Input:
Your role is to assess whether a claim is well-stated or overstated based on the available evidence.

The Input Information will include:
1. Original Claim: The claim under evaluation.
2. Evidence: Research findings, including figures, tables, or other relevant data supporting the claim.

Optional. review comment: Reviewer feedback relevant to the claim's validity.

The claim should be assesed with the following:
Score: Evaluate the claim on a scale of 1 to 5, where:
- 1: Well-stated (accurately reflects the evidence without exaggeration)
- 5: Overstated (exaggerates findings' significance, magnitude, or certainty)

Justification: Provide a detailed explanation that includes:
- Instances of exaggerated wording, insufficient experiments, lack of experimental details, gaps in knowledge, weak grounding in evidence, or lack of acknowledgment of limitations.
- Direct citations from relevant evidence that substantiate your reasoning.
- If review comment is included only take into account the relevant comments towards the claim.

The claim to be assessed is:

This objective is to ensure that the new model will be consistent with the previous optimal ones on the old tasks, thus avoiding catastrophic forgetting.

The review comment to be evaluated is:

The authors proposed a new regularizer for continual learning to tackle the catastrophic forgetting problem. The proposed method minimizes the KL-divergence between the prediction of previous models and current models on the stored samples of previous tasks. The idea is straightforward and sounds technical. Experiments show the effectiveness of the methods compared to state-of-the-art. Although the idea sounds interesting and the experiments look promising, the novelty of the paper seems to be limited. In addition, the experiments setting needs to be improved as well. In the following, you have detailed comments.

1. It seems to be an extension of Learning without Forgetting (LwF) Li & Hoiem 2017 with simply on the examples of previous tasks in the memory. LwF only regularizes on the current task. It is not clear what is the difference between the proposed method and LwF except this.
2. The authors fixed many critical hyper-parameters: temperature(5), learning rate(0.05), epochs(10). The author should report the results for all methods with these hyper-parameters chosen on the validation set.
3. The authors presented how they split the training and testing data. Please be clear how you split the validation set.
4. The authors argued that sample quality does not affect the proposed methods. Then the authors should show the variance from different random sampling.

The evidence to be evaluated is:Consider the continual learning problem where the learner \(\mathbf{\theta}\) receives a sequence of T tasks \(\mathcal{T}=\{\mathcal{T}_{1},\mathcal{T}_{2},\ldots,\mathcal{T}_{T}\}\), each of which consists of a training data \(\mathcal{D}_{t}\). At any given time \(t\), only task \(\mathcal{T}_{t}\) are presented to the learner \(\mathbf{\theta}\) and it has to learn to solve the current task without access to the training data of any previous tasks. The goal of continual learning is to train the learner such that it can solve the not only current task \(\mathcal{T}_{t}\) but also all the previous tasks \(\mathcal{T}_{k<t}\). To avoid forgetting previous knowledge, when learning task \(\mathcal{T}_{t}\), the learner \(\mathbf{\theta}\) minimize the multi-task learning objective as \[\tilde{\mathcal{L}}_{t}(\mathbf{\theta})=\mathcal{L}_{t}(\mathbf{\theta},\mathcal{D}_ {t})+\lambda_{t-1}\mathcal{L}_{t-1}(\mathbf{\theta},\mathcal{D}_{t-1})+\ldots+ \lambda_{1}\mathcal{L}_{1}(\mathbf{\theta},\mathcal{D}_{1}), \tag{1}\] where the loss \(\mathcal{L}_{i}(\mathbf{\theta},\mathcal{D}_{i})\) is usually the negative log-likelihood \(-\sum_{\mathbf{x},y\in\mathcal{D}_{i}}\log p(y|\mathbf{x};\mathbf{\theta})\) of the corresponding training data and \(\lambda_{i}\) regulates the importance of task \(\mathcal{T}_{i}\) to all the tasks observed so far. In continual learning, at time \(t\), we do not have access to the data \(\mathcal{D}_{k<t}\) to evaluate the loss \(\mathcal{L}_{k<t}\), thus, we need to approximate the past losses \(\mathcal{L}_{k<t}(\mathbf{\theta},\mathcal{D}_{k})\) when minimizing \(\mathcal{L}_{t}(\mathbf{\theta},\mathcal{D}_{t})\).

\[\tilde{\mathcal{L}}_{t}(\mathbf{\theta})=\mathcal{L}_{t}(\mathbf{\theta},\mathcal{D}_ {t})+\sum_{i<t}\lambda_{k}(\mathbf{\theta}-\mathbf{\theta}_{i}^{*})^{T}\mathbf{F}_{i}(\bm {\theta}-\mathbf{\theta}_{i}^{*}), \tag{3}\]

According to eqn. 3, we can minimize the negative log-likelihood of \(\mathbf{\theta}\) on the previous tasks without explicitly storing the training samples because the data distribution are already captured in the Fisher information.

Another approach to achieve the approximated multi-task learning objective in eqn. 1 is based on the observation that at the beginning of task \(\mathcal{T}_{t}\), we initialize \(\mathbf{\theta}=\mathbf{\theta}_{t-1}^{*}\), whose value already minimized the lost \(\tilde{\mathcal{L}}_{t-1}\). From this, if we can consider eqn. 1 as minimizing the loss \(\mathcal{L}_{t}(\mathbf{\theta})\) with the constraints that previous losses are not allowed to increase after every mini-batch update, which is the GEM objective. In particular, GEM optimize the following problem \[\begin{array}{ll}\text{Minimize}_{\mathbf{\theta},\mathcal{D}_{t}}&\mathcal{L}_ {t}(\mathbf{\theta})\\ \text{subject to}&\mathcal{L}_{k}(\mathbf{\theta},\mathcal{M}_{k})\leq\mathcal{L}_ {k}(\mathbf{\theta}_{t-1}^{*},\mathcal{M}_{k}),\forall k<t,\end{array} \tag{5}\]where the loss \(\mathcal{L}_{k}\) is evaluated at the memory \(\mathcal{M}_{k}\) that stores some of the training samples from task \(\mathcal{T}_{k}\). GEM further converts the constraints in eqn. 5 into the gradient constraints of the losses and solve them using quadratic programming.

For example, Learning without Forgetting (LwF) (Li & Hoiem, 2017) proposes a form of distillation loss by minimizing the KL-divergence of past models and current model on the newly observed tasks. By changing the losses \(\mathcal{L}_{k<t}\), we can recover different solutions for continual learning in the literature.

In general, if we have too many misclassified samples or label noise, the decision boundary on the previous task might be altered despite minimizing the EWC or GEM objective. This phenomenon happens based on an assumption that we usually do not train the model to have 0 error on the training set to avoid overfitting and we only use a relatively small data portion to estimate the Fisher or store as memory. Therefore, EWC and GEM might not correctly penalize the current model from changing the past tasks' optimized decision boundaries. To tackle the aforementioned problem, we propose to minimize the KL-divergence between the current model and the previous optimal models on the corresponding tasks. This can also be interpreted as distilling the knowledge from \(\mathbf{\theta}_{i}^{*}\) to \(\mathbf{\theta}\) on the corresponding data \(\mathcal{D}_{i}\). In particular, for each task \(\mathcal{T}_{i}\), we store some of the samples \(\mathbf{x}_{i}\) and the prediction of the optimal model \(f_{\mathbf{\theta}_{i}^{*}}(\mathbf{x}_{i})\) in a memory \(\mathcal{M}_{i}\); then the Activity Regularization (AR) objective for task \(\mathcal{T}_{t}\) is \[\tilde{\mathcal{L}}_{t}(\mathbf{\theta})=\mathcal{L}_{t}(\mathbf{\theta})+\sum_{i<t} \sum_{\mathbf{x}\in\mathcal{M}_{i}}\lambda_{i}KL\bigg{(}\frac{1}{\tau}f_{\mathbf{ \theta}_{i}^{*}}(\mathbf{x})\bigg{\|}\ \frac{1}{\tau}f_{\mathbf{\theta}}(\mathbf{x})\bigg{)}, \tag{7}\]

Our goal is to ensure that the new model \(\mathbf{\theta}\) behaves similarly with \(\mathbf{\theta}_{i}^{*}\) on the previous task \(\mathcal{T}_{t}\), thus we call this constraint as activity regularization.

We refer to this algorithm as "Deterministic Activity Regularization" (DAR) which imposes regularization constraints for all the previous tasks at every learning iteration.

7:\(g_{\mathbf{\theta}}\leftarrow\nabla_{\mathbf{\theta}}\left[\sum_{\mathbf{x},y\in\mathcal{D }}\mathcal{L}_{t}(\mathbf{\theta},(\mathbf{x},y))+\sum_{i<t}\sum_{\mathbf{x}\in\mathcal{M} _{i}}\lambda_{i}KL\bigg{(}\frac{1}{\tau}f_{\mathbf{\theta}_{i}^{*}}(\mathbf{x})\ \bigg{\|}\ \frac{1}{\tau}f_{\mathbf{\theta}}(\mathbf{x})\bigg{)}\right]\)

To address the weaknesses of DAR, we propose the Stochastic Activity Regularization (SAR), which randomly samples a past task and apply activity regularization on each gradient update when learning a new task.

Then, when learning a new task, for each mini batch to update \(\mathbf{\theta}\), we apply AR on all of the previous memory.

As a result, SAR is much more robust as it improves the model's plasticity, alleviates catastrophic forgetting and has greater scalability.

The memory \(\mathcal{M}_{i}\) is used to store some training samples of the previous tasks which is used to regularize the current model when learning new tasks.

8:\(g_{\mathbf{\theta}}\leftarrow\nabla_{\mathbf{\theta}}\left[\sum_{\mathbf{x},y\in\mathcal{ D}}\mathcal{L}_{t}(\mathbf{\theta},(\mathbf{x},y))+\sum_{\mathbf{x}\in\mathcal{M}_{i}} \lambda_{i}KL\bigg{(}\frac{1}{\tau}f_{\mathbf{\theta}_{i}^{*}}(\mathbf{x})\ \bigg{\|}\ \frac{1}{\tau}f_{\mathbf{\theta}}(\mathbf{x})\bigg{)}\right]\)

The goal of continual learning is to build a learner that can continuously learn from a stream of tasks while still maintain its previously acquired knowledge.

**Structural regularization** approaches employ penalties on the network's parameters, encourage the important parameters of previous tasks to not change when learning the new task. The parameter importance can be estimated by calculating the Fisher information as in EWC (Kirkpatrick et al., 2017) or directly measure how changes in each parameter will affect the change in the loss function as proposed in SI (Zenke et al., 2017).

Learning without Forgetting (LwF) (Li & Hoiem, 2017) proposes to minimize the difference in KL-divergence of the current and previous models on the new tasks, which is a form of knowledge distillation penalty. Similarly, Less-forgetting learning (LF) (Jung et al., 2019) proposes to minimize the \(\ell_{2}\) norm of the two prediction in stead of KL-divergence as LwF. Motivated from Hebbian learning (Hebbs, 1949), MAS (Aljundi et al., 2018a) proposes to penalize the \(\ell_{2}\) norm of the model's prediction.

Our approaches, DAR and SAR, distills past models on their correctly trained data, thus maintaining the past knowledge for new models.

Finally, **ensemble learning** techniques addresses catastrophic forgetting by having a dedicated sub-network for each task.

After training a task, the corresponding sub-network is frozen and will not allowed to change, thus completely immune to catastrophic forgetting.

Catastrophic forgetting is avoided by using a form of structural regularizer. Ensemble learning approaches are usually resistant to catastrophic forgetting as the cost of either unbounded growth of model's size or cannot be applied to different network architectures.

For EWC, at the end of training, we randomly sample \(N=200\) samples to estimate the diagonal Fisher matrix and store the optimal model \(\mathbf{\theta}_{i}^{*}\) for each task.

SI performs better than EWC thanks to its Fisher approximation strategy on the whole data. GEM attains 91.8% accuracy, higher than both EWC, SI and VCL due the its strong constraints.

Both DAR and SAR are consistently better than GEM at all memory size. Only at 2500 samples, GEM performs better than our methods at 200 samples. At 2500 samples, both DAR and SAR achieves over 96% accuracy, showing great scalability. Across all methods, performance consistently improves when the memory size increases. When only a small amount of samples are available in the memory, e.g. 200 and 500, GEM may overfit to these samples, leading to the performance degrade on the whole past data sets.

In both cases, our methods show advantage over GEM, maintaining performance on previous task by enforcing model's consistency.

noisy memory can easily cause catastrophic forgetting in EWC and GEM while SI is more resistant to noise since it approximates the Fisher on all observed samples of past tasks.

We report the averaged accuracy after 10 tasks in fig. 1(a), where EWC and SI are not as effective as alleviating forgetting as in previous experiments because this benchmark is more challenging.

We propose an approximated multi-task learning framework that can alleviate catastrophic forgetting, an important challenge in continual learning and show that several previous solutions in the literature are special cases of this framework.

To overcome this problem, we propose the deterministic activity regularization for continual learning, DAR, that enforces the new model's behaviour to be consistent with the optimal models on each of the previous tasks. As the result, DAR is much more resistant to noise in the memory since its goal is to maintain the previously optimized decision boundary.

Finally, we show that DAR and SAR can be applied on both MLP and CNN, effectively alleviate catastrophic forgetting.

Input: Figure 1: Comparison of different memory size on the permuted MNIST experiments.

Input: Figure 2: Performance results on the split CIFAR-100 benchmark.

Input:
You should:
1. Carefully review the claim and the text and image evidence. Summarize how the evidence influenced your evaluation of the claim's validity and strength and briefly explain whether the claim is wellstated or overstated on a scale of 1 to 5 (up to 100 words).
2. Provide the final score (choose from 1 to 5), in the format: <score>{score}</score>.

Output:
After reviewing the claim and evidence, I find the claim is well-supported by extensive mathematical formulation and experimental results. The evidence shows:

1. Clear mathematical framework demonstrating how the model maintains consistency with previous tasks through KL-divergence minimization
2. Empirical results in Figures 1 and 2 showing both DAR and SAR methods effectively prevent catastrophic forgetting compared to baselines
3. Experiments across different memory sizes and tasks demonstrating consistent performance
4. Detailed explanation of how the approach maintains previous task knowledge through activity regularization

The claim is precise and accurately reflects what the evidence demonstrates without exaggeration. The authors provide comprehensive theoretical and experimental support for their objective of maintaining model consistency to avoid catastrophic forgetting.

<score>1</score>


Input:
Your role is to assess whether a claim is well-stated or overstated based on the available evidence.

The Input Information will include:
1. Original Claim: The claim under evaluation.
2. Evidence: Research findings, including figures, tables, or other relevant data supporting the claim.

Optional. review comment: Reviewer feedback relevant to the claim's validity.

The claim should be assesed with the following:
Score: Evaluate the claim on a scale of 1 to 5, where:
- 1: Well-stated (accurately reflects the evidence without exaggeration)
- 5: Overstated (exaggerates findings' significance, magnitude, or certainty)

Justification: Provide a detailed explanation that includes:
- Instances of exaggerated wording, insufficient experiments, lack of experimental details, gaps in knowledge, weak grounding in evidence, or lack of acknowledgment of limitations.
- Direct citations from relevant evidence that substantiate your reasoning.
- If review comment is included only take into account the relevant comments towards the claim.

The claim to be assessed is:

We first propose an approximated multi-task learning framework that unifies a family of popular regularization based continual learning methods.

The evidence to be evaluated is:To avoid forgetting previous knowledge, when learning task \(\mathcal{T}_{t}\), the learner \(\mathbf{\theta}\) minimize the multi-task learning objective as \[\tilde{\mathcal{L}}_{t}(\mathbf{\theta})=\mathcal{L}_{t}(\mathbf{\theta},\mathcal{D}_ {t})+\lambda_{t-1}\mathcal{L}_{t-1}(\mathbf{\theta},\mathcal{D}_{t-1})+\ldots+ \lambda_{1}\mathcal{L}_{1}(\mathbf{\theta},\mathcal{D}_{1}), \tag{1}\] where the loss \(\mathcal{L}_{i}(\mathbf{\theta},\mathcal{D}_{i})\) is usually the negative log-likelihood \(-\sum_{\mathbf{x},y\in\mathcal{D}_{i}}\log p(y|\mathbf{x};\mathbf{\theta})\) of the corresponding training data and \(\lambda_{i}\) regulates the importance of task \(\mathcal{T}_{i}\) to all the tasks observed so far. In continual learning, at time \(t\), we do not have access to the data \(\mathcal{D}_{k<t}\) to evaluate the loss \(\mathcal{L}_{k<t}\), thus, we need to approximate the past losses \(\mathcal{L}_{k<t}(\mathbf{\theta},\mathcal{D}_{k})\) when minimizing \(\mathcal{L}_{t}(\mathbf{\theta},\mathcal{D}_{t})\). A common choice is to employ Laplace propagation (Eskin et al., 2004) of the negative log posterior \[-\log p(\mathbf{\theta}|\mathcal{D}_{i})\approx-\log p(\mathbf{\theta}_{i}^{*}| \mathcal{D}_{i})+\frac{1}{2}(\mathbf{\theta}-\mathbf{\theta}_{i}^{*})^{T}\mathbf{F}_{i}( \mathbf{\theta}-\mathbf{\theta}_{i}^{*}), \tag{2}\] where \(\mathbf{\theta}_{i}^{*}\) is a mode of the log posterior and \(\mathbf{F}_{i}\) is the empirical Fisher information matrix evaluated at \(\mathbf{\theta}=\mathbf{\theta}_{i}^{*}\). If we assume an uniform prior on \(\mathbf{\theta}\), the posterior coined with the likelihood 1, thus by substituting eqn. 2 to 1, the objective for training task \(\mathcal{T}_{t}\) becomes

\[\tilde{\mathcal{L}}_{t}(\mathbf{\theta})=\mathcal{L}_{t}(\mathbf{\theta},\mathcal{D}_ {t})+\sum_{i<t}\lambda_{k}(\mathbf{\theta}-\mathbf{\theta}_{i}^{*})^{T}\mathbf{F}_{i}(\bm {\theta}-\mathbf{\theta}_{i}^{*}), \tag{3}\] where the terms \(-\log p(\mathbf{\theta}_{i}^{*}|\mathcal{D}_{i})\) are omitted because they are constant with respect to \(\mathbf{\theta}\).

Eqn. 3 results in a wide range of methods for continual learning, each of which proposes a different method to approximate the Fisher information matrix. For example, Elastic Weight Consolidation (EWC) (Kirkpatrick et al., 2017) assumes the Fisher is diagonal and approximates it using the identity: \[\mathbf{F}_{t}\approx\text{diag}\left(\sum_{n=1}^{N}\left.\left(\nabla_{\mathbf{ \theta}}\log p(\mathbf{y}_{t}^{(n)}|\mathbf{x}_{t}^{(n)};\mathbf{\theta})\right)^{2} \right|_{\mathbf{\theta}=\mathbf{\theta}_{T}}\right) \tag{4}\] Instead of using the Fisher's identity, Synaptic Intelligence (SI) (Zenke et al., 2017) directly measures the sensitivity of a parameter \(\mathbf{\theta}_{t}^{k}\) to the loss function at task \(\mathcal{T}_{t}\) through out the whole training trajectory. Zenke et al. (2017) also shows that this estimation is equivalent to measuring the full Fisher matrix on whole data set under certain choices of the loss function. Another approach to achieve the approximated multi-task learning objective in eqn. 1 is based on the observation that at the beginning of task \(\mathcal{T}_{t}\), we initialize \(\mathbf{\theta}=\mathbf{\theta}_{t-1}^{*}\), whose value already minimized the lost \(\tilde{\mathcal{L}}_{t-1}\). From this, if we can consider eqn. 1 as minimizing the loss \(\mathcal{L}_{t}(\mathbf{\theta})\) with the constraints that previous losses are not allowed to increase after every mini-batch update, which is the GEM objective. In particular, GEM optimize the following problem \[\begin{array}{ll}\text{Minimize}_{\mathbf{\theta},\mathcal{D}_{t}}&\mathcal{L}_ {t}(\mathbf{\theta})\\ \text{subject to}&\mathcal{L}_{k}(\mathbf{\theta},\mathcal{M}_{k})\leq\mathcal{L}_ {k}(\mathbf{\theta}_{t-1}^{*},\mathcal{M}_{k}),\forall k<t,\end{array} \tag{5}\]where the loss \(\mathcal{L}_{k}\) is evaluated at the memory \(\mathcal{M}_{k}\) that stores some of the training samples from task \(\mathcal{T}_{k}\). GEM further converts the constraints in eqn. 5 into the gradient constraints of the losses and solve them using quadratic programming. Finally, the type of losses \(\mathcal{L}_{k<t}\) can be flexibly changed to other loss rather than the negative log-likelihood. For example, Learning without Forgetting (LwF) (Li & Hoiem, 2017) proposes a form of distillation loss by minimizing the KL-divergence of past models and current model on the newly observed tasks. By changing the losses \(\mathcal{L}_{k<t}\), we can recover different solutions for continual learning in the literature.

To tackle the aforementioned problem, we propose to minimize the KL-divergence between the current model and the previous optimal models on the corresponding tasks. This can also be interpreted as distilling the knowledge from \(\mathbf{\theta}_{i}^{*}\) to \(\mathbf{\theta}\) on the corresponding data \(\mathcal{D}_{i}\).

Our goal is to ensure that the new model \(\mathbf{\theta}\) behaves similarly with \(\mathbf{\theta}_{i}^{*}\) on the previous task \(\mathcal{T}_{t}\), thus we call this constraint as activity regularization.

We refer to this algorithm as "Deterministic Activity Regularization" (DAR) which imposes regularization constraints for all the previous tasks at every learning iteration.

Similar to approximated multi-task learning methods, DAR employs a regularization constraint on every of the past tasks when learning a new one, which requires calculating the KL-divergence resulting in an additional forward pass on the memory, and thus results in additional computational cost when training the models.

3: Observe the data set \(\mathcal{D}_{t}\) 4:\(\mathcal{M}_{t}\leftarrow\) add random samples from \(\mathcal{D}_{t}\) to the memory. 5:for k = 1,...,\(n_{iter}\)do 6: Sample \(\mathcal{D}=\{\mathbf{x}^{(n)},y^{(n)}\}_{n=1}^{N}\) a batch from training data \(\mathcal{D}_{t}\) 7:\(g_{\mathbf{\theta}}\leftarrow\nabla_{\mathbf{\theta}}\left[\sum_{\mathbf{x},y\in\mathcal{D }}\mathcal{L}_{t}(\mathbf{\theta},(\mathbf{x},y))+\sum_{i<t}\sum_{\mathbf{x}\in\mathcal{M} _{i}}\lambda_{i}KL\bigg{(}\frac{1}{\tau}f_{\mathbf{\theta}_{i}^{*}}(\mathbf{x})\ \bigg{\|}\ \frac{1}{\tau}f_{\mathbf{\theta}}(\mathbf{x})\bigg{)}\right]\) 8:\(\mathbf{\theta}\leftarrow\mathbf{\theta}-\alpha\cdot\text{SGD}(\mathbf{\theta},g_{\mathbf{ \theta}})\) 9:return\(\mathbf{\theta}\) ```**Algorithm 1** Deterministic Activity Regularization (DAR) algorithm for Continual Learning. To address the weaknesses of DAR, we propose the Stochastic Activity Regularization (SAR), which randomly samples a past task and apply activity regularization on each gradient update when learning a new task. Algorithm 1 summarizes the proposed SAR algorithm for continual learning.

Then, when learning a new task, for each mini batch to update \(\mathbf{\theta}\), we apply AR on all of the previous memory.

The memory \(\mathcal{M}_{i}\) is used to store some training samples of the previous tasks which is used to regularize the current model when learning new tasks. Using a memory for continual learning has been explored by previous works Nguyen et al. (2018) and Lopez-Paz et al. (2017). Although more sophisticated methods such as the greedy \(K\)-center algorithm can be used to select data points that spread through out the input space (Nguyen et al., 2018), in this work, even random sampling works well with our methods and can achieve state-of-the-art results.

7: Sample an index i from \([0,\dots,t-1]\) 8:\(g_{\mathbf{\theta}}\leftarrow\nabla_{\mathbf{\theta}}\left[\sum_{\mathbf{x},y\in\mathcal{ D}}\mathcal{L}_{t}(\mathbf{\theta},(\mathbf{x},y))+\sum_{\mathbf{x}\in\mathcal{M}_{i}} \lambda_{i}KL\bigg{(}\frac{1}{\tau}f_{\mathbf{\theta}_{i}^{*}}(\mathbf{x})\ \bigg{\|}\ \frac{1}{\tau}f_{\mathbf{\theta}}(\mathbf{x})\bigg{)}\right]\) 9:\(\mathbf{\theta}\leftarrow\mathbf{\theta}-\alpha\cdot\text{SGD}(\mathbf{\theta},g_{\mathbf{ \theta}})\)

Prior works can be broadly categorized into 4 main approaches: (1) structural regularization, (2) functional regularization, (3) Bayesian learning and (4) ensemble learning. **Structural regularization** approaches employ penalties on the network's parameters, encourage the important parameters of previous tasks to not change when learning the new task. The parameter importance can be estimated by calculating the Fisher information as in EWC (Kirkpatrick et al., 2017) or directly measure how changes in each parameter will affect the change in the loss function as proposed in SI (Zenke et al., 2017).

**Functional regularization** methods employ the regularizer on the model's output rather than its parameters. Different from structural regularization, functional regularization aims to preserve the learned input-output mapping function of the network. Learning without Forgetting (LwF) (Li & Hoiem, 2017) proposes to minimize the difference in KL-divergence of the current and previous models on the new tasks, which is a form of knowledge distillation penalty. Similarly, Less-forgetting learning (LF) (Jung et al., 2019) proposes to minimize the \(\ell_{2}\) norm of the two prediction in stead of KL-divergence as LwF. Motivated from Hebbian learning (Hebbs, 1949), MAS (Aljundi et al., 2018a) proposes to penalize the \(\ell_{2}\) norm of the model's prediction. Selfless sequential learning (SNI) (Aljundi et al., 2018b) improves MAS by combining with it sparsity in each layer's activation.

While there are some overlapping between Bayesian learning and our approximated multi-task learning framework, the main difference is that our objective allows us to actively control the contribution of previous tasks by setting the importance value \(\lambda_{i}\) for each loss \(\mathcal{L}_{i}\).

In this work, we propose an approximated multi-task learning framework to unify both structural and functional regularization, in which our proposed activity regularization is a special case.

We propose an approximated multi-task learning framework that can alleviate catastrophic forgetting, an important challenge in continual learning and show that several previous solutions in the literature are special cases of this framework.

To overcome this problem, we propose the deterministic activity regularization for continual learning, DAR, that enforces the new model's behaviour to be consistent with the optimal models on each of the previous tasks.

We further improve DAR with a stochastic sampling regularizer (SAR), which balances the model's stability and plasticity, allowing the model to acquire new knowledge, alleviate forgetting and has greater scalability. Through extensive experiments on popular continual learning benchmarks, we show that DAR and SAR consistently outperforms state-of-the-art methods.

Input:
You should:
1. Carefully review the claim and the text and image evidence. Summarize how the evidence influenced your evaluation of the claim's validity and strength and briefly explain whether the claim is wellstated or overstated on a scale of 1 to 5 (up to 100 words).
2. Provide the final score (choose from 1 to 5), in the format: <score>{score}</score>.

Output:
Let me evaluate the claim: "We first propose an approximated multi-task learning framework that unifies a family of popular regularization based continual learning methods."

Summary of evaluation:
The evidence strongly supports this claim. The authors:
1. Present a clear mathematical framework (Equations 1-3) that forms the basis of the approximated multi-task learning approach
2. Demonstrate how existing methods (EWC, SI, GEM, LwF) can be derived from this framework
3. Show the connections between structural and functional regularization approaches
4. Explicitly explain how different loss functions and approximations lead to different continual learning solutions

The claim is well-stated (score 1) as it accurately represents what is demonstrated in the evidence without exaggeration. The mathematical derivations and detailed explanations clearly show how various existing methods can be unified under their proposed framework.

<score>1</score>


Input:
Your role is to assess whether a claim is well-stated or overstated based on the available evidence.

The Input Information will include:
1. Original Claim: The claim under evaluation.
2. Evidence: Research findings, including figures, tables, or other relevant data supporting the claim.

Optional. review comment: Reviewer feedback relevant to the claim's validity.

The claim should be assesed with the following:
Score: Evaluate the claim on a scale of 1 to 5, where:
- 1: Well-stated (accurately reflects the evidence without exaggeration)
- 5: Overstated (exaggerates findings' significance, magnitude, or certainty)

Justification: Provide a detailed explanation that includes:
- Instances of exaggerated wording, insufficient experiments, lack of experimental details, gaps in knowledge, weak grounding in evidence, or lack of acknowledgment of limitations.
- Direct citations from relevant evidence that substantiate your reasoning.
- If review comment is included only take into account the relevant comments towards the claim.

The claim to be assessed is:

We first propose an approximated multi-task learning framework that unifies a family of popular regularization based continual learning methods.

The review comment to be evaluated is:

This paper proposes an approach to mitigate catastrophic forgetting in supervised learning by regularizing activations. The paper views previous techniques (EWC, SI, and GEM) under a multi-task learning lens, and then proposes an additional loss term to minimise the KL between activations from previous and current models, on previous tasks - this is based on a memory which stores some previous samples and their corresponding activations.

I think it is a simple and intuitive approach and a well-written paper. Unfortunately I have a number of concerns that I think preclude publication in the current state.

First, in terms of related work, I believe this is very similar to Learning without forgetting (LwF), with the difference that the KL-divergence is computed on samples kept from the previous tasks. This is briefly mentioned in the paper, but I think it needs to be made more explicit, and LwF should be a baseline in the experiments to clearly indicate the benefit of keeping this data. There is also a relationship to EWC: given the connection between the Fisher information and KL, it can be viewed as minimising the KL divergence in parameter space, rather than in activation space (which is the case here). Also note that EWC uses the true Fisher rather than the empirical, contrary to the derivation in equation (2).
There are also a number of papers that haven’t been cited in the related work [1][2][3][4].

Second, I think the motivation in Section 3.1 could be more convincing. Most importantly, it’s not clear to me that the decision boundary *shouldn’t* change for previously misclassified examples, as this could be an opportunity for backwards transfer.
Further, I don’t think the point in the last paragraph about having a small data portion is relevant, since they are from the same data distribution, and we would expect misclassified samples to be in the same (low) frequency in Fisher estimation as overall. I think the point of this paragraph is just that it is important to consider the entire predictive distribution of previous tasks rather than the probability of the correct class, so this should be stated more clearly and then justified.

Finally, I think the experimental justification could be improved as well. Beyond permuted MNIST (which it has been argued is not as useful as other baselines [4]), only the final performance on split notMNIST / CIFAR-100 is reported. Some comments and questions:
- The accuracies of EWC (and possibly SI) in the table are worse than reported in previous work (eg. [1]), so I think this needs to be examined.
- What is the fine-tuning baseline (I don't believe it is actually clearly defined)? How can it be so low in figure 2a but better in 2b?
- I think plots over time (performance on all tasks) would be much more useful than the final performance in Table 2 and Fig 2.
- Errors and error bars would be beneficial for all results.
- Table 1 should also include the references provided.

Some other comments and questions:
- Compared to eqn (2), eqn (6) is missing the ½ constant.
- Typos in section 5.3: "SI performs better than SI", and VAR instead of SAR.
- Section 2, unclear of meaning of "coined with the likelihood" (should this be “coincide”?)
- The first line should be “Humans have the ability to learn...” In general, I think the introduction could use another proofread for grammar and readability as I saw a few minor things.

[1] Nguyen, Cuong V., et al. "Variational Continual Learning." ICLR, 2018.
[2] Schwarz, Jonathan, et al. "Progress & Compress: A scalable framework for continual learning." ICML, 2018.
[3] Shin, Hanul, et al. "Continual learning with deep generative replay." NIPS, 2017.
[4] Farquhar, Sebastian, and Yarin Gal. "Towards Robust Evaluations of Continual Learning." arXiv, 2018.

The evidence to be evaluated is:To avoid forgetting previous knowledge, when learning task \(\mathcal{T}_{t}\), the learner \(\mathbf{\theta}\) minimize the multi-task learning objective as \[\tilde{\mathcal{L}}_{t}(\mathbf{\theta})=\mathcal{L}_{t}(\mathbf{\theta},\mathcal{D}_ {t})+\lambda_{t-1}\mathcal{L}_{t-1}(\mathbf{\theta},\mathcal{D}_{t-1})+\ldots+ \lambda_{1}\mathcal{L}_{1}(\mathbf{\theta},\mathcal{D}_{1}), \tag{1}\] where the loss \(\mathcal{L}_{i}(\mathbf{\theta},\mathcal{D}_{i})\) is usually the negative log-likelihood \(-\sum_{\mathbf{x},y\in\mathcal{D}_{i}}\log p(y|\mathbf{x};\mathbf{\theta})\) of the corresponding training data and \(\lambda_{i}\) regulates the importance of task \(\mathcal{T}_{i}\) to all the tasks observed so far. In continual learning, at time \(t\), we do not have access to the data \(\mathcal{D}_{k<t}\) to evaluate the loss \(\mathcal{L}_{k<t}\), thus, we need to approximate the past losses \(\mathcal{L}_{k<t}(\mathbf{\theta},\mathcal{D}_{k})\) when minimizing \(\mathcal{L}_{t}(\mathbf{\theta},\mathcal{D}_{t})\). A common choice is to employ Laplace propagation (Eskin et al., 2004) of the negative log posterior \[-\log p(\mathbf{\theta}|\mathcal{D}_{i})\approx-\log p(\mathbf{\theta}_{i}^{*}| \mathcal{D}_{i})+\frac{1}{2}(\mathbf{\theta}-\mathbf{\theta}_{i}^{*})^{T}\mathbf{F}_{i}( \mathbf{\theta}-\mathbf{\theta}_{i}^{*}), \tag{2}\] where \(\mathbf{\theta}_{i}^{*}\) is a mode of the log posterior and \(\mathbf{F}_{i}\) is the empirical Fisher information matrix evaluated at \(\mathbf{\theta}=\mathbf{\theta}_{i}^{*}\). If we assume an uniform prior on \(\mathbf{\theta}\), the posterior coined with the likelihood 1, thus by substituting eqn. 2 to 1, the objective for training task \(\mathcal{T}_{t}\) becomes

\[\tilde{\mathcal{L}}_{t}(\mathbf{\theta})=\mathcal{L}_{t}(\mathbf{\theta},\mathcal{D}_ {t})+\sum_{i<t}\lambda_{k}(\mathbf{\theta}-\mathbf{\theta}_{i}^{*})^{T}\mathbf{F}_{i}(\bm {\theta}-\mathbf{\theta}_{i}^{*}), \tag{3}\] where the terms \(-\log p(\mathbf{\theta}_{i}^{*}|\mathcal{D}_{i})\) are omitted because they are constant with respect to \(\mathbf{\theta}\).

Eqn. 3 results in a wide range of methods for continual learning, each of which proposes a different method to approximate the Fisher information matrix. For example, Elastic Weight Consolidation (EWC) (Kirkpatrick et al., 2017) assumes the Fisher is diagonal and approximates it using the identity: \[\mathbf{F}_{t}\approx\text{diag}\left(\sum_{n=1}^{N}\left.\left(\nabla_{\mathbf{ \theta}}\log p(\mathbf{y}_{t}^{(n)}|\mathbf{x}_{t}^{(n)};\mathbf{\theta})\right)^{2} \right|_{\mathbf{\theta}=\mathbf{\theta}_{T}}\right) \tag{4}\] Instead of using the Fisher's identity, Synaptic Intelligence (SI) (Zenke et al., 2017) directly measures the sensitivity of a parameter \(\mathbf{\theta}_{t}^{k}\) to the loss function at task \(\mathcal{T}_{t}\) through out the whole training trajectory. Zenke et al. (2017) also shows that this estimation is equivalent to measuring the full Fisher matrix on whole data set under certain choices of the loss function. Another approach to achieve the approximated multi-task learning objective in eqn. 1 is based on the observation that at the beginning of task \(\mathcal{T}_{t}\), we initialize \(\mathbf{\theta}=\mathbf{\theta}_{t-1}^{*}\), whose value already minimized the lost \(\tilde{\mathcal{L}}_{t-1}\). From this, if we can consider eqn. 1 as minimizing the loss \(\mathcal{L}_{t}(\mathbf{\theta})\) with the constraints that previous losses are not allowed to increase after every mini-batch update, which is the GEM objective. In particular, GEM optimize the following problem \[\begin{array}{ll}\text{Minimize}_{\mathbf{\theta},\mathcal{D}_{t}}&\mathcal{L}_ {t}(\mathbf{\theta})\\ \text{subject to}&\mathcal{L}_{k}(\mathbf{\theta},\mathcal{M}_{k})\leq\mathcal{L}_ {k}(\mathbf{\theta}_{t-1}^{*},\mathcal{M}_{k}),\forall k<t,\end{array} \tag{5}\]where the loss \(\mathcal{L}_{k}\) is evaluated at the memory \(\mathcal{M}_{k}\) that stores some of the training samples from task \(\mathcal{T}_{k}\). GEM further converts the constraints in eqn. 5 into the gradient constraints of the losses and solve them using quadratic programming. Finally, the type of losses \(\mathcal{L}_{k<t}\) can be flexibly changed to other loss rather than the negative log-likelihood. For example, Learning without Forgetting (LwF) (Li & Hoiem, 2017) proposes a form of distillation loss by minimizing the KL-divergence of past models and current model on the newly observed tasks. By changing the losses \(\mathcal{L}_{k<t}\), we can recover different solutions for continual learning in the literature.

To tackle the aforementioned problem, we propose to minimize the KL-divergence between the current model and the previous optimal models on the corresponding tasks. This can also be interpreted as distilling the knowledge from \(\mathbf{\theta}_{i}^{*}\) to \(\mathbf{\theta}\) on the corresponding data \(\mathcal{D}_{i}\).

Our goal is to ensure that the new model \(\mathbf{\theta}\) behaves similarly with \(\mathbf{\theta}_{i}^{*}\) on the previous task \(\mathcal{T}_{t}\), thus we call this constraint as activity regularization.

We refer to this algorithm as "Deterministic Activity Regularization" (DAR) which imposes regularization constraints for all the previous tasks at every learning iteration.

Similar to approximated multi-task learning methods, DAR employs a regularization constraint on every of the past tasks when learning a new one, which requires calculating the KL-divergence resulting in an additional forward pass on the memory, and thus results in additional computational cost when training the models.

3: Observe the data set \(\mathcal{D}_{t}\) 4:\(\mathcal{M}_{t}\leftarrow\) add random samples from \(\mathcal{D}_{t}\) to the memory. 5:for k = 1,...,\(n_{iter}\)do 6: Sample \(\mathcal{D}=\{\mathbf{x}^{(n)},y^{(n)}\}_{n=1}^{N}\) a batch from training data \(\mathcal{D}_{t}\) 7:\(g_{\mathbf{\theta}}\leftarrow\nabla_{\mathbf{\theta}}\left[\sum_{\mathbf{x},y\in\mathcal{D }}\mathcal{L}_{t}(\mathbf{\theta},(\mathbf{x},y))+\sum_{i<t}\sum_{\mathbf{x}\in\mathcal{M} _{i}}\lambda_{i}KL\bigg{(}\frac{1}{\tau}f_{\mathbf{\theta}_{i}^{*}}(\mathbf{x})\ \bigg{\|}\ \frac{1}{\tau}f_{\mathbf{\theta}}(\mathbf{x})\bigg{)}\right]\) 8:\(\mathbf{\theta}\leftarrow\mathbf{\theta}-\alpha\cdot\text{SGD}(\mathbf{\theta},g_{\mathbf{ \theta}})\) 9:return\(\mathbf{\theta}\) ```**Algorithm 1** Deterministic Activity Regularization (DAR) algorithm for Continual Learning. To address the weaknesses of DAR, we propose the Stochastic Activity Regularization (SAR), which randomly samples a past task and apply activity regularization on each gradient update when learning a new task. Algorithm 1 summarizes the proposed SAR algorithm for continual learning.

Then, when learning a new task, for each mini batch to update \(\mathbf{\theta}\), we apply AR on all of the previous memory.

The memory \(\mathcal{M}_{i}\) is used to store some training samples of the previous tasks which is used to regularize the current model when learning new tasks. Using a memory for continual learning has been explored by previous works Nguyen et al. (2018) and Lopez-Paz et al. (2017). Although more sophisticated methods such as the greedy \(K\)-center algorithm can be used to select data points that spread through out the input space (Nguyen et al., 2018), in this work, even random sampling works well with our methods and can achieve state-of-the-art results.

7: Sample an index i from \([0,\dots,t-1]\) 8:\(g_{\mathbf{\theta}}\leftarrow\nabla_{\mathbf{\theta}}\left[\sum_{\mathbf{x},y\in\mathcal{ D}}\mathcal{L}_{t}(\mathbf{\theta},(\mathbf{x},y))+\sum_{\mathbf{x}\in\mathcal{M}_{i}} \lambda_{i}KL\bigg{(}\frac{1}{\tau}f_{\mathbf{\theta}_{i}^{*}}(\mathbf{x})\ \bigg{\|}\ \frac{1}{\tau}f_{\mathbf{\theta}}(\mathbf{x})\bigg{)}\right]\) 9:\(\mathbf{\theta}\leftarrow\mathbf{\theta}-\alpha\cdot\text{SGD}(\mathbf{\theta},g_{\mathbf{ \theta}})\)

Prior works can be broadly categorized into 4 main approaches: (1) structural regularization, (2) functional regularization, (3) Bayesian learning and (4) ensemble learning. **Structural regularization** approaches employ penalties on the network's parameters, encourage the important parameters of previous tasks to not change when learning the new task. The parameter importance can be estimated by calculating the Fisher information as in EWC (Kirkpatrick et al., 2017) or directly measure how changes in each parameter will affect the change in the loss function as proposed in SI (Zenke et al., 2017).

**Functional regularization** methods employ the regularizer on the model's output rather than its parameters. Different from structural regularization, functional regularization aims to preserve the learned input-output mapping function of the network. Learning without Forgetting (LwF) (Li & Hoiem, 2017) proposes to minimize the difference in KL-divergence of the current and previous models on the new tasks, which is a form of knowledge distillation penalty. Similarly, Less-forgetting learning (LF) (Jung et al., 2019) proposes to minimize the \(\ell_{2}\) norm of the two prediction in stead of KL-divergence as LwF. Motivated from Hebbian learning (Hebbs, 1949), MAS (Aljundi et al., 2018a) proposes to penalize the \(\ell_{2}\) norm of the model's prediction. Selfless sequential learning (SNI) (Aljundi et al., 2018b) improves MAS by combining with it sparsity in each layer's activation.

While there are some overlapping between Bayesian learning and our approximated multi-task learning framework, the main difference is that our objective allows us to actively control the contribution of previous tasks by setting the importance value \(\lambda_{i}\) for each loss \(\mathcal{L}_{i}\).

In this work, we propose an approximated multi-task learning framework to unify both structural and functional regularization, in which our proposed activity regularization is a special case.

We propose an approximated multi-task learning framework that can alleviate catastrophic forgetting, an important challenge in continual learning and show that several previous solutions in the literature are special cases of this framework.

To overcome this problem, we propose the deterministic activity regularization for continual learning, DAR, that enforces the new model's behaviour to be consistent with the optimal models on each of the previous tasks.

We further improve DAR with a stochastic sampling regularizer (SAR), which balances the model's stability and plasticity, allowing the model to acquire new knowledge, alleviate forgetting and has greater scalability. Through extensive experiments on popular continual learning benchmarks, we show that DAR and SAR consistently outperforms state-of-the-art methods.

Input:
You should:
1. Carefully review the claim and the text and image evidence. Summarize how the evidence influenced your evaluation of the claim's validity and strength and briefly explain whether the claim is wellstated or overstated on a scale of 1 to 5 (up to 100 words).
2. Provide the final score (choose from 1 to 5), in the format: <score>{score}</score>.

Output:
Let me evaluate the claim: "We first propose an approximated multi-task learning framework that unifies a family of popular regularization based continual learning methods."

Based on the evidence and review:

1. The reviewer points out this approach has significant similarity to Learning without Forgetting (LwF), with the main difference being KL-divergence computed on stored samples. The reviewer suggests this similarity needs to be made more explicit.

2. The evidence shows that the authors present a mathematical framework in equations 1-5 that does connect several existing methods like EWC, SI, and GEM through approximated multi-task learning. However, there's limited novelty in "first proposing" this unification since similar connections were already known.

3. The claim of being "first" to propose this framework appears overstated given the reviewer's comments about existing related work and similarities to LwF that aren't fully acknowledged.

Score and Justification:
The claim is moderately overstated (3) since while the framework does unify some methods, the novelty claim of being "first" isn't well-supported given the existing literature connections noted by the reviewer.

<score>3</score>


Input:
Your role is to assess whether a claim is well-stated or overstated based on the available evidence.

The Input Information will include:
1. Original Claim: The claim under evaluation.
2. Evidence: Research findings, including figures, tables, or other relevant data supporting the claim.

Optional. review comment: Reviewer feedback relevant to the claim's validity.

The claim should be assesed with the following:
Score: Evaluate the claim on a scale of 1 to 5, where:
- 1: Well-stated (accurately reflects the evidence without exaggeration)
- 5: Overstated (exaggerates findings' significance, magnitude, or certainty)

Justification: Provide a detailed explanation that includes:
- Instances of exaggerated wording, insufficient experiments, lack of experimental details, gaps in knowledge, weak grounding in evidence, or lack of acknowledgment of limitations.
- Direct citations from relevant evidence that substantiate your reasoning.
- If review comment is included only take into account the relevant comments towards the claim.

The claim to be assessed is:

We first propose an approximated multi-task learning framework that unifies a family of popular regularization based continual learning methods.

The review comment to be evaluated is:

The paper addresses the problem of continual learning from a sequence of supervised tasks. The main contribution of the paper are the following. The work:
* puts the problem in uniform general framework in which many of the state-of-the-art methods fit
* identifies some drawbacks of some the state of the art methods (namely EWC- Kirkpatrick et al., 2017 and GEM-Lopez-Paz et al., 2017) .
* proposes two versions of an approach to address these drawbacks
The main identified problem in EWC and GEM is that both methods result in a zero penalty when both the previous and current models misclassify a sample, even if they make different mistakes.
To solve this problem, the authors propose to keep a memory of randomly sampled data from the previous task, and distil the knowledge from the optima of the previous tasks to the current one using a KL penalty. The first version considers computing this penalty from the whole sequence of tasks, while the second randomly selects one task at each iteration and uses it to compute the penalty, paying some overall accuracy for plasticity.

While the paper is clear and well written, I have some concerns about it's quality, originality and significance.

Originality: The work seems to me to be very related to LwF. The main difference is that while LwF uses only the new data for the KL penalty, this paper keeps a memory of previously seen data to compute this loss.

Significance: While the way the authors approached the problem seems well structured and motivated, and is based on a sound observation, the authors limited themselves to experiments where the task data have similar structure. I am not sure how significant the improvement of the method would be in the more challenging and realistic setting where the data comes from different domains.  I am more specifically skeptical about the stochastic version of the algorithm in that case.

Quality:
* The proposed algorithm stores some data from previous tasks along with the outputs of the corresponding optimal model. While knowledge distillation as proposed would result in a non zero penalty when the new model makes different prediction, it still doesn't take advantage of the new information to probably correct the previous models prediction when it is wrong. Why not keeping the ground truth labels instead? This won't increase the memory requirement, and may give better results. It would be interesting to compare against such a method.
* I think selecting the samples to keep in memory randomly could to be suboptimal.  Other selection methods can be considered. A previous work:  iCaRL: Incremental Classifier and Representation Learning, Rebuffi et al. 2017, gives way to select representative samples. It would be interesting to see the effect of such a selection on the results.

Overall, while the paper doesn't present any significant flaw, it doesn't add much to the continual learning literature either, which explains my rating.

The evidence to be evaluated is:To avoid forgetting previous knowledge, when learning task \(\mathcal{T}_{t}\), the learner \(\mathbf{\theta}\) minimize the multi-task learning objective as \[\tilde{\mathcal{L}}_{t}(\mathbf{\theta})=\mathcal{L}_{t}(\mathbf{\theta},\mathcal{D}_ {t})+\lambda_{t-1}\mathcal{L}_{t-1}(\mathbf{\theta},\mathcal{D}_{t-1})+\ldots+ \lambda_{1}\mathcal{L}_{1}(\mathbf{\theta},\mathcal{D}_{1}), \tag{1}\] where the loss \(\mathcal{L}_{i}(\mathbf{\theta},\mathcal{D}_{i})\) is usually the negative log-likelihood \(-\sum_{\mathbf{x},y\in\mathcal{D}_{i}}\log p(y|\mathbf{x};\mathbf{\theta})\) of the corresponding training data and \(\lambda_{i}\) regulates the importance of task \(\mathcal{T}_{i}\) to all the tasks observed so far. In continual learning, at time \(t\), we do not have access to the data \(\mathcal{D}_{k<t}\) to evaluate the loss \(\mathcal{L}_{k<t}\), thus, we need to approximate the past losses \(\mathcal{L}_{k<t}(\mathbf{\theta},\mathcal{D}_{k})\) when minimizing \(\mathcal{L}_{t}(\mathbf{\theta},\mathcal{D}_{t})\). A common choice is to employ Laplace propagation (Eskin et al., 2004) of the negative log posterior \[-\log p(\mathbf{\theta}|\mathcal{D}_{i})\approx-\log p(\mathbf{\theta}_{i}^{*}| \mathcal{D}_{i})+\frac{1}{2}(\mathbf{\theta}-\mathbf{\theta}_{i}^{*})^{T}\mathbf{F}_{i}( \mathbf{\theta}-\mathbf{\theta}_{i}^{*}), \tag{2}\] where \(\mathbf{\theta}_{i}^{*}\) is a mode of the log posterior and \(\mathbf{F}_{i}\) is the empirical Fisher information matrix evaluated at \(\mathbf{\theta}=\mathbf{\theta}_{i}^{*}\). If we assume an uniform prior on \(\mathbf{\theta}\), the posterior coined with the likelihood 1, thus by substituting eqn. 2 to 1, the objective for training task \(\mathcal{T}_{t}\) becomes

\[\tilde{\mathcal{L}}_{t}(\mathbf{\theta})=\mathcal{L}_{t}(\mathbf{\theta},\mathcal{D}_ {t})+\sum_{i<t}\lambda_{k}(\mathbf{\theta}-\mathbf{\theta}_{i}^{*})^{T}\mathbf{F}_{i}(\bm {\theta}-\mathbf{\theta}_{i}^{*}), \tag{3}\] where the terms \(-\log p(\mathbf{\theta}_{i}^{*}|\mathcal{D}_{i})\) are omitted because they are constant with respect to \(\mathbf{\theta}\).

Eqn. 3 results in a wide range of methods for continual learning, each of which proposes a different method to approximate the Fisher information matrix. For example, Elastic Weight Consolidation (EWC) (Kirkpatrick et al., 2017) assumes the Fisher is diagonal and approximates it using the identity: \[\mathbf{F}_{t}\approx\text{diag}\left(\sum_{n=1}^{N}\left.\left(\nabla_{\mathbf{ \theta}}\log p(\mathbf{y}_{t}^{(n)}|\mathbf{x}_{t}^{(n)};\mathbf{\theta})\right)^{2} \right|_{\mathbf{\theta}=\mathbf{\theta}_{T}}\right) \tag{4}\] Instead of using the Fisher's identity, Synaptic Intelligence (SI) (Zenke et al., 2017) directly measures the sensitivity of a parameter \(\mathbf{\theta}_{t}^{k}\) to the loss function at task \(\mathcal{T}_{t}\) through out the whole training trajectory. Zenke et al. (2017) also shows that this estimation is equivalent to measuring the full Fisher matrix on whole data set under certain choices of the loss function. Another approach to achieve the approximated multi-task learning objective in eqn. 1 is based on the observation that at the beginning of task \(\mathcal{T}_{t}\), we initialize \(\mathbf{\theta}=\mathbf{\theta}_{t-1}^{*}\), whose value already minimized the lost \(\tilde{\mathcal{L}}_{t-1}\). From this, if we can consider eqn. 1 as minimizing the loss \(\mathcal{L}_{t}(\mathbf{\theta})\) with the constraints that previous losses are not allowed to increase after every mini-batch update, which is the GEM objective. In particular, GEM optimize the following problem \[\begin{array}{ll}\text{Minimize}_{\mathbf{\theta},\mathcal{D}_{t}}&\mathcal{L}_ {t}(\mathbf{\theta})\\ \text{subject to}&\mathcal{L}_{k}(\mathbf{\theta},\mathcal{M}_{k})\leq\mathcal{L}_ {k}(\mathbf{\theta}_{t-1}^{*},\mathcal{M}_{k}),\forall k<t,\end{array} \tag{5}\]where the loss \(\mathcal{L}_{k}\) is evaluated at the memory \(\mathcal{M}_{k}\) that stores some of the training samples from task \(\mathcal{T}_{k}\). GEM further converts the constraints in eqn. 5 into the gradient constraints of the losses and solve them using quadratic programming. Finally, the type of losses \(\mathcal{L}_{k<t}\) can be flexibly changed to other loss rather than the negative log-likelihood. For example, Learning without Forgetting (LwF) (Li & Hoiem, 2017) proposes a form of distillation loss by minimizing the KL-divergence of past models and current model on the newly observed tasks. By changing the losses \(\mathcal{L}_{k<t}\), we can recover different solutions for continual learning in the literature.

To tackle the aforementioned problem, we propose to minimize the KL-divergence between the current model and the previous optimal models on the corresponding tasks. This can also be interpreted as distilling the knowledge from \(\mathbf{\theta}_{i}^{*}\) to \(\mathbf{\theta}\) on the corresponding data \(\mathcal{D}_{i}\).

Our goal is to ensure that the new model \(\mathbf{\theta}\) behaves similarly with \(\mathbf{\theta}_{i}^{*}\) on the previous task \(\mathcal{T}_{t}\), thus we call this constraint as activity regularization.

We refer to this algorithm as "Deterministic Activity Regularization" (DAR) which imposes regularization constraints for all the previous tasks at every learning iteration.

Similar to approximated multi-task learning methods, DAR employs a regularization constraint on every of the past tasks when learning a new one, which requires calculating the KL-divergence resulting in an additional forward pass on the memory, and thus results in additional computational cost when training the models.

3: Observe the data set \(\mathcal{D}_{t}\) 4:\(\mathcal{M}_{t}\leftarrow\) add random samples from \(\mathcal{D}_{t}\) to the memory. 5:for k = 1,...,\(n_{iter}\)do 6: Sample \(\mathcal{D}=\{\mathbf{x}^{(n)},y^{(n)}\}_{n=1}^{N}\) a batch from training data \(\mathcal{D}_{t}\) 7:\(g_{\mathbf{\theta}}\leftarrow\nabla_{\mathbf{\theta}}\left[\sum_{\mathbf{x},y\in\mathcal{D }}\mathcal{L}_{t}(\mathbf{\theta},(\mathbf{x},y))+\sum_{i<t}\sum_{\mathbf{x}\in\mathcal{M} _{i}}\lambda_{i}KL\bigg{(}\frac{1}{\tau}f_{\mathbf{\theta}_{i}^{*}}(\mathbf{x})\ \bigg{\|}\ \frac{1}{\tau}f_{\mathbf{\theta}}(\mathbf{x})\bigg{)}\right]\) 8:\(\mathbf{\theta}\leftarrow\mathbf{\theta}-\alpha\cdot\text{SGD}(\mathbf{\theta},g_{\mathbf{ \theta}})\) 9:return\(\mathbf{\theta}\) ```**Algorithm 1** Deterministic Activity Regularization (DAR) algorithm for Continual Learning. To address the weaknesses of DAR, we propose the Stochastic Activity Regularization (SAR), which randomly samples a past task and apply activity regularization on each gradient update when learning a new task. Algorithm 1 summarizes the proposed SAR algorithm for continual learning.

Then, when learning a new task, for each mini batch to update \(\mathbf{\theta}\), we apply AR on all of the previous memory.

The memory \(\mathcal{M}_{i}\) is used to store some training samples of the previous tasks which is used to regularize the current model when learning new tasks. Using a memory for continual learning has been explored by previous works Nguyen et al. (2018) and Lopez-Paz et al. (2017). Although more sophisticated methods such as the greedy \(K\)-center algorithm can be used to select data points that spread through out the input space (Nguyen et al., 2018), in this work, even random sampling works well with our methods and can achieve state-of-the-art results.

7: Sample an index i from \([0,\dots,t-1]\) 8:\(g_{\mathbf{\theta}}\leftarrow\nabla_{\mathbf{\theta}}\left[\sum_{\mathbf{x},y\in\mathcal{ D}}\mathcal{L}_{t}(\mathbf{\theta},(\mathbf{x},y))+\sum_{\mathbf{x}\in\mathcal{M}_{i}} \lambda_{i}KL\bigg{(}\frac{1}{\tau}f_{\mathbf{\theta}_{i}^{*}}(\mathbf{x})\ \bigg{\|}\ \frac{1}{\tau}f_{\mathbf{\theta}}(\mathbf{x})\bigg{)}\right]\) 9:\(\mathbf{\theta}\leftarrow\mathbf{\theta}-\alpha\cdot\text{SGD}(\mathbf{\theta},g_{\mathbf{ \theta}})\)

Prior works can be broadly categorized into 4 main approaches: (1) structural regularization, (2) functional regularization, (3) Bayesian learning and (4) ensemble learning. **Structural regularization** approaches employ penalties on the network's parameters, encourage the important parameters of previous tasks to not change when learning the new task. The parameter importance can be estimated by calculating the Fisher information as in EWC (Kirkpatrick et al., 2017) or directly measure how changes in each parameter will affect the change in the loss function as proposed in SI (Zenke et al., 2017).

**Functional regularization** methods employ the regularizer on the model's output rather than its parameters. Different from structural regularization, functional regularization aims to preserve the learned input-output mapping function of the network. Learning without Forgetting (LwF) (Li & Hoiem, 2017) proposes to minimize the difference in KL-divergence of the current and previous models on the new tasks, which is a form of knowledge distillation penalty. Similarly, Less-forgetting learning (LF) (Jung et al., 2019) proposes to minimize the \(\ell_{2}\) norm of the two prediction in stead of KL-divergence as LwF. Motivated from Hebbian learning (Hebbs, 1949), MAS (Aljundi et al., 2018a) proposes to penalize the \(\ell_{2}\) norm of the model's prediction. Selfless sequential learning (SNI) (Aljundi et al., 2018b) improves MAS by combining with it sparsity in each layer's activation.

While there are some overlapping between Bayesian learning and our approximated multi-task learning framework, the main difference is that our objective allows us to actively control the contribution of previous tasks by setting the importance value \(\lambda_{i}\) for each loss \(\mathcal{L}_{i}\).

In this work, we propose an approximated multi-task learning framework to unify both structural and functional regularization, in which our proposed activity regularization is a special case.

We propose an approximated multi-task learning framework that can alleviate catastrophic forgetting, an important challenge in continual learning and show that several previous solutions in the literature are special cases of this framework.

To overcome this problem, we propose the deterministic activity regularization for continual learning, DAR, that enforces the new model's behaviour to be consistent with the optimal models on each of the previous tasks.

We further improve DAR with a stochastic sampling regularizer (SAR), which balances the model's stability and plasticity, allowing the model to acquire new knowledge, alleviate forgetting and has greater scalability. Through extensive experiments on popular continual learning benchmarks, we show that DAR and SAR consistently outperforms state-of-the-art methods.

Input:
You should:
1. Carefully review the claim and the text and image evidence. Summarize how the evidence influenced your evaluation of the claim's validity and strength and briefly explain whether the claim is wellstated or overstated on a scale of 1 to 5 (up to 100 words).
2. Provide the final score (choose from 1 to 5), in the format: <score>{score}</score>.

Output:
I'll evaluate the claim: "We first propose an approximated multi-task learning framework that unifies a family of popular regularization based continual learning methods."

1. Summary of evaluation:
The evidence strongly supports this claim. The paper presents a comprehensive mathematical framework (Equations 1-5) that demonstrates how various existing methods (EWC, SI, GEM, LwF) can be derived from their proposed approximated multi-task learning framework. The reviewer, while critical of other aspects, doesn't dispute this unification contribution. The paper clearly shows how different regularization approaches fit within their framework by varying the loss functions or approximation methods. The claim is well-stated and accurately reflects what the evidence demonstrates.

<score>1</score>

Justification:
- The evidence provides detailed mathematical derivations showing how their framework encompasses different methods:
  - Shows how EWC fits through Fisher matrix approximation (Eq. 4)
  - Demonstrates GEM as a constrained optimization variant (Eq. 5)
  - Explains how LwF's distillation loss fits within the framework
- The reviewer explicitly acknowledges this contribution, stating the work "puts the problem in uniform general framework in which many of the state-of-the-art methods fit"
- The claim avoids overstatement by specifically limiting scope to "regularization based" methods and using "family of" rather than claiming to unify all approaches
- The mathematical framework is thoroughly presented with clear connections to existing methods


Input:
Your role is to assess whether a claim is well-stated or overstated based on the available evidence.

The Input Information will include:
1. Original Claim: The claim under evaluation.
2. Evidence: Research findings, including figures, tables, or other relevant data supporting the claim.

Optional. review comment: Reviewer feedback relevant to the claim's validity.

The claim should be assesed with the following:
Score: Evaluate the claim on a scale of 1 to 5, where:
- 1: Well-stated (accurately reflects the evidence without exaggeration)
- 5: Overstated (exaggerates findings' significance, magnitude, or certainty)

Justification: Provide a detailed explanation that includes:
- Instances of exaggerated wording, insufficient experiments, lack of experimental details, gaps in knowledge, weak grounding in evidence, or lack of acknowledgment of limitations.
- Direct citations from relevant evidence that substantiate your reasoning.
- If review comment is included only take into account the relevant comments towards the claim.

The claim to be assessed is:

We first propose an approximated multi-task learning framework that unifies a family of popular regularization based continual learning methods.

The review comment to be evaluated is:

The authors proposed a new regularizer for continual learning to tackle the catastrophic forgetting problem. The proposed method minimizes the KL-divergence between the prediction of previous models and current models on the stored samples of previous tasks. The idea is straightforward and sounds technical. Experiments show the effectiveness of the methods compared to state-of-the-art. Although the idea sounds interesting and the experiments look promising, the novelty of the paper seems to be limited. In addition, the experiments setting needs to be improved as well. In the following, you have detailed comments.

1. It seems to be an extension of Learning without Forgetting (LwF) Li & Hoiem 2017 with simply on the examples of previous tasks in the memory. LwF only regularizes on the current task. It is not clear what is the difference between the proposed method and LwF except this.
2. The authors fixed many critical hyper-parameters: temperature(5), learning rate(0.05), epochs(10). The author should report the results for all methods with these hyper-parameters chosen on the validation set.
3. The authors presented how they split the training and testing data. Please be clear how you split the validation set.
4. The authors argued that sample quality does not affect the proposed methods. Then the authors should show the variance from different random sampling.

The evidence to be evaluated is:To avoid forgetting previous knowledge, when learning task \(\mathcal{T}_{t}\), the learner \(\mathbf{\theta}\) minimize the multi-task learning objective as \[\tilde{\mathcal{L}}_{t}(\mathbf{\theta})=\mathcal{L}_{t}(\mathbf{\theta},\mathcal{D}_ {t})+\lambda_{t-1}\mathcal{L}_{t-1}(\mathbf{\theta},\mathcal{D}_{t-1})+\ldots+ \lambda_{1}\mathcal{L}_{1}(\mathbf{\theta},\mathcal{D}_{1}), \tag{1}\] where the loss \(\mathcal{L}_{i}(\mathbf{\theta},\mathcal{D}_{i})\) is usually the negative log-likelihood \(-\sum_{\mathbf{x},y\in\mathcal{D}_{i}}\log p(y|\mathbf{x};\mathbf{\theta})\) of the corresponding training data and \(\lambda_{i}\) regulates the importance of task \(\mathcal{T}_{i}\) to all the tasks observed so far. In continual learning, at time \(t\), we do not have access to the data \(\mathcal{D}_{k<t}\) to evaluate the loss \(\mathcal{L}_{k<t}\), thus, we need to approximate the past losses \(\mathcal{L}_{k<t}(\mathbf{\theta},\mathcal{D}_{k})\) when minimizing \(\mathcal{L}_{t}(\mathbf{\theta},\mathcal{D}_{t})\). A common choice is to employ Laplace propagation (Eskin et al., 2004) of the negative log posterior \[-\log p(\mathbf{\theta}|\mathcal{D}_{i})\approx-\log p(\mathbf{\theta}_{i}^{*}| \mathcal{D}_{i})+\frac{1}{2}(\mathbf{\theta}-\mathbf{\theta}_{i}^{*})^{T}\mathbf{F}_{i}( \mathbf{\theta}-\mathbf{\theta}_{i}^{*}), \tag{2}\] where \(\mathbf{\theta}_{i}^{*}\) is a mode of the log posterior and \(\mathbf{F}_{i}\) is the empirical Fisher information matrix evaluated at \(\mathbf{\theta}=\mathbf{\theta}_{i}^{*}\). If we assume an uniform prior on \(\mathbf{\theta}\), the posterior coined with the likelihood 1, thus by substituting eqn. 2 to 1, the objective for training task \(\mathcal{T}_{t}\) becomes

\[\tilde{\mathcal{L}}_{t}(\mathbf{\theta})=\mathcal{L}_{t}(\mathbf{\theta},\mathcal{D}_ {t})+\sum_{i<t}\lambda_{k}(\mathbf{\theta}-\mathbf{\theta}_{i}^{*})^{T}\mathbf{F}_{i}(\bm {\theta}-\mathbf{\theta}_{i}^{*}), \tag{3}\] where the terms \(-\log p(\mathbf{\theta}_{i}^{*}|\mathcal{D}_{i})\) are omitted because they are constant with respect to \(\mathbf{\theta}\).

Eqn. 3 results in a wide range of methods for continual learning, each of which proposes a different method to approximate the Fisher information matrix. For example, Elastic Weight Consolidation (EWC) (Kirkpatrick et al., 2017) assumes the Fisher is diagonal and approximates it using the identity: \[\mathbf{F}_{t}\approx\text{diag}\left(\sum_{n=1}^{N}\left.\left(\nabla_{\mathbf{ \theta}}\log p(\mathbf{y}_{t}^{(n)}|\mathbf{x}_{t}^{(n)};\mathbf{\theta})\right)^{2} \right|_{\mathbf{\theta}=\mathbf{\theta}_{T}}\right) \tag{4}\] Instead of using the Fisher's identity, Synaptic Intelligence (SI) (Zenke et al., 2017) directly measures the sensitivity of a parameter \(\mathbf{\theta}_{t}^{k}\) to the loss function at task \(\mathcal{T}_{t}\) through out the whole training trajectory. Zenke et al. (2017) also shows that this estimation is equivalent to measuring the full Fisher matrix on whole data set under certain choices of the loss function. Another approach to achieve the approximated multi-task learning objective in eqn. 1 is based on the observation that at the beginning of task \(\mathcal{T}_{t}\), we initialize \(\mathbf{\theta}=\mathbf{\theta}_{t-1}^{*}\), whose value already minimized the lost \(\tilde{\mathcal{L}}_{t-1}\). From this, if we can consider eqn. 1 as minimizing the loss \(\mathcal{L}_{t}(\mathbf{\theta})\) with the constraints that previous losses are not allowed to increase after every mini-batch update, which is the GEM objective. In particular, GEM optimize the following problem \[\begin{array}{ll}\text{Minimize}_{\mathbf{\theta},\mathcal{D}_{t}}&\mathcal{L}_ {t}(\mathbf{\theta})\\ \text{subject to}&\mathcal{L}_{k}(\mathbf{\theta},\mathcal{M}_{k})\leq\mathcal{L}_ {k}(\mathbf{\theta}_{t-1}^{*},\mathcal{M}_{k}),\forall k<t,\end{array} \tag{5}\]where the loss \(\mathcal{L}_{k}\) is evaluated at the memory \(\mathcal{M}_{k}\) that stores some of the training samples from task \(\mathcal{T}_{k}\). GEM further converts the constraints in eqn. 5 into the gradient constraints of the losses and solve them using quadratic programming. Finally, the type of losses \(\mathcal{L}_{k<t}\) can be flexibly changed to other loss rather than the negative log-likelihood. For example, Learning without Forgetting (LwF) (Li & Hoiem, 2017) proposes a form of distillation loss by minimizing the KL-divergence of past models and current model on the newly observed tasks. By changing the losses \(\mathcal{L}_{k<t}\), we can recover different solutions for continual learning in the literature.

To tackle the aforementioned problem, we propose to minimize the KL-divergence between the current model and the previous optimal models on the corresponding tasks. This can also be interpreted as distilling the knowledge from \(\mathbf{\theta}_{i}^{*}\) to \(\mathbf{\theta}\) on the corresponding data \(\mathcal{D}_{i}\).

Our goal is to ensure that the new model \(\mathbf{\theta}\) behaves similarly with \(\mathbf{\theta}_{i}^{*}\) on the previous task \(\mathcal{T}_{t}\), thus we call this constraint as activity regularization.

We refer to this algorithm as "Deterministic Activity Regularization" (DAR) which imposes regularization constraints for all the previous tasks at every learning iteration.

Similar to approximated multi-task learning methods, DAR employs a regularization constraint on every of the past tasks when learning a new one, which requires calculating the KL-divergence resulting in an additional forward pass on the memory, and thus results in additional computational cost when training the models.

3: Observe the data set \(\mathcal{D}_{t}\) 4:\(\mathcal{M}_{t}\leftarrow\) add random samples from \(\mathcal{D}_{t}\) to the memory. 5:for k = 1,...,\(n_{iter}\)do 6: Sample \(\mathcal{D}=\{\mathbf{x}^{(n)},y^{(n)}\}_{n=1}^{N}\) a batch from training data \(\mathcal{D}_{t}\) 7:\(g_{\mathbf{\theta}}\leftarrow\nabla_{\mathbf{\theta}}\left[\sum_{\mathbf{x},y\in\mathcal{D }}\mathcal{L}_{t}(\mathbf{\theta},(\mathbf{x},y))+\sum_{i<t}\sum_{\mathbf{x}\in\mathcal{M} _{i}}\lambda_{i}KL\bigg{(}\frac{1}{\tau}f_{\mathbf{\theta}_{i}^{*}}(\mathbf{x})\ \bigg{\|}\ \frac{1}{\tau}f_{\mathbf{\theta}}(\mathbf{x})\bigg{)}\right]\) 8:\(\mathbf{\theta}\leftarrow\mathbf{\theta}-\alpha\cdot\text{SGD}(\mathbf{\theta},g_{\mathbf{ \theta}})\) 9:return\(\mathbf{\theta}\) ```**Algorithm 1** Deterministic Activity Regularization (DAR) algorithm for Continual Learning. To address the weaknesses of DAR, we propose the Stochastic Activity Regularization (SAR), which randomly samples a past task and apply activity regularization on each gradient update when learning a new task. Algorithm 1 summarizes the proposed SAR algorithm for continual learning.

Then, when learning a new task, for each mini batch to update \(\mathbf{\theta}\), we apply AR on all of the previous memory.

The memory \(\mathcal{M}_{i}\) is used to store some training samples of the previous tasks which is used to regularize the current model when learning new tasks. Using a memory for continual learning has been explored by previous works Nguyen et al. (2018) and Lopez-Paz et al. (2017). Although more sophisticated methods such as the greedy \(K\)-center algorithm can be used to select data points that spread through out the input space (Nguyen et al., 2018), in this work, even random sampling works well with our methods and can achieve state-of-the-art results.

7: Sample an index i from \([0,\dots,t-1]\) 8:\(g_{\mathbf{\theta}}\leftarrow\nabla_{\mathbf{\theta}}\left[\sum_{\mathbf{x},y\in\mathcal{ D}}\mathcal{L}_{t}(\mathbf{\theta},(\mathbf{x},y))+\sum_{\mathbf{x}\in\mathcal{M}_{i}} \lambda_{i}KL\bigg{(}\frac{1}{\tau}f_{\mathbf{\theta}_{i}^{*}}(\mathbf{x})\ \bigg{\|}\ \frac{1}{\tau}f_{\mathbf{\theta}}(\mathbf{x})\bigg{)}\right]\) 9:\(\mathbf{\theta}\leftarrow\mathbf{\theta}-\alpha\cdot\text{SGD}(\mathbf{\theta},g_{\mathbf{ \theta}})\)

Prior works can be broadly categorized into 4 main approaches: (1) structural regularization, (2) functional regularization, (3) Bayesian learning and (4) ensemble learning. **Structural regularization** approaches employ penalties on the network's parameters, encourage the important parameters of previous tasks to not change when learning the new task. The parameter importance can be estimated by calculating the Fisher information as in EWC (Kirkpatrick et al., 2017) or directly measure how changes in each parameter will affect the change in the loss function as proposed in SI (Zenke et al., 2017).

**Functional regularization** methods employ the regularizer on the model's output rather than its parameters. Different from structural regularization, functional regularization aims to preserve the learned input-output mapping function of the network. Learning without Forgetting (LwF) (Li & Hoiem, 2017) proposes to minimize the difference in KL-divergence of the current and previous models on the new tasks, which is a form of knowledge distillation penalty. Similarly, Less-forgetting learning (LF) (Jung et al., 2019) proposes to minimize the \(\ell_{2}\) norm of the two prediction in stead of KL-divergence as LwF. Motivated from Hebbian learning (Hebbs, 1949), MAS (Aljundi et al., 2018a) proposes to penalize the \(\ell_{2}\) norm of the model's prediction. Selfless sequential learning (SNI) (Aljundi et al., 2018b) improves MAS by combining with it sparsity in each layer's activation.

While there are some overlapping between Bayesian learning and our approximated multi-task learning framework, the main difference is that our objective allows us to actively control the contribution of previous tasks by setting the importance value \(\lambda_{i}\) for each loss \(\mathcal{L}_{i}\).

In this work, we propose an approximated multi-task learning framework to unify both structural and functional regularization, in which our proposed activity regularization is a special case.

We propose an approximated multi-task learning framework that can alleviate catastrophic forgetting, an important challenge in continual learning and show that several previous solutions in the literature are special cases of this framework.

To overcome this problem, we propose the deterministic activity regularization for continual learning, DAR, that enforces the new model's behaviour to be consistent with the optimal models on each of the previous tasks.

We further improve DAR with a stochastic sampling regularizer (SAR), which balances the model's stability and plasticity, allowing the model to acquire new knowledge, alleviate forgetting and has greater scalability. Through extensive experiments on popular continual learning benchmarks, we show that DAR and SAR consistently outperforms state-of-the-art methods.

Input:
You should:
1. Carefully review the claim and the text and image evidence. Summarize how the evidence influenced your evaluation of the claim's validity and strength and briefly explain whether the claim is wellstated or overstated on a scale of 1 to 5 (up to 100 words).
2. Provide the final score (choose from 1 to 5), in the format: <score>{score}</score>.

Output:
Based on my review of the claim, evidence, and review comments, here is my assessment:

Summary:
The claim about proposing "an approximated multi-task learning framework that unifies a family of popular regularization based continual learning methods" appears to be somewhat overstated. While the evidence shows mathematical formulation connecting different approaches, the reviewer points out that the novelty seems limited and it appears to be primarily an extension of Learning without Forgetting (LwF). The evidence does demonstrate mathematical connections between different methods through Equation 3, but doesn't fully establish a comprehensive unifying framework as claimed.

<score>3</score>

Justification:
1. The evidence shows mathematical relationships between different continual learning approaches (through Equations 1-3), but this appears more as a reformulation rather than a novel unifying framework.

2. The review comment highlights that "it seems to be an extension of Learning without Forgetting (LwF)" with limited novelty, suggesting the claim of proposing a new unifying framework may be overstated.

3. While the evidence demonstrates connections between methods like EWC, SI, and GEM through mathematical formulation, the framework's unifying properties are not comprehensively proven or validated.

4. The paper acknowledges similarities with existing approaches, stating "While there are some overlapping between Bayesian learning and our approximated multi-task learning framework," which suggests the unified framework claim may be too strong.

5. The experimental validation appears incomplete according to the reviewer's comments about fixed hyperparameters and lack of variance reporting, which weakens support for the broad unification claim.


Input:
Your role is to assess whether a claim is well-stated or overstated based on the available evidence.

The Input Information will include:
1. Original Claim: The claim under evaluation.
2. Evidence: Research findings, including figures, tables, or other relevant data supporting the claim.

Optional. review comment: Reviewer feedback relevant to the claim's validity.

The claim should be assesed with the following:
Score: Evaluate the claim on a scale of 1 to 5, where:
- 1: Well-stated (accurately reflects the evidence without exaggeration)
- 5: Overstated (exaggerates findings' significance, magnitude, or certainty)

Justification: Provide a detailed explanation that includes:
- Instances of exaggerated wording, insufficient experiments, lack of experimental details, gaps in knowledge, weak grounding in evidence, or lack of acknowledgment of limitations.
- Direct citations from relevant evidence that substantiate your reasoning.
- If review comment is included only take into account the relevant comments towards the claim.

The claim to be assessed is:

We further develop a stochastic version of AR that randomly samples a task and apply AR in each mini-batch update.

The evidence to be evaluated is:In particular, for each task \(\mathcal{T}_{i}\), we store some of the samples \(\mathbf{x}_{i}\) and the prediction of the optimal model \(f_{\mathbf{\theta}_{i}^{*}}(\mathbf{x}_{i})\) in a memory \(\mathcal{M}_{i}\); then the Activity Regularization (AR) objective for task \(\mathcal{T}_{t}\) is \[\tilde{\mathcal{L}}_{t}(\mathbf{\theta})=\mathcal{L}_{t}(\mathbf{\theta})+\sum_{i<t} \sum_{\mathbf{x}\in\mathcal{M}_{i}}\lambda_{i}KL\bigg{(}\frac{1}{\tau}f_{\mathbf{ \theta}_{i}^{*}}(\mathbf{x})\bigg{\|}\ \frac{1}{\tau}f_{\mathbf{\theta}}(\mathbf{x})\bigg{)}, \tag{7}\]

To address the weaknesses of DAR, we propose the Stochastic Activity Regularization (SAR), which randomly samples a past task and apply activity regularization on each gradient update when learning a new task.

Then, when learning a new task, for each mini batch to update \(\mathbf{\theta}\), we apply AR on all of the previous memory. For SAR, we instead randomly sample an index i of the previous tasks and apply the AR only on that memory.

Although more sophisticated methods such as the greedy \(K\)-center algorithm can be used to select data points that spread through out the input space (Nguyen et al., 2018), in this work, even random sampling works well with our methods and can achieve state-of-the-art results.

6: Sample \(\mathcal{D}=\{\mathbf{x}^{(n)},y^{(n)}\}_{n=1}^{N}\) a batch from training data \(\mathcal{D}_{t}\) 7: Sample an index i from \([0,\dots,t-1]\)

We further improve DAR with a stochastic sampling regularizer (SAR), which balances the model's stability and plasticity, allowing the model to acquire new knowledge, alleviate forgetting and has greater scalability. Through extensive experiments on popular continual learning benchmarks, we show that DAR and SAR consistently outperforms state-of-the-art methods.

Input: Figure 1: Comparison of different memory size on the permuted MNIST experiments.

Input:
You should:
1. Carefully review the claim and the text and image evidence. Summarize how the evidence influenced your evaluation of the claim's validity and strength and briefly explain whether the claim is wellstated or overstated on a scale of 1 to 5 (up to 100 words).
2. Provide the final score (choose from 1 to 5), in the format: <score>{score}</score>.

Output:
I'll evaluate whether the claim about developing a stochastic version of AR is well-stated or overstated.

Summary: The evidence strongly supports the claim. The text explicitly describes the Stochastic Activity Regularization (SAR) method, including detailed mathematical formulation and implementation details (random sampling of previous tasks during mini-batch updates). The experimental results in Figure 1 demonstrate SAR's effectiveness across different memory sizes, consistently outperforming both DAR and GEM baselines. The claim is modest and accurately reflects what is shown in the evidence without exaggeration.

<score>1</score>

Justification:
1. The implementation details are clearly documented in the evidence, showing exactly how tasks are randomly sampled (lines 6-7 in the algorithm)
2. The method's mechanics are thoroughly explained mathematically through equation (7) and supporting text
3. Figure 1 provides empirical validation showing SAR's performance
4. The claim uses appropriate, measured language ("develop a stochastic version") without overstating capabilities
5. The description aligns precisely with what was implemented and tested

The claim is well-stated (score of 1) as it directly reflects the evidence without any exaggeration or overreach.


Input:
Your role is to assess whether a claim is well-stated or overstated based on the available evidence.

The Input Information will include:
1. Original Claim: The claim under evaluation.
2. Evidence: Research findings, including figures, tables, or other relevant data supporting the claim.

Optional. review comment: Reviewer feedback relevant to the claim's validity.

The claim should be assesed with the following:
Score: Evaluate the claim on a scale of 1 to 5, where:
- 1: Well-stated (accurately reflects the evidence without exaggeration)
- 5: Overstated (exaggerates findings' significance, magnitude, or certainty)

Justification: Provide a detailed explanation that includes:
- Instances of exaggerated wording, insufficient experiments, lack of experimental details, gaps in knowledge, weak grounding in evidence, or lack of acknowledgment of limitations.
- Direct citations from relevant evidence that substantiate your reasoning.
- If review comment is included only take into account the relevant comments towards the claim.

The claim to be assessed is:

We further develop a stochastic version of AR that randomly samples a task and apply AR in each mini-batch update.

The review comment to be evaluated is:

This paper proposes an approach to mitigate catastrophic forgetting in supervised learning by regularizing activations. The paper views previous techniques (EWC, SI, and GEM) under a multi-task learning lens, and then proposes an additional loss term to minimise the KL between activations from previous and current models, on previous tasks - this is based on a memory which stores some previous samples and their corresponding activations.

I think it is a simple and intuitive approach and a well-written paper. Unfortunately I have a number of concerns that I think preclude publication in the current state.

First, in terms of related work, I believe this is very similar to Learning without forgetting (LwF), with the difference that the KL-divergence is computed on samples kept from the previous tasks. This is briefly mentioned in the paper, but I think it needs to be made more explicit, and LwF should be a baseline in the experiments to clearly indicate the benefit of keeping this data. There is also a relationship to EWC: given the connection between the Fisher information and KL, it can be viewed as minimising the KL divergence in parameter space, rather than in activation space (which is the case here). Also note that EWC uses the true Fisher rather than the empirical, contrary to the derivation in equation (2).
There are also a number of papers that haven’t been cited in the related work [1][2][3][4].

Second, I think the motivation in Section 3.1 could be more convincing. Most importantly, it’s not clear to me that the decision boundary *shouldn’t* change for previously misclassified examples, as this could be an opportunity for backwards transfer.
Further, I don’t think the point in the last paragraph about having a small data portion is relevant, since they are from the same data distribution, and we would expect misclassified samples to be in the same (low) frequency in Fisher estimation as overall. I think the point of this paragraph is just that it is important to consider the entire predictive distribution of previous tasks rather than the probability of the correct class, so this should be stated more clearly and then justified.

Finally, I think the experimental justification could be improved as well. Beyond permuted MNIST (which it has been argued is not as useful as other baselines [4]), only the final performance on split notMNIST / CIFAR-100 is reported. Some comments and questions:
- The accuracies of EWC (and possibly SI) in the table are worse than reported in previous work (eg. [1]), so I think this needs to be examined.
- What is the fine-tuning baseline (I don't believe it is actually clearly defined)? How can it be so low in figure 2a but better in 2b?
- I think plots over time (performance on all tasks) would be much more useful than the final performance in Table 2 and Fig 2.
- Errors and error bars would be beneficial for all results.
- Table 1 should also include the references provided.

Some other comments and questions:
- Compared to eqn (2), eqn (6) is missing the ½ constant.
- Typos in section 5.3: "SI performs better than SI", and VAR instead of SAR.
- Section 2, unclear of meaning of "coined with the likelihood" (should this be “coincide”?)
- The first line should be “Humans have the ability to learn...” In general, I think the introduction could use another proofread for grammar and readability as I saw a few minor things.

[1] Nguyen, Cuong V., et al. "Variational Continual Learning." ICLR, 2018.
[2] Schwarz, Jonathan, et al. "Progress & Compress: A scalable framework for continual learning." ICML, 2018.
[3] Shin, Hanul, et al. "Continual learning with deep generative replay." NIPS, 2017.
[4] Farquhar, Sebastian, and Yarin Gal. "Towards Robust Evaluations of Continual Learning." arXiv, 2018.

The evidence to be evaluated is:In particular, for each task \(\mathcal{T}_{i}\), we store some of the samples \(\mathbf{x}_{i}\) and the prediction of the optimal model \(f_{\mathbf{\theta}_{i}^{*}}(\mathbf{x}_{i})\) in a memory \(\mathcal{M}_{i}\); then the Activity Regularization (AR) objective for task \(\mathcal{T}_{t}\) is \[\tilde{\mathcal{L}}_{t}(\mathbf{\theta})=\mathcal{L}_{t}(\mathbf{\theta})+\sum_{i<t} \sum_{\mathbf{x}\in\mathcal{M}_{i}}\lambda_{i}KL\bigg{(}\frac{1}{\tau}f_{\mathbf{ \theta}_{i}^{*}}(\mathbf{x})\bigg{\|}\ \frac{1}{\tau}f_{\mathbf{\theta}}(\mathbf{x})\bigg{)}, \tag{7}\]

To address the weaknesses of DAR, we propose the Stochastic Activity Regularization (SAR), which randomly samples a past task and apply activity regularization on each gradient update when learning a new task.

Then, when learning a new task, for each mini batch to update \(\mathbf{\theta}\), we apply AR on all of the previous memory. For SAR, we instead randomly sample an index i of the previous tasks and apply the AR only on that memory.

Although more sophisticated methods such as the greedy \(K\)-center algorithm can be used to select data points that spread through out the input space (Nguyen et al., 2018), in this work, even random sampling works well with our methods and can achieve state-of-the-art results.

6: Sample \(\mathcal{D}=\{\mathbf{x}^{(n)},y^{(n)}\}_{n=1}^{N}\) a batch from training data \(\mathcal{D}_{t}\) 7: Sample an index i from \([0,\dots,t-1]\)

We further improve DAR with a stochastic sampling regularizer (SAR), which balances the model's stability and plasticity, allowing the model to acquire new knowledge, alleviate forgetting and has greater scalability. Through extensive experiments on popular continual learning benchmarks, we show that DAR and SAR consistently outperforms state-of-the-art methods.

Input: Figure 1: Comparison of different memory size on the permuted MNIST experiments.

Input:
You should:
1. Carefully review the claim and the text and image evidence. Summarize how the evidence influenced your evaluation of the claim's validity and strength and briefly explain whether the claim is wellstated or overstated on a scale of 1 to 5 (up to 100 words).
2. Provide the final score (choose from 1 to 5), in the format: <score>{score}</score>.

Output:
After reviewing the claim and evidence, I find that the claim is well-stated. The evidence clearly shows that the authors developed a stochastic version (SAR) by randomly sampling tasks during training, as demonstrated in both the mathematical formulation and algorithmic steps (specifically step 7: "Sample an index i from [0,…,t-1]"). Figure 1 also supports this by showing SAR's implementation with different memory sizes. The review comment doesn't dispute this specific claim. The description matches the actual implementation without exaggeration.

<score>1</score>

Justification:
1. Direct evidence from algorithm shows random task sampling (Step 7)
2. Equation (7) and surrounding text demonstrates the mathematical foundation
3. Implementation results shown in Figure 1 validate the existence and application of this stochastic version
4. The claim is precise and technically accurate without overstatement
5. The statement is purely descriptive of the methodology without making grand claims about performance or superiority


Input:
Your role is to assess whether a claim is well-stated or overstated based on the available evidence.

The Input Information will include:
1. Original Claim: The claim under evaluation.
2. Evidence: Research findings, including figures, tables, or other relevant data supporting the claim.

Optional. review comment: Reviewer feedback relevant to the claim's validity.

The claim should be assesed with the following:
Score: Evaluate the claim on a scale of 1 to 5, where:
- 1: Well-stated (accurately reflects the evidence without exaggeration)
- 5: Overstated (exaggerates findings' significance, magnitude, or certainty)

Justification: Provide a detailed explanation that includes:
- Instances of exaggerated wording, insufficient experiments, lack of experimental details, gaps in knowledge, weak grounding in evidence, or lack of acknowledgment of limitations.
- Direct citations from relevant evidence that substantiate your reasoning.
- If review comment is included only take into account the relevant comments towards the claim.

The claim to be assessed is:

We further develop a stochastic version of AR that randomly samples a task and apply AR in each mini-batch update.

The review comment to be evaluated is:

The paper addresses the problem of continual learning from a sequence of supervised tasks. The main contribution of the paper are the following. The work:
* puts the problem in uniform general framework in which many of the state-of-the-art methods fit
* identifies some drawbacks of some the state of the art methods (namely EWC- Kirkpatrick et al., 2017 and GEM-Lopez-Paz et al., 2017) .
* proposes two versions of an approach to address these drawbacks
The main identified problem in EWC and GEM is that both methods result in a zero penalty when both the previous and current models misclassify a sample, even if they make different mistakes.
To solve this problem, the authors propose to keep a memory of randomly sampled data from the previous task, and distil the knowledge from the optima of the previous tasks to the current one using a KL penalty. The first version considers computing this penalty from the whole sequence of tasks, while the second randomly selects one task at each iteration and uses it to compute the penalty, paying some overall accuracy for plasticity.

While the paper is clear and well written, I have some concerns about it's quality, originality and significance.

Originality: The work seems to me to be very related to LwF. The main difference is that while LwF uses only the new data for the KL penalty, this paper keeps a memory of previously seen data to compute this loss.

Significance: While the way the authors approached the problem seems well structured and motivated, and is based on a sound observation, the authors limited themselves to experiments where the task data have similar structure. I am not sure how significant the improvement of the method would be in the more challenging and realistic setting where the data comes from different domains.  I am more specifically skeptical about the stochastic version of the algorithm in that case.

Quality:
* The proposed algorithm stores some data from previous tasks along with the outputs of the corresponding optimal model. While knowledge distillation as proposed would result in a non zero penalty when the new model makes different prediction, it still doesn't take advantage of the new information to probably correct the previous models prediction when it is wrong. Why not keeping the ground truth labels instead? This won't increase the memory requirement, and may give better results. It would be interesting to compare against such a method.
* I think selecting the samples to keep in memory randomly could to be suboptimal.  Other selection methods can be considered. A previous work:  iCaRL: Incremental Classifier and Representation Learning, Rebuffi et al. 2017, gives way to select representative samples. It would be interesting to see the effect of such a selection on the results.

Overall, while the paper doesn't present any significant flaw, it doesn't add much to the continual learning literature either, which explains my rating.

The evidence to be evaluated is:In particular, for each task \(\mathcal{T}_{i}\), we store some of the samples \(\mathbf{x}_{i}\) and the prediction of the optimal model \(f_{\mathbf{\theta}_{i}^{*}}(\mathbf{x}_{i})\) in a memory \(\mathcal{M}_{i}\); then the Activity Regularization (AR) objective for task \(\mathcal{T}_{t}\) is \[\tilde{\mathcal{L}}_{t}(\mathbf{\theta})=\mathcal{L}_{t}(\mathbf{\theta})+\sum_{i<t} \sum_{\mathbf{x}\in\mathcal{M}_{i}}\lambda_{i}KL\bigg{(}\frac{1}{\tau}f_{\mathbf{ \theta}_{i}^{*}}(\mathbf{x})\bigg{\|}\ \frac{1}{\tau}f_{\mathbf{\theta}}(\mathbf{x})\bigg{)}, \tag{7}\]

To address the weaknesses of DAR, we propose the Stochastic Activity Regularization (SAR), which randomly samples a past task and apply activity regularization on each gradient update when learning a new task.

Then, when learning a new task, for each mini batch to update \(\mathbf{\theta}\), we apply AR on all of the previous memory. For SAR, we instead randomly sample an index i of the previous tasks and apply the AR only on that memory.

Although more sophisticated methods such as the greedy \(K\)-center algorithm can be used to select data points that spread through out the input space (Nguyen et al., 2018), in this work, even random sampling works well with our methods and can achieve state-of-the-art results.

6: Sample \(\mathcal{D}=\{\mathbf{x}^{(n)},y^{(n)}\}_{n=1}^{N}\) a batch from training data \(\mathcal{D}_{t}\) 7: Sample an index i from \([0,\dots,t-1]\)

We further improve DAR with a stochastic sampling regularizer (SAR), which balances the model's stability and plasticity, allowing the model to acquire new knowledge, alleviate forgetting and has greater scalability. Through extensive experiments on popular continual learning benchmarks, we show that DAR and SAR consistently outperforms state-of-the-art methods.

Input: Figure 1: Comparison of different memory size on the permuted MNIST experiments.

Input:
You should:
1. Carefully review the claim and the text and image evidence. Summarize how the evidence influenced your evaluation of the claim's validity and strength and briefly explain whether the claim is wellstated or overstated on a scale of 1 to 5 (up to 100 words).
2. Provide the final score (choose from 1 to 5), in the format: <score>{score}</score>.

Output:
After reviewing the claim and evidence, I find that the claim about developing a stochastic version of AR is well-stated and accurately reflects what is shown in the evidence. The evidence clearly demonstrates:

1. The mathematical formulation showing how SAR randomly samples a past task
2. The algorithmic implementation (steps 6-7) showing the random sampling process
3. Experimental results in Figure 1 comparing DAR and SAR across different memory sizes
4. Clear explanation of how SAR differs from DAR by randomly sampling one previous task per update

While the reviewer raises concerns about the overall methodology, these concerns don't specifically challenge the claim about developing the stochastic version itself.

<score>1</score>