Training Large Neural Networks With Low-Dimensional Error Feedback

Thank you for your feedback. We are encouraged that you find the work well-written and see the potential of our ideas and theory.

1. We couldn’t agree more with you about the overuse (and perhaps misuse) of the term “biological plausibility”. It is not well-defined, unclear, and, more importantly, untrue. Most models that claim biological plausibility probably cannot be implemented by real neurons without significant adjustment and extensions (and we include our model here). However, the term is widely used and has effectively become a jargon. Authors often use relative terms (i.e., “more biologically plausible”), which is also inaccurate. Nevertheless, we are happy to replace that phrasing with a more detailed description. In particular, we have removed unnecessary uses of “biological plausibility” and replaced it with what we actually mean: we study how large neural networks learn using indirect restricted error signals. In the brain, a neural population is unlikely to receive a detailed error signal that originates from an efferent source. Thus, it is crucial to understand how error constraints affect learning and shape neural representations.

2. Regarding your second point, from the machine learning perspective. We believe there is a misunderstanding. You are correct that in linear networks, the error dimensionality increases and is equal to the dimensionality of the output error. In this case, it is unsurprising that we can train with low-rank feedback because the error dimensionality does not change. However, we still gain a lot of insight from the linear networks. Particularly, we see that low-rank feedback matrices must learn or “align” to the error space. This is because they must send the correct error to afferent layers. Developing your idea, studying the dimensionality in the deeper layers of a nonlinear network is potentially possible, assuming random connectivity and using more sophisticated techniques of dynamic mean-field theory or the dynamic cavity method [Clark, Abbott, and Litwin-Kumar. 2023]. This, however, would be the subject of future work.

However, in nonlinear networks, the dimensionality of the error varies between the layers and during training. The Jacobian depends on the activity in every step, effectively increasing the error subspace. Surprisingly, the learning rules we develop using our intuition from linear networks work in the nonlinear case. We believe it is because our basic logic is correct. We’ve elaborated and clarified this point in Section 2.

3. Finally, we emphasize that we do not claim nor did we attempt to derive computationally efficient algorithms or state-of-the-art performance. Instead, we show that large networks can be trained using indirect and restricted error signals if (and only if) the feedback matrix learns (i.e., “aligns” also the efferent error space). Nevertheless, we point out that training with low-rank feedback is not significantly less efficient than BP (unlike other methods of FA). Consider training the weights between two large layers with $M,N\gg1$ Neurons each. This requires learning $O(MN)$ parameters. The low-rank feedback matrix requires learning additional $O(r(M+N))$ parameters, which is negligible in large networks (assuming $r$ is small). Thus, while we do not make any efficiency claims, we can assert that it has not significantly increased compared to BP and would be smaller than other FA-based methods.

References

• Clark, D. G., Abbott, L. F., & Litwin-Kumar, A. (2023). Dimension of activity in random neural networks. Physical Review Letters, 131(11), 118401.

An answer to the specific question

In [Lindsey et al. 2019], the authors observed that network bottlenecks result in a systematic change in the receptive fields, with tighter bottlenecks resulting in more center-surround fields. We find this a fascinating result. Not because of the direct biological implications described by Lindsey et al.—the retinal layer probably does not learn in a supervised manner—but because it relates architecture to receptive fields, offering a powerful tool for computational neuroscience. However, they did not study the mechanism underlying the emergence of the center-surround fields. Moreover, their results raise a conundrum: a bottleneck in the feedforward pass should require more efficient coding, while center-surround is less efficient than Gabor-like filters. The proposition that it is due to the restricted error pathway is perhaps intuitive, but our work is the first time this can be systematically studied. Indeed, we find that restricting the error is sufficient to shape the receptive fields (Fig 5). Furthermore, in the appendix, we show that a network with a bottleneck—similar to that of Lindsey et al.—but with a “full-rank” error that bypasses the bottleneck does not lead to center-surround filters. Our results hold further implications for computational neuroscience; we show that the shape, particularly the symmetry, of tuning curves is affected by the available error signal. We note that restricting the error dimensionality is different from regularizing it. Gunasekar’s work studies the dimensionality of the solution and does not restrict the error during learning. Similarly, Caro et al.'s work studies symmetries in the feedforward and the solution but not the independent effects of the backward pass.

References

• Gunasekar, S., Lee, J. D., Soudry, D., & Srebro, N. (2018). Implicit bias of gradient descent on linear convolutional networks. Advances in neural information processing systems, 31.
• Caro, J. O., Ju, Y., Pyle, R., Dey, S., Brendel, W., Anselmi, F., & Patel, A. B. (2024). Translational symmetry in convolutions with localized kernels causes an implicit bias toward high frequency adversarial examples. Frontiers in Computational Neuroscience, 18.

We thank you for your comments and for taking the time to consider our work. We are pleased you appreciated the relationship between the error feedback and the receptive fields. This question originally prompted our study.

Response to the mentioned weaknesses

1. You correctly noted that our datasets are relatively simple. However, we do not think a more complex dataset will bring further insight. The main message of our work is that if the output error is low-dimensional, then we can train large errors with low-dimensional error feedback if the feedback weights correctly learn the error space. Using image data, we show that this principle extends to nonlinear and convolutional networks and that the error signal shapes the receptive fields. Even training on ImageNet-1k would require a feedback rank of 1000. If successful, which we believe it will be, it will not highlight the effectiveness of low-dimensional feedback. Instead, we elucidate our conclusions that deep networks can learn efficiently using minimal error signals. This is an important finding for studying brain-circuit architecture but not an alternative solution for learning complex visual data.

   Finally, your comment raises a fascinating question: can a network learn a complex dataset, such as ImageNet, with an error signal with a lower output dimensionality? We imagine the top layer would see the full 20k class error feedback, but deeper layers would only see a low-dimensional error, which will be learned. It may be possible due to the structure of the data and labels. However, this question is outside the scope of this work. Nevertheless, we are training a network on ImageNet and will report here if we have interesting findings.

2. Novelty of low-rank gradients. Indeed, there has been work showing that the dynamics of weights during SGD are low dimensional. This aligns with our hypothesis that a low-dimensional error signal is potentially sufficient for training. We have added these references to our introduction. However, previous studies on regularization do not address whether a low-dimensional error can drive the weight dynamics. Regularization and the shape of the loss-landscape [Ma and Ying 2021, Feng and Tu 2021] can constrain dimensionality, but it is not equivalent to a restricted indirect error channel; the latter is of greater importance to neuroscience. Finally, previous studies on the dimensionality of training dynamics do not systematically study the relation between the error signal, the performance, and the receptive fields.

References

• Ma, C., & Ying, L. (2021). On linear stability of sgd and input-smoothness of neural networks. Advances in Neural Information Processing Systems, 34, 16805-16817.
• Feng, Y., & Tu, Y. (2021). The inverse variance–flatness relation in stochastic gradient descent is critical for finding flat minima. Proceedings of the National Academy of Sciences

Thank you for a detailed response!

1. Contribution compared to [Akrout, 2019]

I agree that low-dimensional feedback is not the topic that paper was solving, although I still think that conceptually the most important part of the FA-insipred line of work is the basic Kolen-Pollack algorithm.

In addition, we are not sure how to gauge if adding an additional feedback layer is “less biologically plausible”.

Here's my intuition: for a basic FA-style algorithm, you need a separate error network that mimics all connections in the forward pass. I don't think we have evidence for such connectivity in the cortex. In your case, that network has more hidden layers/neurons (although fewer weights) to implement the same algorithm (backprop), while still requiring one-to-one mapping between forward and backward neurons.

That said, I think your work simultaneously relaxes connectivity constraints by allowing less direct error pathways, which is interesting conceptually.

2. Oja's rule

Thanks for the references. I now recall the [Pehlevan and Chklovskii, 2015] solution, and explaining Oja's rule implementation(s) would definitely strengthen the paper.

However, I disagree that your work is not trying to solve the weight transport problem -- weight transport is exactly why you need to approximate the forward weights. I think your low-rank discussion is a consequence of trying to solve the weight transport problem. (This doesn't make your point weaker, in my opinion, but I think this view better captures your contribution.)

Also, thank you for addressing the small issues/presentation concerns I had.

Overall, I think this work is interesting and contributes to the discussion on potential learning mechanisms in the brain. I'm raising the score from 5 to 8 as most of my concerns have been addressed.

References

• Leinweber, M., Ward, D. R., Sobczak, J. M., Attinger, A., & Keller, G. B. (2017). A sensorimotor circuit in mouse cortex for visual flow predictions. Neuron, 95(6), 1420-1432. (and the relevant erratum)
• Sacramento, J., Ponte Costa, R., Bengio, Y., & Senn, W. (2018). Dendritic cortical microcircuits approximate the backpropagation algorithm. Advances in neural information processing systems, 31.
• Clopath, C., Büsing, L., Vasilaki, E., & Gerstner, W. (2010). Connectivity reflects coding: a model of voltage-based STDP with homeostasis. Nature neuroscience, 13(3), 344-352.
• Turrigiano, G. G. (2008). The self-tuning neuron: synaptic scaling of excitatory synapses. Cell, 135(3), 422-435.
• Pehlevan, C., Hu, T., & Chklovskii, D. B. (2015). A hebbian/anti-hebbian neural network for linear subspace learning: A derivation from multidimensional scaling of streaming data. Neural computation, 27(7), 1461-1495.
• Liao, R., Schwing, A., Zemel, R., & Urtasun, R. (2016). Learning deep parsimonious representations. Advances in neural information processing systems, 29

Answer to specific questions:

1. This is a good question. Quantifying how “center-surround” the receptive fields are is challenging. Since the receptive fields are noisy, quantitative analysis requires training many networks, which is beyond our current capacity due to time limitations. We note that [Lindsey 2019] also did not quantify this observation. Importantly, we emphasize that we did the expected control: we trained a network with a bottleneck in the feedforward while allowing full-rank error propagation (i.e., we used our training but did not restrict the rank). In this case, we saw no center-surround receptive field. These fields emerged when and only when the error was constrained. These results are reported in the appendix.
2. Fixed.
3. Fixed.

We hope that we have conveyed the theoretical significance of our work and its conceptual advance over previous studies of FA methods. As soon as we integrate all comments and proofread, we will upload the corrected PDF.

Thank you for taking the time to read and thoroughly comment on our work. The strengths you mention increase the potential benefits of our applied results. We believe our work also holds novel theoretical insights, and we hope we can convey them with our responses to your comments and questions

Response to the mentioned weaknesses:

1. We emphasize that we address a different problem than [Akrout 2019]. In our model, the low-rank feedback matrix constrains the feedback error dimensionality. This is central to our model since we focus on how networks can learn with a low-dimensional, restricted error signal. Setting $P=I$ is equivalent to [Akrout 2019] is the rank is full (i.e., the rank of $B=QP$ is $\min(N,M)$. Furthermore, we show that if the matrix is low-rank. This learning rule will not work. This observation leads to the key difference from [Akrout 2019] and other previous models: when the error pathway is restricted, the feedback matrix must also learn its input space. In other words, the feedback should not only align with the feedforward weights (as in [Akrout 2019], [Lilicrap 2016], and others) but also align with the error. Our algorithm aligns it both with the weights and the error. In addition, we are not sure how to gauge if adding an additional feedback layer is “less biologically plausible”. We note that this feedback layer will only have $O(d)\ll N$ neurons. Several previous studies has suggested adding interneurons as a mean to implement gradient descent (e.g., Leinweber et al. 2017 or Sacramento et al. 2018)

2. This is a correct and important observation. The learning rule relies on the transposition of the $P$ matrix, the same way that Oja’s rule requires it. Implementation of this rule, as defined in eq. (8) violates synaptic locality due to the $P^TP$ term. Failing to highlight it was an oversight. However, we argue that this point does not reduce the significance of our results:

   1. First, we do not claim we solved the problem of weight transfer. We note that BP suffers from weight transfer, which was the original motivation behind FA. We now clarify this point in the text.
   2. While our update rule involves matrix transposition, it does not require a backward pass like backpropagation, where the signal is propagated backward through the axons. Likewise, it does not require keeping an exact copy of the transposed weights to propagate the error backward.
   3. Possible biological implementations and approximations for Oja’s rule exist. The biological plausibility of the learning rule has been discussed in original work by [Oja 1982], but also more modern variants, including homeostatic plasticity mechanisms [Clopath et al. 2010], synaptic rescaling [Turrigiano 2008], or similarity matching [Pehlevan and Chklovskii, 2015].
   4. Most importantly, the biological plausibility of the learning rule does not change our work's central messages and conclusions. In particular, we show that large neural networks can learn with minimal, restricted error signals if the feedback matrices align with the error space. In that case, we show that it matches BP’s performance. Finally, we show that the error signal dimensionality shapes the receptive fields.

3. Point taken. We have added the results of [Lindsey 2019] in our manuscript for reference.

4. Following your advice, we have clarified and elaborated on how previous work relates to this study. In particular, we point out that [Akrout 2019] and [Liao 2016] have made significant advances, but they have not addressed the problem of restricted error signals. Section 3 now explains that these methods are insufficient for training with a low-rank feedback matrix. Finally, while our learning rule composes only one extra step from the Kolen-Pollack framework, we believe there is a conceptual advancement. Previous studies use plasticity rules to align the feedback matrix with the feedforward weights. Here, we claim that if the error channel is restricted (as in the case of low-rank feedback), the weights must align the row space with the error space. While this is achieved using a single step, it represents an important theoretical idea of what the feedback channel does and is a departure from the Kolen-Pollack framework. Nevertheless, we have removed the word ‘superficially’ from the text.

Answers to specific questions

1. These are valid concerns since none of these methods has variable biological implementations. First, we emphasize that the fully connected nonlinear networks did not use Maxpooling, batchnorm, or dropouts. However, these techniques stabilized and improved training in the full convolutional network. Note that the same is used for the BP we compare. Without these regularization methods, both RAF and BP show poorer but still similar performance. We emphasize that we did not claim or attempt to solve how the brain learns or implements backpropagation. This work focuses on a specific question: can large networks learn using a low-dimensional error signal, which is more likely to be implemented in the brain?
2. Our results state that we can restrict the error dimensionality when the error dimensionality of the output is low. While it sounds trivial, it is not since the credit assignment using backpropagation implies the error in each layer is high-dimensional. We specifically used data sets with low-dimensional readout errors. We also specifically looked at the effects of the output error dimensionality by subsampling the CIFAR-100 dataset. We do not know if our result will extend to LLMs since these produce high-dimensional error signals. If they do, it is not due to the learning dynamics but to degeneracies in the error landscape of these models, which were not the focus of this work.

We have added the references you highlighted in our introduction and discussion and clarified the scope of our work. We are integrating these changes with other comments and will upload a revised version shortly.

Thank you for reading and thinking about our work; we appreciate your input. We are happy you found the strengths of our work as we see them. We believe that these points, as you summarize them, serve as a significant step towards a better understanding of learning in brain circuits.

We are sorry that you find the paper not very well written. We aim our text for a diverse audience from different fields interested in learning representation, specifically because of our work's strong biological motivation and implications. We are aware of the tradeoff that by doing so

Response to the mentioned weaknesses

1. We are sorry to hear that you find the manuscript not well written. Our goal was to make it accessible to a wide spectrum of readership. This was particularly important to us as we focus on biological implications, which may interest readers from various fields and levels. We are aware of the tradeoff that advanced readers may find our language repetitive and perhaps too figurative. Nevertheless, it is intended to be correct and full. We appreciate your specific comments. The caption on Fig. 5 was a mistake we corrected. The repetitive definitions (as for the input $x$) are intended to guide a less familiar audience.
2. You are correct that the paper emphasizes the linear theory. It allows a full solution to the learning dynamics. Crucially, it reveals why the learning fails when the rank is low: the learning dynamics have multiple fixed points, which are not the correct solution. It also helps us identify the learning rule that prevents it—the feedback weights must also "align “ in the error space. Our work is mainly theoretical and explains how neural networks can learn with very limited error signals, which we expect to be the case in the brain. The intent was not to match SoTA performance on complex data sets. We also note that many seminal theoretical papers rely heavily on linear theory to explain the dynamics and behavior of nonlinear networks (e.g., Saxe et al. 2013, Refinetti et al. 2021, or Li and Sompolinsky 2021).
3. We thank you for highlighting these references, which we did not include. However, none of these previous studies have addressed the problem of restricted (low-rank) error pathways, which is important for computational neuroscience. Particularly, we point out that Akrout et al. did not address this problem. Furthermore, we show our manuscript that Akrout and other FA solutions fail with a restricted error pathway as they all lack a crucial ingredient of “aligning” to the error space.

References

• Saxe, A. M., McClelland, J. L., & Ganguli, S. (2013). Exact solutions to the nonlinear dynamics of learning in deep linear neural networks. arXiv preprint arXiv:1312.6120.
• Refinetti, M., Ingrosso, A., & Goldt, S. (2023). Neural networks trained with SGD learn distributions of increasing complexity. International Conference on Machine Learning
• Li, Q., & Sompolinsky, H. (2021). Statistical mechanics of deep linear neural networks: The backpropagating kernel renormalization. Physical Review X

Thank you for tentatively addressing the questions I asked.

1. The answer is not really convincing. My question was aimed to see if you had intuitions about how some operations well-suited for artificial neural networks could have similar counterparts in the learning of the brain. Furthermore, although it may seem self-evident, why does a "low-dimensional error signal is more likely to be implemented in the brain"? Is this a claim or do you have references that study this interesting result?

2. Thank you for your answer. I thought your method could be applied to (relatively) small language models when fine-tuned on classification tasks. This study could balance the experimental part and maybe show that your results stand for image data and not language data, highlighting that different data-dependent learning dynamics exist. That would provide a great insight, though I understand it might not be feasible in the short amount of time you have during this phase.