Paper: Prediction Error-based Classification for Class-Incremental Learning (ICLR-2024)

Reviewer comment (Experiments):
**SOTA Architecture Comparison**: The authors have designed the PEC predictor models to align with the parameter count of reference literature networks. Specifically for the evaluation of SOTA methods, the PEC authors employ simpler network architectures for SVHN and MNIST, while they employ the standard Resnet18 for Cifar10, Split-Cifar100, and the miniImagenet dataset. These latter tasks are considered more challenging in comparison to SVHN and MNIST. My primary concern revolves around the use of the standard Resnet18 that the authors employ for evaluating SOTA methods. Training the standard Resnet18 can be difficult, especially when the number of training iterations is limited (i.e. in an online setting), primarily due to the large number of parameters (11.2M as reported in table 6 of the Appendix B.1 of the paper). In response to this challenge, recent SOTA online approaches rely upon a **reduced** version of Resnet18, which consists of approximately 1.1M parameters [g] [h] [i] [l] [m] [n]. This reduced network is commonly favored for online continual learning due to its computational efficiency and ease of optimization. Utilizing the reduced Resnet18 as a reference is essential to ensure a fair comparison with the existing literature, especially since PEC uses shallow models that are easier to train than Resnet18. It's worth noting that doing so would require the overall number of parameters in the PEC models to be reduced to 1.1 million to match the reference architecture.

Author response:
**(1) SOTA Architecture Comparison**

We thank the Reviewer for making this important point. We are happy to say that we ran additional experiments with the architecture requested by the Reviewer. Before we report on the experiments, let us describe and clarify the variants of ResNet that we are discussing. The 1.1M model mentioned by the Reviewer (let’s call it Narrow ResNet18) has two differences with respect to the original ResNet18: 
1) the initial downscaling (stride 2 in the initial convolution and max pooling) is removed, 
2) the width factor (affecting the number of filters in the convolutional layers across the network) is changed from 64 to 20.

We used in our experiments the variant of ResNet18 from [1] (let’s call it DER-ResNet18), which incorporates the change 1) of Narrow ResNet18 but keeps the width of the original ResNet18. We apologize for not properly describing this in the initial submission; we have now revised the text to include this information, see Section 4.1 (we mark all revised fragments with orange). Importantly, the only difference between Narrow ResNet18 and DER-ResNet18 is width. The depth (which typically makes the optimization harder) does not change between the models. This suggests that the performance with DER-ResNet18 is expected to be better than with Narrow ResNet18, even in the online case. We verify this empirically below.

In our additional experiment, reported in Appendix C.3, we rerun selected baselines with the Narrow ResNet18 architecture, as well as PEC with the adjusted 1.1M parameter budget. Due to limited time, we were not yet able to run all the baselines; however, we already included all rehearsal-free baselines (as they are most directly comparable with PEC) as well as DER++ and X-DER, two of the rehearsal-based methods that performed best in our original experiments. We will include all the baselines for the camera-ready version. Results in Tables 18 & 19 confirm that PEC retains its competitive performance with the smaller parameter budget. It still outperforms other rehearsal-free methods in every case, with wide margins. PEC is also able to outperform challenging rehearsal-based methods in 4 out of 8 cases. Furthermore, we can see that typically, the performance of a given method deteriorates when Narrow ResNet18 is used instead of DER-ResNet18. This provides an important sanity check for our original choice of architecture. We believe that these results fully address the issue raised by the Reviewer.

Finally, we note that in our original submission, in Figure 2 we already partly addressed the issue, as we reported the performance of PEC vs several baselines when changing the parameter counts. For CIFAR-10 (bottom right figure), the left end of the $x$-axis roughly corresponds to the Narrow ResNet18.

Reviewer comment (Experiments):
**Discriminative Exemplar-Free Class Incremental (EFCIL) Comparison**: Comparing PEC, an online exemplar-free method, with other common exemplar-free approaches is crucial. In the main paper, only EWC [a] and Label Trick [b] are evaluated, with EWC designed for offline continual learning and not suitable for online settings. Other common weight regularization methods, such as MAS [c] and SI [d], are typically evaluated in the online continual learning literature, since they allow computing weight importance online. Moreover, LwF [e], which regularizes output with knowledge distillation, is known to outperform weight regularization methods and is suitable for online settings. The absence of this comparison is noteworthy. The only other baseline provided by the authors in the main paper is "Label Trick" [b], which involves training only the head of the current task. Nowadays, this is a common practice in exemplar-free approaches, as it has been shown to work better. All the exemplar-free implemented methods (including LwF) on FACIL (Masana et al. [f]) use this practice. All the exemplar-free methods reported in the main paper must apply this trick. Moreover, in Appendix C.1.1, various EFCIL methods are presented, and it might be beneficial to move some of them to the main paper, implementing the Label Trick where applicable for a more comprehensive evaluation.

Author response:
**(2) Discriminative Exemplar-Free Class Incremental (EFCIL) Comparison**

We are happy to have been able to include additional baselines requested by the Reviewer, namely SI and LwF. If the Reviewer believes it is important, we will add MAS as well in the camera-ready version; the reason for not including it now is that this method was not readily available in our software framework. However, given the results of other regularization-based methods, it seems unlikely that MAS could challenge the performance of PEC. Additionally, we follow the Reviewer’s advice to add Labels Trick on top of all non-rehearsal methods. We now report versions with Labels Trick in our main text Tables 1 & 2, and report vanilla performances in Appendix C.2. After adding the requested baselines, PEC still outperforms rehearsal-free approaches in all cases, with an especially large lead in the one-class-per-task case.

Reviewer comment (Experiments):
**Discriminative Exemplar-Based Class Incremental (EBCIL) Comparison**: PEC is an **Online** method. When comparing it with exemplar-based methods, it is essential to align the compared methods with the current **Online** EBCIL literature. ER, A-GEM, DER++, and ICARL, already reported in the main paper, are common comparisons in the online literature. On the other hand, BiC is an **offline** method because it requires additional epochs on a validation set to calibrate the classifier (thus violating online incremental constraints). X-DER, a recent work, was originally designed in the multi-epoch setting (i.e. offline) but has been adapted in this paper to work in an online setting. To ensure a comprehensive comparison with the current online literature, it's important to consider additional relevant approaches. A straightforward baseline to provide is "ER + LwF". Notably, ER-ACE [g] has recently delivered state-of-the-art results in online continual learning, so it is advisable to include this comparison in the main paper. Other approaches that should be evaluated include SCR [l] and RAR [m]. For a more extensive list of relevant comparisons, refer to Cormerais et al. [n].

Author response:
**(3) Discriminative Exemplar-Based Class Incremental (EBCIL) Comparison**

We note that PEC is an exemplar-free method and we demonstrate that it outperforms exemplar-free methods in multiple cases (all that we consider). It is not our claim that PEC always outperforms exemplar-based methods, and in fact, in the paper, we show some cases where it does not. However, we experimentally show that PEC is still competitive against rehearsal-based methods using small replay buffers, and include strong recent baselines such as DER++ or X-DER. We think this is a major additional merit of our paper. However, we don’t see why this would require us to include an even vaster array of exemplar-based baselines (as PEC is exemplar-free!). 

That being said, we are happy to say that we included one additional baseline requested by the Reviewer, namely ER-ACE (results are included in Tables 1 & 2). This baseline does not change the conclusions of the paper, as PEC outperforms it in 8 out of 10 cases (all but Balanced SVHN experiments).

Reviewer comment (Evaluation):
**Memory Requirements & Inference FLOPS Computation**: Regarding the **Memory Requirements**, PEC requires a linear increase of parameters with the number of classes encountered. While the authors acknowledge this intrinsic challenge in their approach and have discussed it in the limitations section, a more detailed analysis is necessary. Specifically, it's essential to examine how many parameters are utilized by each predictor network per dataset and by the teacher network to understand how much the memory increases across the tasks. As for the **FLOPS comparison**, the authors have not provided an analysis of the number of Floating Point Operations (FLOPS) required by their model. Since online methods are intended to work in environments with hardware resource limitations, it is advisable to disclose the number of FLOPS for both backward and forward passes. For training, FLOPS evaluation can be carried out, as demonstrated by [o]. As regards the inference step, in Section 3.1 the authors say that only two forward passes are needed to obtain the classification scores – one for a merged student, and one for the teacher network. While this is true, considering the online nature of their method and its intended use in resource-constrained settings, it is crucial to provide and compare the overall number of FLOPS required by the forward pass for both teacher and student networks. Therefore, it is necessary to determine how much the FLOPS of PEC method increases compared to current state-of-the-art methods.

Author response:
**(4) Memory Requirements & Inference FLOPS Computation**

Regarding memory requirements, parameter counts for all methods are disclosed in Appendix B.2, paragraph “Parameter counts”.

Regarding FLOPS, in the revised version we added a report on the number of multiply-accumulate operations (MACs) used for a single inference forward pass, for PEC vs discriminative approaches, see Appendix B.2. For convenience, we include the results here as well:

| Method         | MNIST | SVHN | CIFAR-10 | CIFAR-100 | miniImageNet |
|----------------|-------|------|----------|-----------|--------------|
| PEC            | 4.5M  | 352M | 352M     | 300M      | 1.9G         |
| discriminative | 90K   | 556M | 556M     | 556M      | 3.9G         |

We see that except for the MNIST case, PEC uses fewer operations than the discriminative baseline. Intuitively, this is because PEC does a substantial part of its computation in a fully connected layer which has a much smaller compute-to-parameters density compared to convolutional layers which mostly constitute the ResNet architecture.
Unfortunately, we could not report numbers on backpropagation as we were not able to quickly find appropriate software for this. However, we note that in this case, the advantage of PEC should be even higher, as the backpropagation is not performed for the teacher, which constitutes a large part of our overall computation graph.

Reviewer comment (Theory):
**Buffer Balancing and Equal Budget**: The authors assert that their approach is afflicted by class imbalancing. However, they do not provide any theoretical motivation for this claim; they hypothesize that the issue may be linked to the number of gradient updates, as mentioned in Section 4.4 of the main paper. In cases where no theoretical motivation is available, it is necessary to conduct a detailed analysis that elucidates the factors contributing to this behavior. To address the class imbalancing issue, the authors introduce two strategies in the experimental section: Buffer Balancing and Equal Budget Strategies. While the easier Equal Budget Strategy is well-explained in the paper (and in supplementary Section B.2), the **Buffer balancing**, based on CMA-ES optimization, is considerably more complex and difficult to grasp. Therefore, additional details on how this optimization algorithm is used for online-class incremental learning should be provided to enhance understanding. Finally, a natural question arises: How well do existing methods perform in the class imbalancing setting compared to PEC?

Author response:
**(5) Buffer Balancing and Equal Budget**

As for the first part of the issue, we show for both SVHN and CIFAR-10 that (1) performance for the unbalanced version is bad, (2) performance for the balanced version is good, (3) if we remove the “imbalance” (e.g. by using our Equal budgets strategy), then the method recovers the good performance. Together, this constitutes evidence that class imbalance can cause the vanilla version of PEC to perform badly. We believe the fact that we clearly describe limitations like this one contributes to the scientific standard of our work; we think that a deeper analysis, although interesting, would be out of scope.

As for explaining balancing strategies, we tried our best to do it in the original version of the paper; in the revised version, we include slight changes in Section 4.4. Regarding Buffer balancing, if the predictions need to be performed during the online training, the balancing procedure could be done either before each prediction or from time to time, to trade off the quality of computed balancing scalars and the computational complexity. We are happy to answer further questions if there are any.

Reviewer comment (Experiments):
The "efficiency" claimed in the contribution is less empirically supported. Furthermore, due to the individual design for student networks, is this a good plus when the number of classes increases?

Author response:
**The “efficiency” claimed in the contribution is less empirically supported.**: Firstly, typically PEC can achieve a given performance level with a lower number of epochs or parameters than other methods (e.g. most closely related VAE-GC), as shown in Figure 2 in our original submission. Secondly, to emphasize efficiency, we perform our main experiments in a challenging single-pass-through-data setting. Improvements of PEC over baselines are especially pronounced in this efficiency-focused setting and are typically larger than in multi-epoch settings, see Table 15 in the revised Appendix (Table 14 in the original submission). Finally, in the revised manuscript, Appendix B.2, paragraph “Number of operations needed”, we report the number of operations needed for inference in the case of PEC vs discriminative methods and show that here PEC is more efficient computation-wise in 4 of 5 considered cases.

Reviewer comment (Writing):
The notations are bad. In algorithms, the number of classes is $N$ but later it is denoted by $C$ in equation (1). In proposition (2), I believe $N$ denotes the sample size but it was denoted as the number of classes in Algorithm 1.

Author response:
**Inconsistency in notation.**: We thank the Reviewer for pointing this out. In the revised manuscript we have made the notation consistent: we now consistently use $C$ to denote the number of classes and $N$ to denote the number of samples. We also made a few other changes (all indicated in orange in the revised manuscript) to improve the consistency in notation throughout the paper.

Reviewer comment (Theory):
The proposition (2) is problematic. The $\epsilon$ should relate to $\gamma$ and $N$, otherwise, it sounds your approximator can achieve any $\epsilon$ under the current samples and probability tolerance. Additionally, why there is no $\gamma$ in the proof of proposition (2)?

Author response:
**Proposition (2) is problematic.** 

***The $\epsilon$ should relate to $\gamma$ and $N$.***: Rather than $\epsilon$ being a function of $\gamma$ and $N$, we just consider an appropriately powerful family of approximators (that depends on $\epsilon$, $\gamma$, and $N$) so that the assumptions are met. That means that to get better and better approximations, we would need to consider more powerful approximators (e.g., bigger neural networks).

***Why is there no $\gamma$ in the proof of Proposition (2)?***: There is $\gamma$ in the proof. Right before the inequality formulas, we look at the situation where two inequalities jointly hold, individual ones having probabilities at least $1 - \gamma$ and $1 - \delta$. We conclude that the joint event has the probability of at least $1 - \delta - \gamma$. We edited the proof so that this is more clearly explained.

Reviewer comment (Theory):
As the authors pointed out $H$ is the Gaussian process which is approximated/replaced by the network $h$. Thus, there should be another term to quantify this approximation error, which is dependent on the width of the neural network. However, I cannot see anything implying this information in the theorems.

Author response:
***Should there be a term to quantify the error of approximating a GP $H$ by a network $h$?***: There is no term in Proposition 1 or Proposition 2 to quantify the approximation error of replacing the Gaussian Process $H$ by the network $h$ because in these two propositions, we connect GP-PEC($B$) to the GP posterior variance classification rule. In the paragraph above Proposition 1 (page 5), we in turn connect the practical method PEC to GP-PEC($B$) by setting $B=1$ and replacing the GP teacher $H$ with a neural network teacher $h$. We do not provide a theory that quantifies errors coming from that step (we clarify this in the revised Section 3.1). Instead, we justify this replacement by noting that wide neural networks with random initialization approximate GPs, and we provide empirical justification by showing that the performance of PEC improves by increasing the width of the teacher network but not by increasing the depth (see Section 4.5). Note that we do not claim that the theoretical section explains every aspect of our method. Rather, it provides additional motivation and interpretation of our approach. Supplementing this, we provide a strong empirical section demonstrating that our method works well in practice.

Reviewer comment (Theory):
“If, on the other hand, some out-of-distribution input x is considered, then the error … is unlikely to be small” in Section 3.1: Even though seen $\boldsymbol x$ can push $g_{\theta^c}$ and $h_\phi$ to be closer by learning $\theta^c$, does this necessarily mean that the difference $\| g\_{\theta^c}(x)-h\_\phi(x) \|$ for OOD $\boldsymbol x$ is unlikely to be small?

Author response:
**Is it guaranteed that the difference $|g_{\theta^c}(x)-h_{\phi}(x)|$ for OOD $x$ is unlikely to be small?**: Firstly, it has been demonstrated that neural networks struggle to generalize out-of-distribution [1], despite their good in-distribution performance; this gives intuitive support for our statement. Secondly, if we replaced here $h$ with a sample from a GP $H$, then Proposition 1 says that the expectation of this error is lower bounded by the variance of the GP posterior at the out-of-distribution data point. Such variance also should not be small because GP posterior only “learns” from the points that are distant to $x$.

Reviewer comment (Evaluation):
Why does PEC have identical performances on different task splits as in Tables 1 and 2?

Author response:
**Why does PEC have identical performances in Tables 1 and 2?**: The performance of PEC does not depend on the task split (i.e., how many classes there are in each task) because the class-specific student networks of PEC are only updated on examples from one class. Combined with the fact that we always use batch size = 1 for PEC, this means that it does not matter for PEC whether each task only contains one class (as in Table 1) or multiple classes (as in Table 2). This was explained in the last paragraph of Appendix B.2 (page 14), but we realize that we did not clearly refer to this explanation from the main text. In the revised manuscript we now refer to this explanation from the captions of both Table 2 and Figure 2.