in the neural tangent kernel ( NTK ) regime . In more detail , by extending a statistical mechanical analysis of kernel ridge-less regression on single-task training ( Bordelon et al. , 2020 ) to continual learning , we investigate the learning curve , i.e. , the dependence of generalization error on sample size or number of tasks . We focus on the continual learning with explicit task boundaries . Suppose that the model continually learns data generated by two similar teacher functions , which we call target functions , from one task to another . All input samples are generated from the same distribution in an i.i.d . manner , and each task has output samples ( labels ) generated by its own target function . In the NTK regime , the training of the network becomes independent among heads , and learning is formulated as a regression of each head on a corresponding target function . Our main contributions are summarized as follows . First , we revealed characteristic transitions from negative to positive transfer depending on the target similarity . More similar targets above a specific critical value can achieve positive knowledge transfer ( i.e. , better prediction on the subsequent task than training without the first task ) . Compared to this , backward transfer ( i.e. , prediction on the previous task ) has a large critical value , and subtle dissimilarity between targets causes negative transfer . The error can rapidly increase , which clarifies that catastrophic forgetting is literally catastrophic ( Section 4.1 ) . Second , we considered continual learning of the same target function . Even for the continual learning in iterative tasks , the trained model ’ s performance improves or deteriorates in a non-monotonic way . This depends on the sample size of each task ; for equal sample sizes , we can guarantee that the generalization error monotonically decreases from task to task ( Section 4.2 for two tasks & Section 5 for more tasks ) . Unbalanced sample sizes , however , deteriorates generalization ( Section 4.3 ) . We refer to these improvement and deterioration of generalization as self-knowledge transfer and forgetting , respectively . Finally , we empirically confirmed that self-knowledge transfer and forgetting actually appear in continual learning of multi-layer perceptron ( MLP ) and ResNet-18 ( Section 5.1 ) . Thus , this study sheds light on fundamental understanding and the universal behavior of continual learning in over-parameterized learning machines . 2 RELATED WORK . Method of analysis . As an analysis tool , we use the replica method originally developed for statistical mechanics . Statistical mechanical analysis enables us typical-case evaluation , that is , the average evaluation over data samples or parameter configurations . It sometimes provides us with novel insight into what the worst-case evaluation has not captured ( Abbaras et al. , 2020 ; Spigler et al. , 2020 ) . The replica method for kernel methods has been developed in Dietrich et al . ( 1999 ) , and recently in Bordelon et al . ( 2020 ) ; Canatar et al . ( 2021 ) . These recent works showed excellent agreement between theory and experiments on NTK regression , which enables us to quantitatively understand sample-size dependencies including implicit spectral bias , double descent , and multiple descent . Continual learning . Continual learning dynamics in the NTK regime that we investigate was first formulated by Bennani et al . ( 2020 ) ; Doan et al . ( 2021 ) , though the evaluation of generalization remains unclear . They derived an upper bound of generalization via the Rademacher complexity , but it includes naive summation over single tasks and seems conservative . In contrast , our typical-case evaluation enables us to find such rich behaviors as negative/positive transfer and self-knowledge transfer/forgetting we newly find . The continual learning in the NTK regime belongs to so-called single-head setting ( Farquhar & Gal , 2018 ) , and it allows the model to revisit the previous classes ( target functions ) in subsequent tasks . This is complementary to earlier studies on incremental learning of new classes and its catastrophic forgetting ( Ramasesh et al. , 2020 ; Lee et al. , 2021 ) , where each task includes different classes and does not allow the revisit . Note that the basic concept of continual learning is not limited to incremental learning but allows the revisit ( McCloskey & Cohen , 1989 ; Kirkpatrick et al. , 2017 ) . Under the limited data acquisition or resources of memory , we often need to train the same model with the same targets ( but different samples ) from task to task . Therefore , the setting allowing the revisit seems reasonable . 3 PRELIMINARIES . 3.1 NEURAL TANGENT KERNEL REGIME . We summarize conventional settings of the NTK regime ( Jacot et al. , 2018 ; Lee et al. , 2019 ) . Let us consider a fully connected neural network f = uL given by ul = σwWlhl−1/ √ Ml−1 + σbbl , hl = φ ( ul ) ( l = 1 , ... , L ) , ( 1 ) where we define weight matrices Wl ∈ RMl×Ml−1 , bias terms bl ∈ RMl , and their variances σ2w and σ2b . We set random Gaussian initialization Wl , ij , bl , i ∼ N ( 0 , 1 ) and focus on the mean squared error loss : L ( θ ) = ∑N µ=1 ‖yµ − f ( xµ ) ‖2 , where the training samples { xµ , yµ } Nµ=1 are composed of inputs xµ ∈ RD normalized by ‖xµ‖2 = 1 and labels yµ ∈ RC . The set of all parameters is denoted as θ , and the number of training samples is N . Assume the infinite-width limit for hidden layers ( Ml →∞ ) , finite sample size , and depth compared with the width . The gradient descent dynamics with a certain learning rate then converges to a global minimum sufficiently close to the random initialization . This is known as the NTK regime , and the trained model is explicitly obtained as f ( c ) ( x′ ) = f ( c ) 0 ( x ′ ) + Θ ( x′ , X ) Θ ( X ) −1 ( y ( c ) − f ( c ) 0 ( X ) ) ( c = 1 , ... , C ) . ( 2 ) We denote the NTK matrix as Θ , arbitrary input samples ( including test samples ) as x′ , and the set of training samples as X . The indices of f mean 0 for the model at initialization and c for the head of the network . Entries of NTK Θ ( x′ , x ) are defined by∇θf0 ( x′ ) ∇θf0 ( x ) > . We write Θ ( X ) as an abbreviation for Θ ( X , X ) . The trained network is equivalent to a linearized model around random initialization ( Lee et al. , 2019 ) , that is , f ( c ) = f ( c ) 0 +∇θf ( c ) 0 ∆θ with ∆θ = θ − θ0 = C∑ c=1 ∇θf ( c ) 0 ( X ) > Θ ( X ) −1 ( y ( c ) − f ( c ) 0 ( X ) ) . ( 3 ) While over-parameterized models have many global minima , the NTK regime ’ s dynamics implicitly select the above θ , which corresponds to the L2 min-norm solution . Usually , we ignore f0 by taking the average over random initialization . The trained model ( 2 ) is then equivalent to the kernel ridge-less regression ( KRR ) . NTK regime also holds in various architectures including ResNets and CNNs ( Yang & Littwin , 2021 ) , and the difference only appears in the NTK matrix . Although we focus on the fully connected network in synthetic experiments , the following NTK formulation of continual learning and our theoretical results hold in any architecture under the NTK regime . Continual learning . We denote the set of training samples in the n-th task as ( Xn , yn ) ( n = 1 , 2 , ... , K ) , a model trained in a sequential manner from task 1 to task n as fn and its parameters as θn . That is , we train the network initialized at θn−1 for the n-th task and obtain fn . Assume the number of tasks K is finite . The trained model within the NTK regime is then given as follows ( Bennani et al. , 2020 ; Doan et al. , 2021 ) : fn ( x ′ ) = fn−1 ( x ′ ) + Θ ( x′ , Xn ) Θ ( Xn ) −1 ( yn − fn−1 ( Xn ) ) , ( 4 ) θn − θn−1 = ∇θf0 ( Xn ) > Θ ( Xn ) −1 ( yn − fn−1 ( Xn ) ) . ( 5 ) We omit the index c because each head is updated independently . The model fn completely fits the n-th task , i.e. , yn = fn ( Xn ) . The main purpose of this study is to analyze the generalization performance of the sequentially trained model ( 4 ) . At each task , the model has an inductive bias of KRR in the function space and L2 min-norm solution in the parameter space . The next task uses this inductive bias as the initialization of training . The problem is whether this inductive bias helps improve the generalization in the subsequent tasks . Remark ( independence between different heads ) . For a more accurate understanding of the continual learning in the NTK regime , it may be helpful to remark on the heads ’ independence , which previous studies did not explicitly mention . As in Eq . ( 2 ) , all heads share the same NTK , and f ( c ) depends only on the label of its class y ( c ) . While the parameter update ( 3 ) includes information of all classes , the c-th head can access only the information of the c-th class1 . For example , suppose that the n-th task includes all classes except 1 , i.e. , { 2 , ... , C } . Then , the model update on the the class 1 at the n-th task , i.e. , f1n − f1n−1 , becomes ∇θf ( 1 ) 0 ( x′ ) ( θn − θn−1 ) = ∇θf ( 1 ) 0 ( x ′ ) ∑ c=2 ∇θf ( c ) 0 ( Xn ) > Θ ( Xn ) −1 ( y ( c ) n − f ( c ) n−1 ) = 0 because ∇θf ( c ) 0 ∇θf ( c′ ) > 0 = 0 ( c 6= c′ ) in the infinite-width limit ( Jacot et al. , 2018 ; Yang , 2019 ) . Thus , we can deal with each head independently and analyze the generalization by setting C = 1 without loss of generality . This indicates that in the NTK regime , interaction between different heads do not cause knowledge transfer and forgetting . One may wonder if there are any non-trivial knowledge transfer and forgetting in such a regime . Contrary to such intuition , we reveal that when the subsequent task revisits previous classes ( targets ) , the generalization shows interesting increase and decrease . 1We use the term ” class ” , although the regression problem is assumed in NTK theory . Usually , NTK studies solve the classification problem by regression with a target y ( c ) = { 0 , 1 } .

Summary: This paper provides a theoretical analysis for sequential training based on related NTK results.  It shows the similarity between the target functions is a key factor for forward and backward transfer, and when the target functions are the same the samples size of each task is a key factor to forward and backward transfer. It also shows even a slight difference between the target functions can cause catastrophic forgetting and the forgetting might still happen for the same target function when the sample size of a later task increases.