Model Sensitivity Aware Continual Learning

Authors: zhenyi  wang
heng  huang
Venue: NeurIPS_cc_2024_Conference
Published: 2024-05-14 15:36:34

Abstract: Continual learning (CL) aims to adapt to non-stationary data distributions while retaining previously acquired knowledge. However, CL models typically face a trade-off between preserving old task knowledge and excelling in new task performance. Existing approaches often sacrifice one for the other. To overcome this limitation, orthogonal to existing approaches, we propose a novel perspective that views the CL model ability in preserving old knowledge and performing well in new task as a matter of model sensitivity to parameter updates. \textit{Excessive} parameter sensitivity can lead to two drawbacks: (1) significant forgetting of previous knowledge; and (2) overfitting to new tasks. To reduce parameter sensitivity, we optimize the model's performance based on the parameter distribution, which achieves the worst-case CL performance within a distribution neighborhood. This innovative learning paradigm offers dual benefits: (1) reduced forgetting of old knowledge by mitigating drastic changes in model predictions under small parameter updates; and (2) enhanced new task performance by preventing overfitting to new tasks. Consequently, our method achieves superior ability in retaining old knowledge and achieving excellent new task performance simultaneously.
Importantly, our approach is compatible with existing CL methodologies, allowing seamless integration while delivering significant improvements in effectiveness, efficiency, and versatility with both theoretical and empirical supports.

Main paper: 
 introduction: 
 Continual learning (CL) embodies a dynamic approach aimed at adapting to non-stationary data distributions that evolve over time. However, in pursuit of this goal, CL encounters a significant challenge: the trade-off between preserving previously acquired knowledge and effectively learning new knowledge. As the model assimilates new information, it often swiftly erases previously learned knowledge, resulting in catastrophic forgetting (CF) on past tasks [44, 54] . Effectively addressing CF during CL is essential to preserve previously acquired information. On the other hand, effectively learning new information is equally crucial for CL models to adapt to new tasks and environments.Existing approaches in CL often face a dilemma: they either prioritize preserving old knowledge or excelling in new task performance, often at the expense of the other. When a CL model prioritizes preserving old knowledge, it focuses on retaining information from previous tasks while minimizing interference or forgetting. However, excessive emphasis on old knowledge can limit the model's ability to adapt to new tasks. Conversely, when a model prioritizes new task performance, it aims to quickly adapt to new tasks or data distributions. Yet, this emphasis on new tasks can potentially degrade performance on previously learned tasks.To overcome the aforementioned limitations, orthogonal to existing approaches, we introduce the concept of model sensitivity and approach the challenge of balancing old knowledge retention and new task performance in CL from the perspective of model parameter sensitivity. When a CL model exhibits high sensitivity to parameter changes, it leads to two significant issues: (1) Increased Forgetting: Excessive sensitivity in model parameters can cause abrupt and substantial changes in model predictions with minor parameter adjustments during CL. This phenomenon results in significant forgetting of previous tasks. (2) Diminished New Task Performance: High sensitivity in model parameters can also result in severe overfitting on new tasks. Overfitting occurs when a model memorizes the training data instead of generalizing patterns that can be applied to unseen data. High parameter sensitivity means that even minor alterations in the training data can induce substantial modifications in the learned model. This renders the model excessively tailored to the training data and reduces its adaptability to new, unseen data, consequently leading to suboptimal performance on new tasks.To reduce the CL model parameter sensitivity under model updates, we aim to ensure that even minor alterations in model parameters do not substantially impair CL model performance. This is accomplished by optimizing the model's performance based on the worst-case scenario of parameter distributions within a distribution neighborhood. However, finding the optimal worst-case CL model parameter distribution is challenging since the space of all possible distributions within the neighborhood is an infinite-dimensional space [32] . To efficiently solve this problem, we parameterize the optimal worst-case CL model parameter distribution as Gaussian distribution. We propose a natural-gradient descent (NGD) method to efficiently inference the mean and covariance of the Gaussian distribution since NGD incorporates the information geometry of the parameter space by adapting the step size based on the curvature of the cost function. This adaptive approach leads to faster convergence compared to conventional gradient descent methods, particularly in highdimensional spaces where the curvature exhibits notable variations. This is especially beneficial for CL models. However, calculating the natural gradient is computationally expensive due to the explicit calculation of Fisher information matrix (FIM). We thus update the worst-case CL parameters in the expectation parameter space, rather than the traditional natural parameter space, of the Gaussian distribution, thereby eliminating the need for explicit calculation of the FIM.Our method offers dual benefits: (1) Reduced Forgetting: By mitigating parameter sensitivity and avoiding drastic changes in model prediction, our approach effectively reduces the loss of previously learned task knowledge. (2) Improved New Task Performance: Through decreased parameter sensitivity, the model becomes less susceptible to overfitting on new task training data. This reduced vulnerability to minor fluctuations fosters the learning of more generalized patterns rather than memorizing specific examples. As a result, the model demonstrates enhanced generalization capabilities on new tasks. Therefore, our method simultaneously achieves superior performance in retaining previously learned knowledge and excelling in new task performance.We provide a thorough theoretical analysis for our method. Firstly, the theory illustrates that our approach implicitly reduces the variance of loss against different parameter variations, thereby indicating reduced model parameter sensitivity. Secondly, our method tightens the generalization bound of CL models, suggesting enhanced generalization. Furthermore, our extensive experiments across multiple datasets, compared to various state-of-the-art (SOTA) baseline methods, reveal substantial enhancements in overall performance across all learned tasks, backward transfer, and new task test accuracy. These results indicate significantly enhanced CL model ability in preserving old knowledge and achieving better performance on new task with our method. Additionally, our proposed approach seamlessly integrates with existing CL methodologies, functioning as a versatile plug-in. This demonstrates the effectiveness, efficiency, and versatility of our method.Our contributions can be summarized as follows:• We tackle the challenge of both retaining old task knowledge and excelling in new task in CL from a novel perspective by mitigating model parameter sensitivity.• We introduce a novel CL approach aimed at reducing model parameter sensitivity by optimizing CL model performance under the worst-case parameter distribution within a distribution neighborhood. Additionally, we propose an efficient learning algorithm to identify the worst-case parameter distribution.• We provide comprehensive theoretical analyses that substantiate our method's ability to decrease model parameter sensitivity and improve model generalization.• Extensive experiments conducted across multiple datasets demonstrate the efficacy and versatility of our proposed method. 
related works: 
 CL aims to learn non-stationary data distributions without forgetting previously learned knowledge.The CL scenarios can be further categorized into three scenarios: task-incremental learning (Task-IL), domain-incremental learning (Domain-IL) and class-incremental learning (Class-IL) [67] . Task-IL and Class-IL are most representative scenarios in CL, we thus focus on these two scenarios.Existing approaches for CL can be categorized into five classes: (1) regularization-based methods incorporate regularization terms either in model weights or outputs into the loss function to mitigate catastrophic forgetting when learning new tasks, including [28, 62, 84, 55, 11, 1, 22, 10, 39] ; (2) memory replay-based methods address the challenge of catastrophic forgetting by explicitly storing and replaying a subset of past experiences (samples from previous tasks) while learning new tasks, including [40, 57, 15, 7, 51, 68, 3, 8, 75, 4, 74, 76, 61, 78, 77, 83, 36, 73, 72] ; (3) gradient-projectionbased methods aim to mitigate catastrophic forgetting by projecting gradient updates onto subspaces that minimize interference with previously learned tasks, including [13, 17, 60, 71, 38, 52, 82] ;(4) architecture-based methods involve dynamically adapting and modifying the neural network architecture to accommodate new tasks while preserving performance on previously learned tasks, including [41, 63, 34, 23] ; (5) Bayesian-based methods leverage principles from Bayesian inference to manage the uncertainty and learning of new tasks while preserving knowledge from previous tasks, including [48, 58, 30, 25, 21, 49, 66, 59] .In contrast to existing methods, which often necessitate a trade-off between retaining old knowledge and learning new knowledge, sacrificing one for the other, our approach takes a different path. It sets itself apart from these existing methods by offering an orthogonal solution that preserves old task knowledge while simultaneously enhancing new task performance. This novel perspective is achieved by reducing parameter sensitivity.Connection with existing flat-minima/SWAD approaches: (1) Connection and difference with sharpness-aware minimization (SAM) [18, 27, 45] related approach: Our method is fundamentally different from SAM-based CL in two aspects. (i) Deterministic vs. Probabilistic Approach: SAM uses a fixed deterministic neighborhood, which can be restrictive in practice since updates are constrained within a fixed ball. In contrast, our method employs a probabilistic distributional approach, offering two distinct advantages: (a) The distributional neighborhood is more flexible and covers a broader range of parameter variations by sampling from a neighborhood distribution, and (b) Stochastic Gradient Descent (SGD) introduces noise during CL. Our distributional approach accounts for this noise, making it a more realistic model in practice and providing stronger guarantees against parameter sensitivity. (ii) Uniform vs. Parameter-specific sensitivity without explicit calculation of FIM: SAM uniformly updates all parameters, overlooking the varying importance and sensitivity of each parameter in the context of CL. Our method, on the other hand, considers these differences and treats parameters uniquely through the natural gradient without needing to explicitly calculating the FIM. This distinction is crucial for CL, as each parameter has different sensitivity to forgetting-a factor that SAM does not address. (2) Connection and difference with model averaging flatness seeking approach: SWA [24] and SWAD [9] , which aim to achieve flatter minima by averaging multiple models during training. However, these approaches are memory-intensive and inefficient for CL, as they require storing multiple sets of model parameters, which compromises memory efficiency. 
method: 
 In this section, we first present the preliminary in section 3.1 and then present the model sensitivity aware continual learning in section 3.2.Continual Learning Setup The standard CL problem involves learning a sequence of T tasks, represented asD tr = {D tr 1 , D tr 2 , • • • , D tr T }.The training dataset D tr k for the k th task contains a collection of triplets:(x k i , y k i ) n k i=1, where x k i denotes the i th data example specific to task k, y k i represents the associated data label for x k i . The primary objective is to train a neural network function, parameterized by θ, denoted as g θ (x). The goal is to achieve good performance on the test datasets from all the learned tasks, represented asD te = {D te 1 , D te 2 , • • • , D te T }, while ensuring that knowledge acquired from previous tasks is not forgotten. The CL loss function is defined as the following:EQUATIONwhere L CE (x, y; θ) is the current task cross-entropy loss function. L f (θ) is the forgetting-mitigation loss, e.g., memory-replay, weight-regularization and gradient-projection loss, etc. ζ is a constant that balances the weight between the loss of the new task and the loss of the previous tasks.The exponential family distribution [70] is defined as:EQUATIONWhere := denotes a definition. In existing literature [70] , ϕ are called the natural parameters for defining the distribution, P ϕ (θ). h(θ) is the base measure, Ω(θ) is the sufficient statistic, Z(ϕ) := log h(θ)exp(⟨ϕ, Ω(θ)⟩)dθ is the log-partition function, ⟨, ⟩ denotes the dot product between two vectors. We denote the expectation parameters as λ := E P ϕ (θ) Ω(θ). We can write multivariate Gaussian distribution as canonical form of exponential family as:EQUATIONEQUATIONTherefore, the correspondence between f (θ; µ, Σ) and exponential family distribution in Eq.( 2) can be expressed as the following:EQUATIONEQUATIONDerivations details of Eq.( 6) can be found in Appendix B.1. In the following section, we use exponential family distributions to parameterize the worst-case of CL model parameter distribution since this enables us to efficiently calculate the natural gradient in the expectation parameter space λ without needing to explicitly calculate the Fisher information matrix (FIM) in natural parameter space ϕ.Learning Objective Specifically, we propose the following CL learning objective to reduce the CL parameter sensitivity under model parameter updates:EQUATIONs.t. U = {U : D KL (U, V) ≤ ϵ}where U denotes the uncertainty set. D KL (U, V) denotes the KL divergence between the current CL model parameter distribution V and the neighbour CL model parameter distribution U. ϵ is a small constant. max U∈U E θ∼U(θ) L CL (θ) aims to find the worst-case CL model parameter distribution within a neighbourhood. We choose probabilistic distributional neighbourhood due to two-fold reasons: (1) the distributional neighbourhood covers more flexible parameter space; and (2) widely used SGD method incurs update noise during CL, thereby distributional neighbourhood provides stronger guarantee against parameter sensitivity. It is important to note that the outer minimization is performed with respect to µ, the expectation of θ, since during inference, only µ is used as the model parameter for predictions.Objective for Learning the Worst-Case CL Parameter Distribution We convert the constrained inner maximization optimization in Eq. ( 7) into the following unconstrained optimization to find the worst-case CL model parameter distribution.EQUATIONwhere α > 0 is a constant. However, solving Eq. ( 8) is intractable since the optimization target is in an infinite-dimensional function space [32] . For computation efficiency, we set the current CL model parameter distribution as V(θ) = N (θ|µ 0 , Σ 0 ), where µ 0 and Σ 0 denote the mean vector and covariance matrix, respectively. We set the neighbourhood distribution as U(θ) = N (θ|µ, Σ), where µ and Σ denote the mean vector and covariance matrix, respectively. To further improve computational efficiency, we constrain the covariance matrix to be diagonal matrix, i.e., Σ = diag(σ 2 ) and Σ 0 = diag(ρ 2 ). We denote the density function of U(θ) and V(θ) as u(θ) and v(θ), respectively. We express the loss function in Eq. ( 8) as the following:EQUATIONBy parameterizing the distribution U as exponential family distribution in Eq. ( 4), our goal is to learn the parameters ϕ in Eq. ( 5) with natural gradient descent (NGD) [42] as the following equation:EQUATION)where F is the FIM. We opt for NGD because it adjusts the step size according to the curvature of the cost function, making convergence faster than traditional gradient descent methods. This is especially advantageous in high-dimensional spaces where the curvature and parameter-wise sensitivity vary significantly, benefiting CL models. However, computing the natural gradient is computationally intensive due to the need to calculate the FIM. To address this, we develop an efficient update method in the dual space, specifically the expectation parameter space λ, rather than the natural parameter space ϕ, eliminating the need for explicit FIM calculation. In the following, we will use L(ϕ) and L(λ) interchangeably, as they represent the same loss function only parameterized in different spaces.We leverage the relation between NGD in natural parameter space and gradient descent in expectation parameter space (in Appendix A.1), NGD can be performed without explicitly computing the FIM. This update in its dual space leads to significantly more efficient parameter updates and promising computational advantages.NGD for Efficiently Finding the Worst-Case Gaussian Distribution In the following, we present specific algorithms for updating the µ and Σ with NGD to find the worst-case Gaussian distribution, i.e., U * := arg min U H(U). We can get the following updates for mean µ and diagonal covariance Σ = diag(σ 2 ) (detailed derivations can be found in Appendix B):EQUATIONEQUATIONBy plug-in Σ = diag(σ 2 ) and Σ 0 = diag(ρ 2 ) into the above equations, we can obtain the following updates:EQUATIONEQUATIONIn practice, we set α = 1.0 to reduce the reliance on hyperparameters. However, computing the diagonal Hessian matrix ∇ 2 θθ L CL (θ) in Eq. ( 14) is a computationally challenging task. Following [42] , we efficiently approximate the Hessian as the following:EQUATION)where N is the number of training data points for the current task, L CL j (θ) denotes the loss function for the data point j, θ k denotes the k th element of the model parameter θ. It is crucial to note that this Hessian approximation is computed only once after learning each task and involves calculating only the diagonal elements, i.e., Σ = diag(σ 2 ). As a result, the overall computational cost throughout the continual learning process remains low. Additionally, this update mechanism maintains the same number of learnable parameters as existing methods, ensuring fair comparisons. This is because, during the learning of each task, only the mean parameters of the Gaussian distribution are updated.Learning Algorithm We name our method as Model sensitivity Aware Continual Learning (MACL). The detailed algorithm is present in Algorithm 1.1: REQUIRE: model parameters θ, CL model learning rate β, worst-case Gaussian learning rate η, number of CL tasks T , number of CL steps K for each task, distribution neighbourhood regularization strengths α = 1.0. Randomly initialized diagonal covariance matrix, i.e., diag(σ 2 ). 2: for n = 1 to T do 3:for i = 1 to K do 4: calculate the CL loss function according to Eq. (1) 5:update the worst-case Gaussian mean µ (i.e., θ) byθ ′ i = θi + ησ 2 n [∇ θ L CL (θi) -(θi -θ0)ρ -2 ] 6:sample parameters from the worst-case CL model parameter distribution.θ ′ = θ ′ i + σnζ, where ζ ∼ N (0, I) 7: update CL model parameters θi+1 = θ ′ -β∇ θ L CL (θ ′ ) 8:end for 9:update the worst-case Gaussian covariance σ byσ -2 n+1 = (1 -η)σ -2 n + η[-∇ 2 θθ L CL (θ) + ρ -2 ] 10:where the Hessian is calculated by∇ 2 θ k θ k L CL (θ) = 1 N j=N j=1 [∇ θ k L CL j (θ)]2 according to Eq. ( 15) 11: end for 
theoretical analysis: 
 In this section, we build the theoretical connection between MACL and parameter sensitivity in Theorem 4.2 and the generalization analysis in Theorem 4.3. Due to the space limitations, we provide the theorem proof in Appendix A.2. Let's first look at the inner maximization problem in Eq. (7) .max U∈U L CL (θ)dU(θ), s.t. U = {U : D KL (U, V) ≤ ϵ} (16) Lemma 4.1. D KL (U, V) = u(θ) log( u(θ) v(θ) )dθ ≤ (u(θ)-v(θ)) 2 v(θ) dθ Theorem 4.2. Assume || 1 v(θ) || ∞ dθ ≤ M ,we can obtain the following conclusion for Eq. ( 16):EQUATION)where L CL (θ) := L CL (θ)dV(θ). V ar(L CL (θ)) denotes the variance of L CL (θ) with respect to different model parameters variations, i.e., V ar(L CL (θ)) = E(L CL (θ) -L CL (θ)) 2 = (L CL (θ) -L CL (θ)) 2 dθ.In this context, V ar(L CL (θ)) serves as a measure of the CL model's sensitivity to parameter updates. Essentially, a smaller loss variance indicates lower parameter sensitivity in the CL model. However, directly optimizing the loss variance within the parameter distribution neighborhood is impractical, as it requires computing the loss variation across a large number of different sets of CL model parameters and training data points. In contrast, our method (MACL) offers an efficient and effective alternative. MACL implicitly minimizes the loss variance across different model parameter variations by optimizing CL performance solely on the worst-case CL model parameter distribution. In the following, inspired by UDIL [64] , we further provide the following generalization bound for CL: Theorem 4.3 (Generalization bound of MACL). Let q be the number of CL model parameters and n be the number of training data points. The CL loss L CL (θ) ≤ C (C is a constant). With high probability of 1 -δ, the following bound holds:E θ∼N (µ,Σ) i=T i L CL Di (θ) ≤ max U∈U E θ∼U L CL (θ) + C N T + ζ i=T -1 i=1 N i + (18) τ 2 ( √ q + 2 log(N T + ζ i=T -1 i=1 N i )) 2 + R + 2 log( N T +ζ i=T -1 i=1 Ni δ ) 4(N T + ζ i=T -1 i=1 N i -1)Where τ is a constant. We denote the number of data examples for task 1, • • • , T -1 in the memory buffer M during training on task T as N 1 , N 2 , • • • , N T -1 when using memory replay based approach or the number of training data points when using regularization based approach. L CL Di (θ) denotes the CL loss on the data from data distribution D i of task i (generalization error), i.e., it is defined as: L CL Di (θ) := E (x,y)∼Di L(x, y, θ). L CL (θ) denotes the empirical CL loss as Eq. ( 1). N (µ, Σ) denotes the CL model parameter posterior distribution parameterized with Gaussian distribution. Generalization bound implication: (1) When using a memory-replay approach, the number of samples from new tasks often exceeds the number of samples in the memory buffer, causing data imbalance. This imbalance, where fewer samples from previous tasks are stored, affects the second and third terms in the generalization bound. The bound suggests that as the number of samples in the memory buffer increases (i.e., i=T -1 i=1 N i ↑), these terms tighten, leading to a tighter generalization upper bound. This is because lim x→∞ [h(x) := log x x ] = 0, meaning the generalization improves with a larger buffer, aligning with the intuition that more memory buffer data leads to better performance.(2) In the regularization-based approach, ζ i=T -1 i=1 N i is treated as the effective sample size for previous tasks since the loss is approximated in the absence of earlier data. The parameter ζ controls the trade-off between learning the new task and retaining knowledge from past tasks. A larger ζ increases regularization, preventing the model from deviating too much from the parameters learned on previous tasks. This leads to higher empirical loss on the new task (first term), but tighter bounds (second and third terms), indicating that knowledge from previous tasks is retained effectively. This prioritizes stability over learning flexibility for the new task. 
setup: 
 Datasets We conduct experiments on several datasets, including CIFAR10 (10 classes), CIFAR100 (100 classes) [29] , and Tiny-ImageNet (200 classes) [80] , to assess the effectiveness of MACL in task incremental learning (Task-IL) and class incremental learning (Class-IL). In addition, we also conduct experiments on 5-dataset [79, 5] , CUB200 [69] and ImageNet-R [20] (in Appendix). Following the approach in [7] , we split the CIFAR-10 dataset into five tasks, each with two distinct classes. We divided the CIFAR-100 dataset into ten tasks, each containing ten classes. We split the Tiny-ImageNet dataset into ten tasks, each comprising twenty classes. More dataset statistics can be found in Appendix E.1.Baselines We compare to the following various SOTA CL methods. (1) Regularization-based methods, including oEWC [62] , synaptic intelligence (SI) [84] , Learning without Forgetting (LwF) [35] , Classifier-Projection Regularization (CPR) [10] . (2) Bayesian-based methods, including NCL [25] . (3) Architecture-based methods, including HAT [63] . (4) Memory-based methods, including ER [15] , A-GEM [14] , iCaRL [55] , GSS [2] , HAL [12] , DER++ [7] , ER-ACE [8] and LODE [36] .(5) Gradient-projection-based methods: Gradient Projection Memory (GPM) [60] .Implementation Details Following [7] , we use ResNet18 [19] as the backbone network for all the CL datasets and compared baseline methods. For the baselines that are included in the open-source code of DER++ [7] , we use the same hyperparameters provided in DER++ [7] for the compared methods. For the baselines not included in the open-source code of DER++, e.g., GPM, LODE, etc, we use the open-source code from their original paper for comparisons. For the hyperparameters in our method, we set α = 1.0 across all the datasets to minimize the model's dependence on hyperparameters. For η, we set η = 1e -5 for CIFAR10 and CIFAR100, and η = 1e -6 for Tiny-ImageNet. The η is selected from the range of [1e -4, 1e -5, 1e -6, 1e -7]. Following [7, 14] , the hyperparameter is determined through the validation sets split from the training sets from the first three tasks. Similar to [7] , we train all the CL models using the standard SGD optimizer to update the CL model. The batch size and replay buffer batch size are set to 32. We use a single NVIDIA A5000 GPU with 24GB memory to run the experiments. Each experiment result is averaged for 10 runs with mean and standard deviation.We evaluate the performance of different CL methods with (1) overall accuracy; (2) new task accuracy; and (3) backward transfer in the following.Overall Accuracy (ACC) ACC is the average accuracy across the entire task sequence. We present the results on CIFAR10, CIFAR-100 and Tiny-ImageNet in Table 1 . We can observe that our method substantially improve over various SOTA baseline methods up to 3% to 4% on CIFAR100, TinyIma-geNet by integrating MACL with existing CL methods. This overall performance improvement is attributed to the reduced parameter sensitivity. New Task Accuracy To evaluate the new task performance of the proposed CL method, we evaluate the new task performance during CL by integrating MACL with DER++ and GPM in Figure 1 .The results show that MACL can significantly improves the new task performance for different CL methods, indicating that reducing the model parameter sensitivity is beneficial to improve new task performance during CL.Figure 1 : new task performance during CL.Backward Transfer Backward transfer (BWT) quantifies the degree of forgetting observed on previously learned tasks. When BWT > 0, it indicates that learning the current new task positively influences the performance on previously learned tasks.Conversely, when BWT ≤ 0, it signals that learning the current new task may result in forgetting previously acquired knowledge. We evaluate BWT in Table 2 . We can observe that our method significantly improves BWT by up to 5% through integrating MACL with existing CL methods. This indicates that reducing parameter sensitivity can substantially reduce forgetting on previously learned knowledge. These empirical analysis also verify our theoretical analysis that our method implicitly improves the stability by reducing loss variance. Effect of Memory Size To assess the impact of varying memory buffer sizes, we present the results in Table 3 . The results demonstrate that compared to different baseline methods, our MACL plug-in also enhances the performance of baseline methods with a memory size of 2000. Benefit of NGD To evaluate the benefits of using NGD over gradient descent (GD) for calculating the worst-case Gaussian distribution, we present comparison results in Table 6 in Appendix D.2. The results show that NGD outperforms GD because NGD better captures parameter importance, which helps preserve old knowledge while effectively adapting to new tasks.Efficiency Evaluation To assess the efficiency of our proposed method, we compare the running time of integration of different CL methods with MACL and corresponding CL methods alone on CIFAR100, as shown in Table 15 in Appendix D.8. The results indicate that incorporating MACL increases the computational cost by only 55% to 61% compared to the corresponding CL methods alone. This demonstrates the high efficiency of our method, as it introduces only small additional training cost.To evaluate the impact of different architectures, we compared various approaches using both ViT and ResNet32. For the ResNet32 experiments, we followed the setup in [85] , integrating MACL with MEMO [86] and comparing it to MEMO alone, using a memory buffer size of 2000 on CIFAR100. Additionally, we conducted experiments with a pre-trained Vision Transformer (ViT) [16] , specifically the vit-base-patch16-224 model pre-trained on ImageNet1K. On CIFAR100, we integrated MACL with DER++, using a memory size of 500, and demonstrated that using a pre-trained ViT significantly improves CL performance. Moreover, combining MACL with DER++ further enhances CL performance with the pre-trained ViT. The results are presented in the Appendix.Long Task Sequence To assess the effectiveness of the proposed approach across varying task lengths, we conducted experiments by splitting Tiny-ImageNet into sequences of 10 and 20 tasks. The Task-IL and Class-IL results for integrating DER++ with MACL, using a memory buffer size of 500, are presented in Table 4 . These results demonstrate that even with longer task sequences, our method still significantly outperforms DER++. Online CL Under the online CL setting, we evaluate the effectiveness of the proposed approach on CIFAR100 and Tiny-ImageNet by comparing with MKD [46] and PCR [37] . The results are put in the Appendix.To assess the effectiveness of MACL on the 5-Datasets benchmark [79, 5] , which includes CIFAR-10, MNIST [33] , Fashion-MNIST [81] , SVHN [47] , and notMNIST [6] , we conducted experiments. This dataset provides a diverse range of CL tasks. We performed experiments on 5-Datasets, using a memory buffer size of 500, with MACL. The detailed results are provided in the Appendix.We further evaluate the effectiveness of MACL on CUB200 [69] and ImageNet-R [20] , the results are shown in the Appendix.Question: Does the paper provide open access to the data and code, with sufficient instructions to faithfully reproduce the main experimental results, as described in supplemental material?Answer: [Yes]Justification: Code will be released.Guidelines:• The answer NA means that paper does not include experiments requiring code. • Providing as much information as possible in supplemental material (appended to the paper) is recommended, but including URLs to data and code is permitted. 
conclusion: 
 In this paper, we address the challenge of balancing learning new tasks while preserving knowledge from previous ones in continual learning. We propose a model sensitivity-aware continual learning method that enhances both the model's ability to retain old knowledge and improve performance on new tasks. Specifically, our goal is to reduce model parameter sensitivity by optimizing CL performance for the worst-case parameter distribution within the neighborhood of the current model's parameter distribution. This approach improves stability in preserving old knowledge and mitigates overfitting on new tasks. We provide a comprehensive theoretical analysis of the proposed method, and extensive experiments on multiple datasets demonstrate its effectiveness, efficiency, and versatility.Limitation Discussion Our method introduces additional training cost compared to existing continual learning approaches.Question: Does the paper specify all the training and test details (e.g., data splits, hyperparameters, how they were chosen, type of optimizer, etc.) necessary to understand the results?Answer: [Yes]Justification: We provided detailed implementation details regarding training and test details, data splits, hyperparameters and type of optimizer.Guidelines:• The answer NA means that the paper does not include experiments.• The experimental setting should be presented in the core of the paper to a level of detail that is necessary to appreciate the results and make sense of them. • The full details can be provided either with the code, in appendix, or as supplemental material. 
experiment statistical significance: 
 Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?Answer: [Yes]Justification: We provided results standard deviation with multiple experiment runs.Guidelines:• The answer NA means that the paper does not include experiments.• The authors should answer "Yes" if the results are accompanied by error bars, confidence intervals, or statistical significance tests, at least for the experiments that support the main claims of the paper. • The factors of variability that the error bars are capturing should be clearly stated (for example, train/test split, initialization, random drawing of some parameter, or overall run with given experimental conditions). • The method for calculating the error bars should be explained (closed form formula, call to a library function, bootstrap, etc.) • The assumptions made should be given (e.g., Normally distributed errors). • It should be clear whether the error bar is the standard deviation or the standard error of the mean. • It is OK to report 1-sigma error bars, but one should state it. The authors should preferably report a 2-sigma error bar than state that they have a 96% CI, if the hypothesis of Normality of errors is not verified. • For asymmetric distributions, the authors should be careful not to show in tables or figures symmetric error bars that would yield results that are out of range (e.g. negative error rates). • If error bars are reported in tables or plots, The authors should explain in the text how they were calculated and reference the corresponding figures or tables in the text. 
experiments compute resources: 
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 Appendix: Our work advances continual learning, which is beneficial to develop more adaptable and efficient AI. Our work has no negative societal impacts.A.1 Duality in Natural Gradient Descent for Exponential Family Distribution Theorem A.1. Gradient of the loss L(λ) with respect to the expectation parameter λ, i.e., ∇ λ L(λ), is equal to the natural gradient with respect to natural parameter ϕ, i.e., F -1 ∇ ϕ L(ϕ). This can be expressed as the following:EQUATIONIn particular, NGD in natural parameter space can be equivalently performed through gradient descent with respect to the expectation parameters as the following:EQUATION)where η is the learning rate and F is the Fisher information matrix (FIM).Proof. The exponential family distribution is defined as the following:EQUATIONAccording to the expectation of the score function is 0, we can obtain the followingEQUATIONTherefore,EQUATIONwhere the first equality is due to the fact that the expectation of the score function is zero.We then derive the Fisher information matrix (FIM) as the following:EQUATIONEQUATIONEQUATIONEQUATIONEQUATIONwhere := denotes defined as. Then,EQUATIONNext,EQUATIONMore general results on manifold can be found in [53] .EQUATION≤ log u(θ) 2 v(θ) dθ (by Jensen's inequality) (34)EQUATIONEQUATIONwhere the last equality is becauseEQUATIONSince u(θ)dθ = v(θ)dθ = 1Theorem A.3. Assume || 1 v(θ) || ∞ dθ ≤ M ,we can obtain the following conclusion for Eq. ( 16):EQUATION)where L CL (θ) := L CL (θ)dV(θ). We denote the variance of the random variableL CL (θ) as V ar(L CL (θ)) = E(L CL (θ) -L CL (θ)) 2 = (L CL (θ) -L CL (θ)) 2 dθ.Proof. We define a new distribution Z := U -V.EQUATIONBy Lemma 4.1 and Hölder's inequality, we can obtain the following:EQUATIONEQUATIONEQUATIONTherefore,EQUATION)EQUATIONEQUATION)The equality holds when the following condition holds:EQUATION)where a is a constant.EQUATIONThe last equality is because(u(θ) -v(θ))dθ = u(θ)dθ -v(θ)dθ = 1 -1 = 0Therefore, we can obtain the following conclusion:EQUATION)In this context, the CL loss variance across various sets of model parameters V ar(L CL (θ)) serves as a measure of the CL model's sensitivity to parameter updates. Essentially, a smaller loss variance indicates lower parameter sensitivity in the CL model. However, directly optimizing the loss variance within the parameter distribution neighborhood is impractical, as it requires computing the loss variance across a large number of different sets of CL model parameters and training data points. In contrast, our method (MACL) offers an efficient and effective alternative. MACL implicitly minimizes the loss variance across different model parameter variations by optimizing CL performance solely on the worst-case CL model parameter distribution.We denote the prior distribution as V(θ) = N (µ p , Σ p ) and posterior distribution as U(θ) = N (µ s , Σ s ) Theorem A.4 (Generalization bound of MACL). Let q be the number of CL model parameters and n be the number of training data points. The CL loss L CL (θ) ≤ C (C is a constant). With high probability of 1 -δ, the following bound holds:E θ∼N (µ,Σ) i=T i L CL Di (θ) ≤ max U∈U E θ∼U L CL (θ) + C N T + ζ i=T -1 i=1 N i + (53) τ 2 ( √ q + 2 log(N T + ζ i=T -1 i=1 N i )) 2 + R + 2 log( N T +ζ i=T -1 i=1 Ni δ ) 4(N T + ζ i=T -1 i=1 N i -1)Where τ is a constant. We denote the number of data examples for task 1, • • • , T -1 in the memory buffer M during training on task T as N 1 , N 2 , • • • , N T -1 when using memory replay based approach or the number of training data points when using regularization based approach. L CL Di (θ) denotes the CL loss on the data from data distribution D i (generalization error), i.e., it is defined as: L CL Di (θ) := E (x,y)∼Di L(x, y, θ). L CL (θ) denotes the empirical CL loss as Eq. ( 1). N (µ, Σ) denotes the CL model parameter posterior distribution parameterized with Gaussian distribution.Proof. We apply the PAC-Bayes theorem [43] that for any prior distribution, with probability 1 -δ over the CL training dataset T , the following bound holds:EQUATIONThe KL divergence between posterior and prior distribution can be calculated as the following:EQUATIONWe assume the following inequality:EQUATIONTherefore,EQUATIONAccording to [50] , we have the following identity:For a random variable θ ∼ N (µ, Σ)EQUATIONwhere T r denotes the trace of A matrix. Therefore, according to Eq. ( 60), we have the following two 61 and 62.EQUATIONWe set γ = Σ -1 2 p (µ s -µ p ). Then, ||γ|| 2 = (µ s -µ p ) T Σ -1 p (µ s -µ p ). If γ ∼ N (0, τ 2 I), according to [31] , we have the following inequality with probability of 1 -EQUATIONThen we partition the space of µ s into two disjoint area that satisfy(µ s -µ p ) T Σ -1 p (µ s -µ p ) ≤ 2ϵ -R and (µ s -µ p ) T Σ -1 p (µ s -µ p ) > 2ϵ -R. (1) In the case of (µ s -µ p ) T Σ -1 p (µ s -µ p ) ≤ 2ϵ -R,we take the maximum loss over µ s , we have the following inequality:EQUATION(2) For the case of (µ s - 1) and ( 2), we can obtain the following generalization bound:µ p ) T Σ -1 p (µ s -µ p ) > 2ϵ -R, we have L CL T (θ) ≤ C Combining case (EQUATIONWe have the following bound with probability of 1 -1 n :EQUATIONEQUATIONThen, we can obtain the following generalization bound with probability of 1 -1 n :EQUATIONτ 2 ( √ q + 2 log(N T + ζ i=T -1 i=1 N i )) 2 + R + 2 log( N T +ζ i=T -1 i=1 Ni δ ) 4(N T + ζ i=T -1 i=1 N i -1)In this theorem, we provide the theoretical guarantee for the generalization analysis of our proposed method. This bound indicates by optimizing the MACL loss, our method tighten/reduce the generalization error of the CL method, thus improving the overall performance of our method.According to the definition of expectation, we can obtain the following equation:EQUATIONAccording to the definition of covariance matrix,EQUATIONEQUATIONBy rearranging the above equation, we can obtain the following:EQUATIONThen, the conclusion follows:EQUATIONGradient of Loss L(λ) With Respect to λ Taking gradient with respect to λ as the following:EQUATIONIn Eq. ( 76), the second equality is because the identity:EQUATIONEQUATIONIn Eq. ( 77), the second equality is because the identity: ∂µ ∂λ 2 = 0, ∂Σ ∂λ 2 = 1 (by Eq. ( 75)) According to Eq. ( 5), we set the natural parameters as:EQUATION• (1) NGD with respect to ϕ 2 : According to Eq. (20 and 77), NGD with respect to ϕ 2 can be obtained as:EQUATIONThen, obtain the following update:EQUATION• (2) NGD with respect to ϕ 1 : Similarly, according to Eq. (20 and 76), NGD with respect to ϕ 1 can be obtained as:EQUATIONBy simplifying and rearranging Eq. ( 81), the following update for µ:EQUATIONMean and Covariance Updates Derivations Following the results in [56, 26] , we can obtain the following equation:EQUATIONEQUATIONThen, we only need to calculate E θ∼u(θ) ∇ θ L(µ, Σ) and E θ∼u(θ) ∇ 2 θθ L(µ, Σ). Here, since we assumed a general CL Gaussian distribution for the current CL parameter distribution, i.e., V(θ) = N (θ|µ 0 , Σ 0 ) and neighbourhood distribution, i.e., U(θ) = N (θ|µ, Σ). The detailed derivations for the gradient are present in the following:EQUATIONEQUATIONEQUATIONPlug-in the gradient derivation into Eq. (82 and 80), we can obtain the following results:EQUATIONEQUATIONFinally, by using single sample from distribution U with density θ ∼ u(θ) to approximate the expectation. By plug-in Σ = diag(σ 2 ) and Σ 0 = diag(ρ 2 ) into the above equations, we can obtain the following updates:EQUATIONC Baseline Details• EWC [28] : EWC endeavors to alleviate forgetting in continual learning through the utilization of a weighted weight regularization technique based on the Fisher information matrix.• CPR [10] : Drawing on neural networks with wide local minima and principles from information theory, CPR introduces an extra regularization term. This term aims to maximize the entropy of a classifier's output probabilities, thereby reaching wider local minima to enhance generalization.• GPM [60] : A CL model acquires new skills by adjusting its parameters through gradient steps that move orthogonal to the gradient subspaces considered vital for previous tasks. The Gradient Projection Memory (GPM) establishes these subspaces by analyzing network activations following the completion of each task using Singular Value Decomposition (SVD), then preserves them in memory.• HAT [63] : HAT is a task-driven hard attention mechanism that retains information from prior tasks while ensuring it doesn't interfere with the current task's learning process.• A-GEM [14] : AGEM aims to guarantee that, at every training step, the average loss of episodic memory over past tasks does not rise, thus mitigating the risk of forgetting previously acquired knowledge.• Gradient-Based Sample Selection (GSS-Greedy) [2] : The goal is to populate the memory buffer with a diverse set of examples, using the data gradient as a feature for sample selection.For comparison, we opt for the efficient GSS-Greedy version.• ER [15] : This method stores a subset of examples from previous tasks using reservoir sampling [15] . During each iteration, we randomly replay a subset of examples from the memory buffer.• DER++ [7] : This method combines experience replay with knowledge distillation to further improve the effectiveness of experience replay.• ER-ACE [8] : They discovered that ER causes significant overlap between the representations of newly added classes and previous ones, resulting in highly disruptive parameter updates. From this empirical analysis, they proposed a new method to address this issue by protecting the learned representations from drastic adaptations when incorporating new classes. Their approach uses an asymmetric update rule that pushes new classes to adapt to the older ones, rather than the reverse. This technique is particularly effective at task boundaries, where much of the forgetting typically happens.• LODE [36] : They conducted an in-depth analysis of the impacts of distinguishing between new and old classes, as well as among new classes, finding that these two learning objectives result in varying degrees of forgetting. Consequently, combining these objectives negatively affects the performance of the CL model. To address this, LODE separates the two objectives for new tasks by decoupling the loss associated with them. This approach allows LODE to assign different weights to each objective, leading to better performance compared to methods that use a coupled loss. We conducted an experiment integrating MACL with the SOTA prompt-based CL method, CODA-Prompt [65] . Our method operates on the parameters of prompt components and corresponding keys/attention vectors. We conducted experiment on the recent CL datasets of ImageNet-R and CUB200 with pre-trained Vision Transformer (ViT), i.e., vit-base-patch16-224 as the backbone following the codebase of DER++. The results (memory size of 500) are shown in the following table. , with an open-source dataset or instructions for how to construct the dataset). (d) We recognize that reproducibility may be tricky in some cases, in which case authors are welcome to describe the particular way they provide for reproducibility.In the case of closed-source models, it may be that access to the model is limited in some way (e.g., to registered users), but it should be possible for other researchers to have some path to reproducing or verifying the results. 


Decision: This work considers model sensitivity for improving continual learning. An optimization approach based on parameter distribution is introduced to reduce forgetting and overfitting. While there seems to be a room for improvement, the authors provided sufficient updates to address the concerns regarding empirical evidence. Therefore, an acceptance is recommended. 

For the camera-ready submission, please make sure to improve the manuscript as much as possible to reflect the reviewers' feedback and concerns. In particular, please make sure to have the data distribution $D$ well defined and to have the generalization bound addressed sufficiently comprehensively. Updated empirical results should be reflected at least in the appendix with proper comments in the main text. 

Even though not commented by the reviewers, it might be also helpful to address the connection of this work to the existing flat-minima/SWAD related works.

Reviews: review: paper_summary: paper_summary: The paper proposes a new perspective on stability-plasticity trade-off in continual learning, centered on controlling model sensitivity to model updates. The goal is then to ensure that alteration in model parameters does not negatively impact the CL performance of the model. 

In order to solve this challenging task, the authors propose to optimise model’s performance based on the worst-case scenario of parameter distributions within a distribution neighborhood. Given the intractability of this problem due to the infinite dimension of the space of all possible distribution within this neighborhood. The authors solve this issue by modelling the model parameters by a Gaussian distribution. They then propose to solve a min-max problem, with a CL objective equal to the sum of a cross entropy loss and  a forgetting mitigation loss (which can come from a replay method, a regularization term, a gradient projection loss, ...) within a neighborhood based on the KL-divergence with the current CL parameter distribution. In order to efficiently solve this optimization problem: 
* The authors leverage natural gradient descent in the distribution parameter space. 
* They develop an mirror-descent method in the dual space alleviating the need for computing the Fisher information matrix. 

The authors support their method with a theoretical analysis and some experiments. The theoretical analysis shows a reduction in loss variance, indicating a lower model parameter sensitivity, and tighter generalization bounds. The experiments were conducted using a ResNet18 backbone, Split CIFAR and Tiny-ImageNet benchmarks, and adding the proposed approach to a large number of CL algorithms. They also include ablation study to analyse the hyperparameters, study the effect of memory size, comparing natural gradient descent to SGD and analysing the method efficiency.
strength_weakness: strength_weakness: The main weakness of the paper is the lack of diversity in the evaluation settings, and this in multiple aspects: 

* The evaluation is limited to benchmarks that lack in diversity in the data distribution. I invite the authors to look at other benchmarks (e.g. 5-datasets (used in different recent works, e.g. Learning to Prompt for Continual Learning, Wang et al. 2022), Nevis'22 (Bornschein et al. 2023), ... 

* The evaluation is limited to relatively short sequences, it would be interesting to analyse the effect of the sequence length. 

* The evaluation is limited to the task aware scenario, although it is a priori also suitable for the more challenging Online Continual Learning setting. 

* The evaluation is limited to a single architecture that is trained from scratch. It would be interesting to analyse the impact of architecture, and of initialization on the method. 

The authors also analyse the efficiency of the method in one setting only. It would be interesting to add other methods, as the additional cost can be a fix one, and the small additive cost in the analyzed setting can be due to the high cost of the original CL algorithm. 

Despite these limits, I think the work already provides interesting perspective and insights.
questions: questions: * Is modelling the current CL parameters with a standard normal distribution realistic? Isn't it too restrictive? What if we start the CL process with an already pretrained model? 

* Can this modelling assumption be changed with a more general Gaussian distribution? What would be the impact on the derivation? 

* Can the method be added to the recent family of learning to prompt techniques for CL?
limitations: limitations: The work is of foundational nature. I think discussing potential negative societal impact is out of scope.
paper_summary: paper_summary: This paper presents a min-max optimization framework targeting at the model sensitivity for continual learning, where the authors claim that it can mitigate the abrupt change of the model parameters and thus simultaneously alleviates the problem of catastrophic forgetting of the past knowledge and overfitting to the current task. By assuming the posterior distribution of the model parameters as multivariate gaussian, it achieves effective min-max optimization. Some experimental results are are presented to demonstrate the model's better performance.
strength_weakness: strength_weakness: 1. The paper starts with treating the current continual learning objective as a whole without discussing the interplays among the different tasks or the data scarcity problem of the previous tasks, which is not a usual case. The oversimplification of assuming a $\mathcal{L}^{\text{CL}}(\theta)$ can be problematic in the sense that we generally want to treat different tasks differently (past vs. current) so that old knowledge retention and new knowledge acquisition can be achieved at the same time. To further illustrate my point, consider some other arbitrary learning scenarios, this "model sensitivity awareness" can be seamlessly applied. So please elaborate on why CL specifically benefits from this algorithm. 
2. It is not clear what the generalization bound provided by theorem 4.3 implies. It still contains the proxy training target for continual learning $\mathcal{L}^{\text{CL}}$ and seems not very informative about the CL algorithm.
3. The organization and presentation of the paper is not confusing, especially for Section 3.2. There are theorems, together with too detailed derivation of the intermediate results, I would suggest the author re-organize the material and only summarize the most important results in the main paper, and put the not-so-relevant content in the appendix. 
4. The experimental results seem not rigorously verified. As far as I know, the CIL results on CIFAR100 dataset should start with 40-50% overall accuracy but not this low in the paper (too many references and I will just skip providing them). Please provide more demonstration on this.
questions: questions: 1. I find the formulation of the fundamental optimization problem in this work confusing. In Eq. 7,
$$
\min_\theta\max_{U\in\mathcal{U}} E_{\theta\sim U(\theta)}[\mathcal{L}^{\text{CL}}(\theta)],
$$ 
$$ 
\text{s.t.}\quad \mathcal{U} = \\{U: D_{\text{KL}}(U, V) < \epsilon \\},
$$
the outer minimization over $\theta$ is not taking any effect in the inner maximization as it is integrated out over $U$. I assume this $V$ should be dependent on $\theta$? But in the main paper, the authors claim $V=\mathcal{N}(0, I)$. Please elaborate on this.
limitations: limitations: Yes. The authors addressed the limitations in Section 6.
paper_summary: paper_summary: The paper presents a new method to address catastrophic forgetting and improve the learning ability of a new task in continual learning. They attribute these two objectives to parameter sensitivity. To address this problem, they propose to minimize the performance on the worst-case CL parameter distribution within the neighborhood of the current CL model. The approach is integrated with existing CL methods and evaluated on three continual learning benchmarks CIFAR10, CIFAR100, and Tiny-ImageNet.
strength_weakness: strength_weakness: - It is not clear whether the assumption of CL model parameters following a normal distribution could be valid for all models.

- I found it hard to easily understand the relation between the parameter sensitivity and the worst-case CL model. 

- The presentation could be improved. Some sections are wordy. Those can be concise and some of the theoretical proofs in the appendix can be moved to the main paper to easily follow.

- I would expect to see forgetting reported in Table 3.

- Most baselines are a bit old.
questions: questions: - It seems that you mostly focus on integrating the method with replay-based approaches. Is there a motivation behind it?

- Do the same findings generalize to a long sequence of tasks?

- From Table 1 and Table 2, it seems that the methods improve new task learning more than reducing catastrophic forgetting. Do you have some thoughts on that?
limitations: limitations: Yes.
paper_summary: paper_summary: The paper "Model Sensitivity Aware Continual Learning" introduces a novel approach to continual learning (CL) that addresses the trade-off between retaining previously acquired knowledge and excelling in new task performance. Traditional CL models often face the dilemma of catastrophic forgetting or overfitting to new tasks. The proposed method mitigates model parameter sensitivity to updates, ensuring that minor parameter changes do not significantly impact the model's performance. This is achieved by optimizing the model's performance based on the worst-case scenario within a parameter distribution neighborhood. The approach is compatible with existing CL methodologies, offering substantial improvements in retaining old knowledge and enhancing new task performance, supported by theoretical and empirical evidence.
strength_weakness: strength_weakness: The computational complexity of the proposed method, particularly the calculation of the Fisher Information Matrix, might be a concern for large-scale applications.
While the method is demonstrated to be effective in offline CL scenarios, its applicability to online CL is not explored.
The hyperparameter sensitivity, especially regarding the learning rate η, requires careful tuning, which might limit the method's usability in different settings without additional hyperparameter optimization.
The datasets used in this paper are outdated and could consider more recent datasets.
questions: questions: Can the authors provide more insights into the computational overhead introduced by the proposed method, particularly in large-scale datasets or real-time applications?
Have the authors considered any strategies for optimizing hyperparameters in a more automated fashion to improve the method's usability across different datasets and tasks?
Are there any plans to extend this approach to online continual learning scenarios? If so, what challenges do the authors anticipate?
limitations: limitations: The authors have adequately addressed the limitations of their work, particularly in the context of offline CL scenarios. However, exploring the method's applicability to online CL could provide a more comprehensive understanding of its potential. Additionally, a more detailed discussion on the computational complexity and strategies to mitigate it would be beneficial.

Comments: title: Further Clarification
Thank you!
Answer to rebuttal
Further Clarification
Further Clarification (2/2)
Further Clarification (1/2)
Reviewer-author discussion period will end soon.
looking forward to your reply
Thank you!
Official Comment by Reviewer kb2F
Thank you!
Response to Rebuttal by Authors
comment: Thank you very much for your response and support! We sincerely appreciate your valuable suggestions! We will certainly revise our paper according to your comments.


**Q1** I think this result should be included in the revision to make the original theorem more specific and not an overly generalized one.

**A1** Thank you for your suggestions! We will ensure the improved theorem be included in the revision.

**Q2** data absence generalization bound: how is it reflected in the generalization bound or does it affect the generalization bound at all?

**A2** We denote the number of training data examples for task $1, \cdots, T-1$  when training on task $T$ as $N_1, N_2, \cdots, N_{T-1}$. We also denote the number of training examples for the new task $T$ as $N_T$. The generalization bound, considering data absence, can be formulated as follows:

$L _ {T}^{CL}(\theta) \approx L _ {T_{new}}(\theta) + \lambda L _ {T_{old}^{appr}}(\theta)$

$\mathbb{E}_{\theta \sim \mathcal{N}(\mu, \Sigma)} L _ {D}^{CL}(\theta) \leq \max _ {\mathbb{U} \in \mathcal{U}} \mathbb{E} _ {\theta \sim \mathbb {U}}  L^{CL} _ {T}(\theta)  + \frac{C}{N _T + \lambda \sum _ {i=1}^{i=T-1} N_i} +     \sqrt{\frac{\tau ^2(\sqrt{q} + \sqrt{2 \log (N _ T + \lambda \sum _ {i=1} ^{i=T-1} N _ i})^2 + R + 2 \log(\frac{N _ T + \lambda \sum _ {i=1} ^{i=T-1} N _ i}{\delta})}{4(N _ T + \lambda \sum _ {i=1} ^{i=T-1} N _ {i})}}$  

 We denote $\lambda \sum _ {i=1} ^{i=T-1} N _ {i}$ as the effective sample size from previous tasks since the loss function on the training data from previous tasks is approximated due to the absence of data from earlier tasks. The parameter $\lambda$ reflects the trade-off between learning the new task and retaining knowledge from previous tasks. A larger $\lambda$ imposes stronger regularization on the first term on the RHS of the generalization bound, ensuring that the parameters learned for the new task do not deviate significantly from those learned on previous tasks. Consequently, the first term increases, indicating that the overall empirical training loss rises. Meanwhile, the second and third terms on the RHS decrease. The third term, representing the simplified KL divergence between the posterior and prior, decreases, indicating that the posterior distribution for the new task remains close to the parameter distribution from previous tasks. This occurs because we prioritize preserving knowledge from earlier tasks, which in turn constrains the learning process for the new task.

**Q3** a more general case would be considering imbalance and absence at the same time, encompassing both of them in one generalization bound.

**A3**  Yes, we can have a more general theory bound as below: 

$\mathbb{E}_{\theta \sim \mathcal{N}(\mu, \Sigma)} L _ {D}^{CL}(\theta) \leq \max _ {\mathbb{U} \in \mathcal{U}} \mathbb{E} _ {\theta \sim \mathbb {U}}  L^{CL} _ {T}(\theta)  + \frac{C}{N _T + \lambda \sum _ {i=1}^{i=T-1} N_i} +     \sqrt{\frac{\tau ^2(\sqrt{q} + \sqrt{2 \log (N _ T + \lambda \sum _ {i=1} ^{i=T-1} N _ i})^2 + R + 2 \log(\frac{N _ T + \lambda \sum _ {i=1} ^{i=T-1} N _ i}{\delta})}{4(N _ T + \lambda \sum _ {i=1} ^{i=T-1} N _ {i})}}$  

when $\lambda = 1$, and $L^{CL}_{T}(\theta)=L _{T _{new}}(\theta)+\alpha L _{\mathcal{M}}(\theta)$, the effective sample size from previous tasks is equal to the memory buffer size. The generalization bound corresponds to memory replay CL approach.

when $\lambda = \beta$, and $L _ {T}^{CL}(\theta) \approx L _ {T_{new}}(\theta) + \lambda L _ {T_{old}^{appr}}(\theta)$, the bound correspond to regularization-based approach, the effective sample size from previous tasks is equal to $\lambda-$ weighted sample size from previous tasks since the loss function  on the training data from previous tasks is approximated due to the absence of data from earlier tasks. The generalization bound corresponds to data absense of CL approach.
Thank you for your updates! We deeply appreciate your support!
I appreciate the extensive answers of the authors to all the reviews, and I find that many of the added results strengthen the paper. I thank the authors for their efforts, and raise my score accordingly.
Thank you for your response. 

I'm now happy with the data imbalance case, and it shares certain level of the similarity of the bound presented in the previous work [1]. I think this result should be included in the revision to make the original theorem more specific and not an overly generalized one. 

I am still curious about the case of data absence: 
- how is it reflected in the generalization bound or does it affect the generalization bound at all?  
- a more general case would be considering imbalance and absence at the same time, encompassing both of them in one generalization bound. 

I have raised my rating to 5 accordingly. I think this submission can be a significant contribution to the community in the future if a set of improved empirical results and more accurate theorems stated. I would sincerely thank the authors for responding to my questions and I look forward to an improved manuscript in the near future (regardless of whether it is accepted). 

**References**
- [1] Shi, Haizhou, and Hao Wang. "A unified approach to domain incremental learning with memory: Theory and algorithm." Advances in Neural Information Processing Systems 36 (2024).
Thank you for your suggestions! We greatly appreciate your feedback! We promise we will add the new results and discussion on the ResNet-32 and MEMO baselines in the revision.


**Q1**  does $\mathcal{L}_ {\mathcal{D}}^{CL}(\theta)$ in the LHS represents $\sum_{i=1}^T\mathcal{L}_{\mathcal{D_i}}(\theta)$?

**A1**: Yes, it represents the $\sum_{i=1}^T\mathcal{L}_{\mathcal{D_i}}(\theta)$

**Q2** does $\mathcal{L}_{\mathcal{T}}^{CL}(\theta)$ consider the data imbalance and even data absence of the previous tasks?

**A2**:

* **data imbalance**:

**A**: In memory-replay scenarios, the dataset $T$ consists of both the new task data $\mathcal{T}_{new}$ and the memory buffer data $\mathcal{M}$. The underlying assumption in memory-replay-based approaches is that the memory buffer $\mathcal{M}$ effectively approximates the data distribution of previous tasks.

The empirical CL loss function can be defined as:  $L^{CL}_{T}(\theta)=L _{T _{new}}(\theta)+\alpha L _{\mathcal{M}}(\theta)$, 

 where $L_{T_{new}}(\theta)$ denotes the new task empirical training loss and $L_{\mathcal{M}}(\theta)$ is the expected loss on memory buffer data. In other words, the expectation of the loss function on memory buffer, i.e., $L_{\mathcal{M}}(\theta)$, serves as an approximation of the empirical training loss from previous tasks.





We denote the number of data examples for task $1, \cdots, T-1$ in the memory buffer $\mathcal{M}$ when training on task $T$ as $N_1, N_2, \cdots, N_{T-1}$. We also denote the number of training examples for task $T$ as $N_T$. The generalization bound, accounting for data imbalance, can be expressed as follows:




$\mathbb{E}_{\theta \sim \mathcal{N}(\mu, \Sigma)} L _ {D}^{CL}(\theta) \leq \max _ {\mathbb{U} \in \mathcal{U}} \mathbb{E} _ {\theta \sim \mathbb {U}}  L^{CL} _ {T}(\theta)  + \frac{C}{N _T + \sum _ {i=1}^{i=T-1} N_i} +     \sqrt{\frac{\tau ^2(\sqrt{q} + \sqrt{2 \log (N _ T + \sum _ {i=1} ^{i=T-1} N _ i})^2 + R + 2 \log(\frac{N _ T + \sum _ {i=1} ^{i=T-1} N _ i}{\delta})}{4(N _ T + \sum _ {i=1} ^{i=T-1} N _ {i-1})}}$  


The data imbalance issue is reflected in the second and third terms of the generalization bound. This bound suggests that as the amount of data in the memory buffer increases (i.e., $\sum_{i=1}^{i=T-1} (N_i-1)  \uparrow$), the second and third terms in the generalization bound decrease. This is because $\lim_{x\to\infty} [h(x):=\frac{\log x}{x}] = 0$, and $h(x)$ is a decreasing function as $x$ increases over the interval $[c, \infty]$. Consequently, the generalization error decreases, which aligns with the intuitive understanding that a larger memory buffer leads to better performance.


* **data absense**

**A**:   $\mathcal{L}_{\mathcal{T}}^{CL}(\theta)$  can incorporate data absense of previous tasks. For example, when the CL model learns on the task 2, the ideal task loss can be expressed as: 
   
   $\mathcal{L} _ {\mathcal{T}} ^{CL}(\theta) = \mathcal{L} _ {1}(\theta) + \mathcal{L} _ {2}(\theta)$, where $\mathcal{L} _ {1}(\theta)$ denotes the old task 1 empirical training loss and $\mathcal{L}_{2}(\theta)$ denotes the new task 2 empirical training loss.   
   
   
When the data from Task 1 is unavailable while learning Task 2, we can approximate the loss function $\mathcal{L}_{1}(\theta)$ using a second-order Taylor expansion as follows:
  
  
  $\mathcal{L} _ {1}(\theta) \approx \mathcal{L} _ {1}(\theta_1^*) + \nabla_{\theta}\mathcal{L}_{1}(\theta_1^*)^T(\theta - \theta_1^*) +\frac{1}{2}(\theta - \theta_1^*)^TH(\theta - \theta_1^*)$
  
  where $H$ is the Hessian matrix and $\theta_1^*$ is the optimal solution for task 1. Since the gradient $\nabla_ {\theta}\mathcal{L}_ {1}(\theta_1^*)$ achieves approximately optimal at $\theta_1^*$, $\nabla_{\theta}\mathcal{L}_{1}(\theta_1^*) \approx 0$. Then, 
  
  $\mathcal{L} _ {1}(\theta) \approx\mathcal{L}_{1}(\theta_1^*) +\frac{1}{2}(\theta - \theta_1^*)^TH(\theta - \theta_1^*)$
  
The above equation is similar to EWC approach.
  
  Therefore, $\mathcal{L}_{\mathcal{T}}^{CL}(\theta)$ can incorporate the case that the data from previous tasks are unavailable.
Thank you for the response. I am mostly convinced by the experimental results you show in the rebuttal and I highly recommend you to report the ResNet-32 results for CIL in the revised manuscript since the current Table is too far away from the SOTA methods. 

As for the theory, I am still not convinced. If the data distribution is defined as $\mathcal{D}=\bigcup\_{i}\mathcal{D}\_{i}$, then the definition of the dataset $\mathcal{T} = \{(x\_i, y\_i)\}\_{i=1}^{N} \sim \mathcal{D}$ is broken, since this sampling is not reflecting what we have in CL: usually the memory and the current data samples are not evenly sampled, which is the data imbalance (for an example, please refer to UDIL [1], where they provide a generalization bound considering that). To be more specific, in Theorem 4.3,
- does $\mathcal{L}\_{\mathcal{D}}^{CL}(\theta)$ in the LHS represents $\sum\_{i=1}^T\mathcal{L}\_{\mathcal{D\_i}}(\theta)$?
- does $\mathcal{L}\_{\mathcal{T}}^{CL}(\theta)$ consider the data imbalance and even data absence of the previous tasks?

**References**
- [1] Shi, Haizhou, and Hao Wang. "A unified approach to domain incremental learning with memory: Theory and algorithm." Advances in Neural Information Processing Systems 36 (2024).
This response continues the previous discussion.

**Q5. Broken MinMax Problem in Eq. 7** Sorry I still don't get it. In the inner maximization, the expectation over $\theta$ is evaluated under the distribution $U$, it has nothing to do with the $\theta$ you have in the outer minimization. I am okay with the inner maximization being a "sensitivity penalty", can you please elaborate more on the outer $\theta$?





**A**: Thanks for your question!   We would like to clarify that the inner expectation of the loss function can be formulated as $L(\mu, \Sigma)$, which is demonstrated in the following equations. The outer minimization is to optimize with respect to the mean of $\theta$, i.e., $\mu$. Therefore, the outer minimization takes effects on the inner maxmization. This is to make the presentation more concise. We will clarify this in the revision.  


The inner maximization can be converted into the following minimization optimization:

\begin{align} 
\min_{U} [H(U) = -\mathbb{E}_{\theta \sim U(\theta)} L^{CL}(\theta) + \alpha KL(U, V)]
\end{align}


In our paper, we choose $\mathbb{U}(\theta) = \mathcal{N}(\theta|\mu, \Sigma)$, where $\mu$ and $\Sigma$ denote the mean and covariance, respectively. and $u(\theta)$ denotes  the density function. The above equation can be further formulated to be the following equation:

\begin{align} 
L(\mu, \Sigma) := \mathbb{E}_{\theta \sim u(\theta)} [- L^{CL}(\theta) + \alpha [\log u(\theta) - \log v(\theta)]]
\end{align}

The outer minimization is with respect to $\mu$, which is the expectation of $\theta$.  This is because during inference, we only use $\mu$ as the model parameter to perform prediction. 


Additional Questions

**Q1** Throughout the introduction of the method (Section 3.2), there is no effective reference to existing literature. Where was the concept of "Model Sensitivity" first defined and what are existing techniques? If this concept if completely new and so as the methodology of this paper, please indicate it in the main body.

**A1**: Thank you for your suggestions! To our best knowledge, the concept of "Model Sensitivity" as we have defined it in our paper is indeed novel, and we have developed this methodology as an original contribution to the field. We will update the manuscript to highlight the novelty of this concept and methodology more explicitly.



**Q2** To me this method looks alike the idea of SAM (Sharpness-Aware Minimization) applied to CL [3]. Please discuss the relationship between this method and SAM-based CL methods.


**A2** Our method is fundamentally different from SAM-based CL [3] in several aspects.   


* **Deterministic vs. Probabilistic Approach**: SAM uses a fixed deterministic neighborhood, which can be restrictive in practice since updates are constrained within a fixed ball. In contrast, our method employs a probabilistic distributional approach, offering two distinct advantages: (a) The distributional neighborhood is more flexible and covers a broader range of parameter variations by sampling from a neighborhood distribution, and (b) Stochastic Gradient Descent (SGD) introduces noise during Continual Learning (CL). Our distributional approach accounts for this noise, making it a more realistic model in practice and providing stronger guarantees against parameter sensitivity.

* **Uniform vs. Parameter-Specific Updates**: SAM uniformly updates all parameters, overlooking the varying importance and sensitivity of each parameter in the context of CL. Our method, on the other hand, considers these differences and treats parameters uniquely through the natural gradient with the Fisher Information Matrix (FIM). This distinction is crucial for CL, as each parameter has different sensitivity to forgetting—a factor that SAM does not address.
Thank you for your feedback! Currently, we provide more clarifications as below. 

**Q1+Q2 [Oversimplified CL Loss and Generalization Bound]** The loss function we use (for optimization and theoretical analysis) always have an underlying assumption, which is the data distribution it is evaluated. CL loss is composed of two parts, as mentioned by the authors, $\mathcal{L}_ {\text{old}}(\theta)$ and $\mathcal{L}_ {\text{new}}(\theta)$, which are typically evaluated under different data distributions: replay buffer populated from the old data distribution $\mathcal{D}_ {\text{old}}$ and the current-task dataset populated from the current data distribution $\mathcal{D}_ {\text{new}}$. Hence it raises a serious question to the main theorem (Theorem 4.3): what is the data distribution defined $\mathcal{D}$ here? A trivial weighted average of $\mathcal{D}_ {\text{old}}$ and $\mathcal{D}_ {\text{old}}$ is not okay as the losses are evaluated separately. Therefore I would question the validity of the theory in this paper as it to me is serious oversimplification. 

**A**: $\mathcal{D}$ is defined as the CL underlying data distribution, i.e.,  $\mathcal{D}= \bigcup_{i=1}^{i=M} \mathcal{D_i}$, where $D_i$ is the data distribution for task $i$ and $M$ denotes the number of CL tasks. The generalization error **$L_{\mathcal{D}}^{CL}(\theta)$** contains the loss on unseen data, i.e., beyond $\mathcal{D}_ {\text{old}}$ and $\mathcal{D}_ {\text{new}}$, is defined as the loss on the entire CL task sequence: $L_{\mathcal{D}}^{CL}(\theta)​ = \sum_{i=1}^{M} L_ {(x, y)\sim \mathcal{D_i}} (x,y, \theta)$.  Therefore, the generalization loss we define is not an oversimplification but follows the standard definition. $\mathcal{D}_ {\text{old}}$ and $\mathcal{D}_ {\text{new}}$ are new and old task **empirical training samples** from distribution $\mathcal{D}$, respectively.



**Q4. Low Performance of CIL models** Are there any specific reasons for choosing these two papers as the baselines? Specifically, most of the existing work uses ResNet-32 as the backbone for CIFAR100 [1] and why do you use ResNet-18? iCaRL [2], a baseline from 7 years ago, can achieve 40+ accuracy on CIFAR100, with is conducted under the exact setting as you mentioned. The numbers in the Table are not the state-of-the-art.

**A** 

**why choose these two papers as baseline** We would like to clarify that we choose these two papers since they all follow the same codebase of DER++. This would enable us to conduct consistent comparisons.  

**why choose these ResNet-18** We choose ResNet-18, since this architecture is widely adopted in CL works, e.g., DER++, ER-ACE, LODE, etc.   Furthermore, our method does not depend on specific backbones. To illustrate, we integrate our method (MACL) with MEMO [2] on CIFAR100 using ResNet-32 by following the implementation in [1]. The results are shown in the following table. The dataset is split into 10 tasks, where each task has 10 classes and exemplar size is 2000. Our MACL can further improve the base method performance. We will add these discussion about the ResNet-32 results in the revision.



| Method | Class-IL | 
| -------- | -------- | 
| MEMO    | 58.49     | 
| MEMO + MACL    | **59.61**     | 


**iCaRL can achieve 40+ accuracy on CIFAR100**. We would like to clarify that our method (MACL) is orthogonal to existing approaches and can be seamlessly integrated with various methods to enhance their performance. In our submission, we demonstrate its integration with DER++, ER-ACE, and LODE as examples, but MACL is not limited to these methods. To further illustrate its effectiveness, we conducted an additional experiment integrating MACL with iCaRL with memory size of 500. The results, shown below, demonstrate that our method outperforms iCaRL on its own.

| Method |  Class-IL |
| -------- | -------- | 
|   iCaRL   |      44.16 $\pm$  1.53   |
|   iCaRL + MACL   |   **48.27 $\pm$ 0.95**      |


Reference: 

[1] Class-Incremental Learning: A Survey. TPAMI 2024

[2] A Model or 603 Exemplars: Towards Memory-Efficient Class-Incremental Learning. ICLR 2023.
Dear reviewer, authors have provided their rebuttal. Can you please check it and provide your response? Reviewer-author discussion period will end very soon. Thanks. 
- AC
Thank you for your response. After the rebuttal, some of my concerns are solved, but not all of them. Here I would like to kindly ask for further clarification. 

**Q1+Q2 [Oversimplified CL Loss and Generalization Bound]** The loss function we use (for optimization and theoretical analysis) always have an underlying assumption, which is the data distribution it is evaluated. CL loss is composed of two parts, as mentioned by the authors, $\mathcal{L}\_{\text{old}}(\theta)$ and $\mathcal{L}\_{\text{new}}(\theta)$, which are typically evaluated under different data distributions: replay buffer populated from the old data distribution $\mathcal{D}\_{\text{old}}$ and the current-task dataset populated from the current data distribution $\mathcal{D}\_{\text{old}}$. **Hence it raises a serious question to the main theorem (Theorem 4.3): what is the data distribution defined $\mathcal{D}$ here?** A trivial weighted average of $\mathcal{D}\_{\text{old}}$ and $\mathcal{D}\_{\text{new}}$ is not okay as the losses are evaluated separately. Therefore I would question the validity of the theory in this paper as it to me is serious oversimplification. 


**Q4. [Low Performance of CIL models]** Are there any specific reasons for choosing these two papers as the baselines? Specifically, most of the existing work uses ResNet-32 as the backbone for CIFAR100 [1] and why do you use ResNet-18? iCaRL [2], a baseline from 7 years ago, can achieve 40+ accuracy on CIFAR100, with is conducted under the exact setting as you mentioned. The numbers in the Table are not the state-of-the-art. 

**Q5. [Broken MinMax Problem in Eq. 7]** Sorry I still don't get it. In the inner maximization, the expectation over $\theta$ is evaluated under the distribution $U$, it has nothing to do with the $\theta$ you have in the outer minimization. I am okay with the inner maximization being a "sensitivity penalty", can you please elaborate more on the outer $\theta$? 

**Additional Questions** 
- Throughout the introduction of the method (Section 3.2), there is no effective reference to existing literature. Where was the concept of "Model Sensitivity" first defined and what are existing techniques? If this concept if completely new and so as the methodology of this paper, please indicate it in the main body. 
- To me this method looks alike the idea of SAM (Sharpness-Aware Minimization) applied to CL [3]. Please discuss the relationship between this method and SAM-based CL methods. 

**References**
- [1] Zhou, Da-Wei, et al. "Class-Incremental Learning: A Survey." IEEE Transactions on Pattern Analysis and Machine Intelligence (2024).
- [2] Rebuffi, Sylvestre-Alvise, et al. "icarl: Incremental classifier and representation learning." Proceedings of the IEEE conference on Computer Vision and Pattern Recognition. 2017.
- [3] Mehta, Sanket Vaibhav, et al. "An empirical investigation of the role of pre-training in lifelong learning." Journal of Machine Learning Research 24.214 (2023): 1-50.
Dear Reviewer YRBc,

Thank you for your thoughtful feedback on our paper! We have done our best to address your concerns and questions. We would greatly appreciate it if you could let us know whether our response has satisfactorily addressed your concerns.  

We look forward to hearing from you.
Thank you for your response! We sincerely appreciate your support!
Thank you for addressing my concerns and performing additional experiments. I will keep my positive score.
Thank you for your feedback!  We really appreciate your support!
Thanks for the detailed answers to my questions. I am impressed by the additional complexity analysis, parameter analysis, and more recent data experiments. I will be happy to see them in your future paper. For now, I will keep my score.