We thank the reviewer for their detailed feedback.

q2: Could you include some more motivation on why BELL is needed and why researchers should use it rather than other benchmarks or tasks.

Other CL benchmarks, such as split-CIFAR and CLEAR[2], can differ by the continual learning setup they assume (e.g. class-incremental, domain-incremental etc.). As a result, their sequences implicitly simultaneously evaluate a subset of CL properties, e.g. plasticity, stability and perceptual transfer, which are all summarized by a single number --- the final average accuracy. This makes it difficult to determine which of these properties is lacking from a CL algorithm.

It is important that the benchmarks which we use be able to measure whether algorithms have specific CL properties, like those we list in Table 4. This would guide future research in the field by making the desiderata explicit and facilitating the comparison of different CL algorithms against it.

CTrL [1] is a step in this direction. However, it evaluates perceptual transfer between two problems by shuffling the original problem’s labels, and it evaluates latent transfer between MNIST images with two different background colors. As such, CTrL cannot evaluate perceptual and latent transfer between disparate tasks and input domains.
BELL is a further step in that direction, which addresses CTrL’s shortcomings by creating sequences of compositional problems. The problems’ compositionality allows us to create sequences which evaluate perceptual and latent transfer across disparate tasks and input domains. It also allows us to create more challenging sequences for evaluating perceptual transfer --- S^out* and S^out**. BELL also introduces new sequences which evaluate new CL properties --- S^few which evaluates few-shot transfer and S^sp which evaluates latent transfer across different input spaces.

Overall, BELL should be used by researchers in order to diagnose the CL properties of a CL algorithm, which would improve our ability to compare different approaches.

wC, wS: Making the paper more self-contained, incorporating additional motivation for using BELL

Thank you for the suggestion, we will add more information and motivation about the benchmark suite to the main text.

wO1: LMC also introduced a benchmark for compositional generalization based on colored-mnist that is not mentioned when introducing BELL.

Thank you for pointing this out, we will change the text to mention the LMC sequence for compositional generalization.

LMC introduced a sequence of tasks for evaluating compositional generalization. In it, each problem is derived from MNIST and is defined by one background-foreground color selection and one MNIST label pair. This leads to a limited variety between problems -- input domains are always somewhat similar as they all use MNIST digits, and tasks are restricted to be distinguishing between MNIST classes.

This sequence appears to be the most similar to BELL’s S^few which evaluates a model’s ability to achieve few-shot transfer by recomposing previous knowledge, with S^few involving disparate input domains and output tasks.

wO2, q1: Running PICLE on LMC's compositional color MNIST.

We agree that these results would be nice to have. It's unlikely that we will have time to get these results by the end of the discussion period, but we will try and report back if we manage.

q3: Why is $\mathbf{\pi}^\*$ not used in Algorithm 2

In Algorithm 2, $\mathbf{\pi}^\*$ is the given list of previous solutions. Our search through PT paths does not need it, which is reflected in it being unused in Algorithm 2. We currently have $\mathbf{\pi}^\*$ as an input in order to keep the inputs to Algorithms 2 and 3 the same. However, we now appreciate that this can cause confusion and will remove $\mathbf{\pi}^\*$ from the inputs of Algorithm 2.

References:
[1] Veniat, T., Denoyer, L. and Ranzato, M.A., 2020. Efficient continual learning with modular networks and task-driven priors. arXiv preprint arXiv:2012.12631.
[2] Lin, Z., Shi, J., Pathak, D. and Ramanan, D., 2021, August. The clear benchmark: Continual learning on real-world imagery. In Thirty-fifth conference on neural information processing systems datasets and benchmarks track (round 2).

Thank you for your responses. After reading them and those to other reviewers I have decided to raise my score to 8.

We thank the reviewer for their feedback.

q1: Quantitative evaluation of the generative quality of the proposed method.

For our purposes, the employed generative model needs to be useful for distinguishing between inputs sampled from different distributions. As a result, we have not explored the generative quality of the samples. Given our use of efficient Gaussian approximations, we expect that it is poor. Instead, in Appendix I, we have provided a quantitative evaluation of our method’s ability to distinguish between different input domains. The results indicate that using random projections improves the discriminative capabilities of our generative model.

q2: Baselines in Tables 1-2 appear classic. Can the authors comment on comparing with more recent works?

Our work aims to improve modular CL methods. Therefore, we compare to the latest state-of-the-art modular CL algorithms, namely: MNTDP-D [1] (2020) and LMC [2] (2021). We also compare to an additional modular CL algorithm -- HOUDINI [3]. For completeness, as done in previous modular CL work [1, 2], we use O-EWC [4, 5] as a classic regularization-based CL algorithm, and ER [6] as a replay-based method, which was shown to have competitive performance despite its simplicity.

Tables 1-2 do not report the performance of LMC since it performed poorly, despite our efforts to adjust it to the BELL benchmark suite.

q3: Can the authors compare the computational overhead of their method against the baselines?

Figure 3a of our submission compares the computation demand of PICLE to those of the baselines, evaluated in FLOPs. It can be seen that since PICLE uses slightly more resources than MNTDP-D since it considers a larger search space. However, it can also be observed that PICLE’s computational requirements scale well with the number of solve problems.

References:
[1] Veniat, T., Denoyer, L. and Ranzato, M.A., 2020. Efficient continual learning with modular networks and task-driven priors. arXiv preprint arXiv:2012.12631.
[2] Ostapenko, O., Rodriguez, P., Caccia, M. and Charlin, L., 2021. Continual learning via local module composition. Advances in Neural Information Processing Systems, 34, pp.30298-30312.
[3] Valkov, L., Chaudhari, D., Srivastava, A., Sutton, C. and Chaudhuri, S., 2018. Houdini: Lifelong learning as program synthesis. Advances in neural information processing systems, 31.
[4] Schwarz, J., Czarnecki, W., Luketina, J., Grabska-Barwinska, A., Teh, Y.W., Pascanu, R. and Hadsell, R., 2018, July. Progress & compress: A scalable framework for continual learning. In International conference on machine learning (pp. 4528-4537). PMLR.
[5] Chaudhry, A., Dokania, P.K., Ajanthan, T. and Torr, P.H., 2018. Riemannian walk for incremental learning: Understanding forgetting and intransigence. In Proceedings of the European conference on computer vision (ECCV) (pp. 532-547).
[6] Chaudhry, A., Rohrbach, M., Elhoseiny, M., Ajanthan, T., Dokania, P.K., Torr, P.H. and Ranzato, M.A., 2019. On tiny episodic memories in continual learning. arXiv preprint arXiv:1902.10486.

We thank the reviewer for their feedback.

w1a: Lack of uncertainty metrics in the table.

As the performance of the different algorithms on the short sequences of the CTrL benchmark is similar, we added the standard deviations of the results shown in Table 3. This can be found in Table 6, Appendix K.1. Does this address your concern?

w1b: Show percentages for up to 1 decimal point.

We agree that this would improve the paper’s readability and will make the changes.

w2: The paper should clearly state that the PICLE method is task-aware

We agree that this is an important property, which provides an interesting distinction between the algorithms. Therefore, we will mention it and add it as a row in Table 4.

w3: Credit to and better distinction from LMC.

We have credited LMC with a similar approach for perceptual and few-shot transfer in the Related Work section, while also outlining the differences. However, we agree with the wording proposed by the reviewer, and will also add it to Section 4 for better visibility. Moreover, we acknowledge that the conceptual differences can be better presented and will amend the text accordingly.

The main conceptual difference is that while LMC also uses a module-specific generative model, PICLE unites the generative model for each layer using a single probabilistic model, which brings numerous advantages. First, our probabilistic model allows us to define a prior over the choice of modules. Second, by changing the probabilistic model, we can incorporate further assumptions in a principled way, for example, we can put a prior over the choice of pre-trained modules for multiple layers, rather than layer-specific priors. Third, the probabilistic model allows us to compute the posterior over the choice of pre-trained module for each layer collectively, rather than individually.

The main implementation difference is that PICLE utilizes a different approximation of a module’s input distribution, using orders of magnitude fewer extra parameters per pre-trained module than LMC.

We thank the reviewer for their feedback. We will correct the reported typos.

Comparing MNTDP-D and PICLE:

We would argue that this is a case where looking at the average accuracy across tasks is misleading. Table 4 and Appendix H of our submission list a set of desirable properties for CL methods. Each sequence from our experiments evaluates a different CL property. Our argument is that MNTDP-D and PICLE share many desirable properties (see Table 4), so they will have similar accuracy on sequences that test those properties. But PICLE has a few desirable properties that MNTDP-D does not.

As a result, MNTDP-D and PICLE are expected to and do perform similarly on the sequences which evaluate: perceptual transfer (S^-, S^out, S^out*), plasticity (S^pl), Stability (S^+). We note that, PICLE is able to demonstrate perceptual transfer on S^out**, Table 1 while MNTDP-D fails to do so, leading to PICLE achieving +10.3 higher transfer on the last problem.

However, PICLE is expected to and does significantly outperform MNTDP-D on few-shot transfer and latent transfer. Since the short sequences were designed to evaluate different transfer learning properties by an algorithm’s performance on the last problem, we consider the amount of transfer to the last problem. For few-shot transfer, PICLE achieves +34.65 higher transfer (S^few, Table 1) than MNTDP. For latent transfer, PICLE also demonstrates much higher transfer: +12.58 (S^in, Table 2), +23.65 (S^sp, Table 2), +16.89 (S^in, Table 3).

We discuss this in Section 6, however, thanks to your comments, we will change the text to state this more clearly.

w1: The quantitative advantage over MNTDP-D in accuracy is marginal on average ~1-3%
q1: “Why are the numbers for MNTDP-D and PICL forward transfer so similar?”

Since PICLE and MNTDP-D share many CL properties, apart from few-shot transfer and latent transfer, their performance --- including accuracy and forward transfer --- on the short sequences is similar, apart from on S^few, S^in, S^sp.

In Table 1, the accuracy is averaged over mostly shared CL properties (apart from few-shot transfer). PICLE’s advantage is only apparent by its superior performance on the last problem of S^few and S^out**. As a result, the average accuracy in Table 1 averages over 42 problems, while PICLE achieves significant improvement over MNTDP-D on 2 of them. This leads to the similar average accuracy which we observe. Similarly, the short sequences of CTrL mostly evaluate CL properties which are shared between MNTDP-D and PICLE, leading to PICLE outperforming MNTDP-D only on the last problem of S^in, and in turn leading similar average accuracies in Table 3.

PICLE’s advantage over MNTDP-D is that it can achieve few-shot and latent transfer. Overall, PICLE achieves much higher transfer on the last problem than MNTDP-D on: perceptual transfer (S^out**, Table 1, +10.3), few-shot transfer (S^few, Table 1, +34.65), latent transfer (S^in, Table 2, +12.58), (S^sp, Table2, +23,65), (S^in, Table 3, +16,89).

w2: The method requires slightly more FLOPs than MNTDP-D (Figure 3 (a))

This is correct. Figure 3(a) shows that PICLE requires slightly more FLOPs, while considering a much bigger bigger search space. The figure also shows that the computational demands of both PICLE and MNTDP-D scale well with the number of problems.

Other questions:

q2: “Algorithm 3 - Do you mean $\ell \in \{ 2, ..., L-1 \}$?”

The $\ell$ defined on line 11 of Algorithm 3 refers to the index of the first pre-trained module. On line 12 we use it to extract the pre-trained modules $\pi’[\ell : L]$. For $\ell = \ell_{min}$ , the length of the pre-trained suffix $\pi’[\ell_{min} : L]$ is $\ell_{min} + 1$. And for $\ell = 2$, the length of the pre-trained suffix $\pi’[2 : L]$ is $L-1$. Therefore, we indeed attempt to transfer ${\ell_{min}+1, ..., L-1}$ pre-trained layers. However, thanks to your comment, we now see that it is confusing to use $\ell$ inside the text to denote length while simultaneously using it in Algorithm 3 to denote an index. We will adjust the text accordingly.