Critical Learning Periods Emerge Even in Deep Linear Networks

We thank the reviewer for the detailed comments and accurate summary of our work. We also thank the reviewer for the concrete presentation suggestions. We describe how we address their comments below.

The authors prove the dependence on depth and data distribution. But how can they claim independence from other factors, like architecture and non-linearities, especially when considering models other than deep linear networks?

The reviewer is correct that the point is misphrased. Indeed, critical periods can depend, in part, on other factors (such as optimization algorithm and non-linearities), which has also been shown empirically in nonlinear networks (Achille et al., 2019). We rephrased the point in our introduction and discussion. We want to emphasize, however, that even ablating those details (nonlinearities and optimization algorithm), as done in our paper, we still observe critical periods that depend on architecture depth and data distribution. This means that the effect of other factors is not necessary for the emergence of critical periods in deep networks, and in general may have a less significant effect than architecture depth and data distribution.

In our revised paper, we ran additional experiments in our multipathway setup where we included nonlinear activation functions (Relu and Tanh) and observed similar trends to in the deep multipathway linear case (Fig 11 and Fig 12 in revision).

Figure 2’s markers should be changed. It is difficult to differentiate between a circle and a circle-triangle overlap. Try plus (+) and cross (x) as markers or a straight line (|) and an oblique line (/).

We thank the reviewer for the suggestion. We modified our multipathway plots accordingly, and we believe it helps with distinguishing between the different pathways.

Codebase not provided.

We will provide our code along with the final version of our paper.

Typos:

We fixed the typos. Thanks for the careful reading and for catching those!

I understood the use of PyTorch’s “.detch()” for blocking the flow of gradients in the deprived branch. However, shifting the desired target output to ensure learning the unexplained component did not make sense to me. Please elaborate.

Detaching is how it is implemented in our code. The alternative (and equivalent) interpretation as shifting the target was provided solely to relate it to a more biologically meaningful process. Specifically, consider a deficit where parameters along pathway A are not updated. Without the deficit the Loss is $L_{normal} = (y - W_A^{tot}x - W_B^{tot}x)^2$, and gradients are updated in both pathways. During the deficit (to pathway A) the loss is equivalent to $L_{def} = (\tilde{y} - W_B^{tot} x )^2$ where $\tilde{y} = y - W_A^{tot}x$, and gradients only occur in pathway B. We referred to $\tilde{y}$ as the shifted output.

Achille, Alessandro, Matteo Rovere, and Stefano Soatto. "Critical learning periods in deep networks." International Conference on Learning Representations. 2019.

We thank the reviewer for their time, feedback, accurate summary of our work, and comments. We address their concerns below.

I think the assumption of the deep linear network is a bit strong. Since in real-world applications, most neural networks require non-linear activations. A deep linear network can be actually approximated by a single-layer linear network.

While it is true that a deep linear network has the same expressive power as a single-layer linear network, we emphasize that the parameterization of a deep linear network leads to complex and depth-dependent non-linear learning dynamics that is sufficient to lead to the emergence of critical periods. By restricting our analyses to deep linear networks, we highlight that the deep parameterization alone (which is shared with non-linear networks) is enough to explain the emergence of critical periods and may indeed be the most important factor underlying them. Indeed, in our updated experiments in our revision paper that we posted, we show that even a non-linear network follows largely the exact same trends as the multipathway linear network in our experiments (Figure 11, Figure 12).

Critical periods in artificial deep neural networks (DNNs) may be due to specificities of the optimization process, such as an annealing learning rate, or from defects in the artificial implementation and training, like ReLU units becoming frozen or gradients vanishing.

We agree and acknowledge this directly in our introduction as one of the main motivations for studying deep linear networks in our paper. In particular we show that deep linear networks trained with constant learning rate and without nonlinearities also exhibit analogous phenomena, and also permit analytical analysis. As we replied to R1, we don't want to claim that in any model depth and structure of the data distribution are the only factors determining the existence of critical periods. In fact, other papers have shown that choice of optimization algorithm or regularization can significantly affect the critical period (Achille et al., 2019). What we show is that depth and structure of the data are a minimal set of requirements that alone can support the emergence of a critical period, and also enable differential equations that precisely characterize the learning dynamics. We clarified the text to better communicate this.

What's the model's generalization ability on more sparse data settings?

We interpret this question to be referring to our matrix completion settings. In Fig. 5 right, we examine reconstruction error as a function of the number of entries. In general matrix completion becomes easier with a larger number of observed entries. When we have a very sparse number of entries (1000 entries) the network obtains a large reconstruction error even without pre-training on a different task (this corresponds to the point deficit removal at 0). In the sparse setting too, the network obtains worse reconstruction by initially training on the higher rank task (other points along the curves). We also found that pre-training on a subset of the observed entries adversely affects generalization (Fig 5, left). We added in an additional experiment to highlight that this effect also depends on the depth of the parameterization, with deeper architectures being more affected (Fig 13).

Achille, Alessandro, Matteo Rovere, and Stefano Soatto. "Critical learning periods in deep networks." International Conference on Learning Representations. 2019.

I thank the author for the detailed response, which has addressed some of my concerns. I have increased the score accordingly.

remind us what "Path Singular Value" is in the caption/change caption e.g .singular value dimensions (confusing that its referred to as 1-5 but labelled 0-4 on plot and called something different).

Path Singular Value refers to the contribution of each pathway to the singular value and corresponds to to $K_a$ (or $K_b$) defined in Sect 3.1. We now better describe this in the text. $K_a + K_b$ sum to the singular values at convergence. In Fig. 3, we only plot the top 5 task singular values – we clarified the text.

I didn't understand the comment "and do not affect performance on the final task" in fig [4]. Does it mean they have equal performance?

We clarified the sentence in our revision. Indeed, it means they lead to equivalent performance on the final task.

Achille, Alessandro, Matteo Rovere, and Stefano Soatto. "Critical learning periods in deep networks." International Conference on Learning Representations. 2019.

Kleinman, Michael, Alessandro Achille, and Stefano Soatto. "Critical Learning Periods for Multisensory Integration in Deep Networks." Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. 2023.

We thank the reviewer for the accurate summary of our work, and their time, feedback, and insightful comments. We appreciate the positive assessment of our research and presentation of our results. We explain below how we address the reviewer’s comments.

Framing of introduction and conclusion:

We appreciate this thoughtful and insightful point. We updated the introduction and discussion to better highlight this nuance – indeed “natural information processing constraints” should not be contrasted with biochemical processes. In our introduction and discussion, we aim to emphasize that architecture and data distribution are sufficient for leading to critical periods in deep artificial networks, and that this offers an intriguing, and different lens for understanding critical periods in biological systems. We plan to further elaborate on this in subsequent revisions subject to space constraints and incorporating other reviewer feedback during the discussion period.

Terms used in equations are not explained when introduced in 3.1 -- I know it's from a citation and you might be short on space, but you will lose a lot of readers here and it's a shame not to continue the clear prose you've had thus far.

We updated our manuscript to better introduce notation (also pointed out by other reviewers).

Around eqn 9, connection of matric completion to what you're trying to do is not clear -- be explicit about how imputing missing values is the same as / allows us to specify task relationships with flexibility.

Thanks, we updated our manuscript to better introduce the matrix completion setting – the changes are in blue text of our revision.

While the relationship of experiments to showing dependence on depth is very clear and well-explained, the insights and relationship of experiments to data distribution is much less clear.

We appreciate the feedback. In general, pre-training on a modified data distribution for some duration can either be helpful or harmful, depending on the relationships between tasks. In the case of critical learning periods, we focus on the case when such pre-training is harmful. Empirically, Achille et al., (2019) found that pre-training on blurred images was harmful for subsequent generalization on normal images, but pre-training on pure noise for the same duration was much less harmful. This suggests that the learning dynamics depend on the complex relationship between the tasks.

In general it’s difficult to define the “complexity of a task” or the relationships between tasks. However, the matrix completion setting gives us the ability to completely specify the task, its complexity, and the relationship between tasks. We find that pre-training on a higher rank task (more complex) leads to worse generalization performance (Fig 5, left). We also found that pre-training on a subset of the observed entries (a partial view of the data, similar to the blurred image experiments) also leads to impaired generalization (Fig 6, left). We added in an experiment to highlight that this effect becomes more pronounced in deeper architectures (Fig 13 of revision).

We updated the text to better motivate these experiments in text.

The main other thing I'd like to see is connections made/discussed with nonlinear networks -- what should we expect to change vs. hold? maybe according to the sources you cite (e.g. Saxe), or ideally, a suite of experiments with e.g. MLPs of increasing depth. Another specific link could be made in Fig. 6 to scaling laws.

We expect deeper networks to be more affected by a temporary deficit, and we now directly discuss this connection in the text. This finding that deeper architectures are more affected by an initially training on a related, partial dataset, is consistent with the empirical results of Achille et al., (2019, in particular see Fig 2, right panel of reference) and Kleinman et al., 2023 (in particular see Fig 5 center panel of reference) who empirically found that deeper convolutional architectures were increasingly affected by initially networks on blurred images before training on regular images. We also ran an additional experiment where we trained on a subset of the observed entries, and we found that while a single layer network is not affected, deeper architectures are increasingly affected by a sufficiently long initial deficit (Figure 13 of revision) We thank the reviewer for the helpful suggestion.

inconsistent use of "artificial neural network", "deep net", "neural net"

We thank the reviewer for the feedback and will better clarify the text. There were times where we wanted to emphasize the distinction between artificial and biological networks, as well as highlight the depths of the architectures, but we will strive for improved consistency and clarity.

We thank the reviewer for their accurate summary of our work and for their time, feedback, and comments. We explain below how we address the reviewer’s comments.

Improved clarity of the ``linear multi-pathway framework" (Sec.3.1.)

In our revised manuscript, we better introduce our notation in Sect 3.1. The updates are highlighted in blue text. We thank the reviewer for pointing this out, and we believe the changes lead to a more readable manuscript.

Only studying deep linear networks seems not enough to clearly understand why deep networks have critical periods, and network depth and data sources cannot guarantee sufficient persuasiveness.

We don't want to claim that in any model depth and structure of the data distribution are the only factors determining the existence of critical periods. In fact, other papers have shown that choice of optimization algorithm or regularization can affect the critical period (Achille et al., 2019, Zilly et al., 2021). What we show instead is that depth and structure of the data are a minimal set of requirements that alone can support the emergence of a critical period, and also enable differential equations that precisely characterize the learning dynamics. This allowed us to better understand how depth, competition between sources, and data distribution affects critical learning periods. We have clarified the text in our revision.

There are fewer categories of experiments, and it would be better if the authors could provide more types of experiments to prove their claims.

We ran additional experiments to further support our claims that we include in our revision. Related to the above question, for the multipathway simulations, we included results using Relu and Tanh nonlinearity and observed similar conclusions where deficits early on as in the deep linear case (Figure 11, and Figure 12 in revision)

For the matrix completion setup, we also ran an additional experiment where we trained on a subset of the observed entries, and we found that while a single layer network is not affected, deeper architectures are increasingly affected by a sufficiently long initial deficit (Figure 13). This finding that deeper architectures are more affected by an initially training on a related, partial dataset, is consistent with previous empirical results of Achille et al (2019; in particular see Fig 2, right panel of reference) and Kleinman et al (2023; in particular see Fig 5, center panel of reference) who empirically found that deeper convolutional architectures (with nonlinearities) were increasingly affected by initially training on blurred images before training on regular images.

For the experiments related to pre-training, I cannot guarantee sufficient correlation with key learning practices. I hope I can explain the motivation for doing this part of the experiment. By the way, this is not to say that this is an obvious weakness.

In neuroscience, critical periods are observed in two settings broadly speaking: (1) sensory deprivation, when the input to one sensor is completely absent, and (2) the related setting when the agent initially learn on a abnormal data distribution (e.g., because of anomalous working of the sensor, or abnormal sensory experience) before having access to proper data later in life. While our multi-pathway experiments are targeted at modeling the competition-based effects coming from the first case, our matrix completion setup where we pre-train on a modified data distribution setting more closely models the second case (the network is pretrained on a different data distribution than the one it sees later in training). The matrix completion setup provides a good simple model to probe this critical period phenomena (how generalization on a final task is affected by pre-training on an initial task) since it allows a notion of generalization, as well as a well defined notion of tasks, their complexity, and their relationships.

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