1. Feedback Alignment with MLP on MNIST can achieve 98% test performance.

As stated in Line 325 (left column), to ensure fairness, we adopted the same architecture (784-1024-512-256-10 FC layers, see Appendix C.2) for all algorithms on MNIST. The architecture used in the paper you mentioned consists of a 256-256-256-256-256-10 FC structure. Differences in network architecture can lead to performance discrepancies, and deeper networks typically achieve higher accuracy. For fair comparison, we avoided additional training technologies (like batch normalization, residual connection) as much as possible because potential biases introduced by these additional techniques may have varying effects across different algorithms. Therefore, it is not surprising that our reproduced test performance is slightly lower than theirs.

2. The second criterion is problematic considering the neuro-modulatory signals in the brain that deliver global error signals to neurons.

Thanks for your suggestion. We are aware of the presence of global signals in the brain, which play an important role in modulating certain mechanisms across neurons. However, there is currently no clear biological evidence that these signals represent errors corresponding to the difference between expected and actual outputs, nor that they precisely capture the direction and magnitude required for updating each neuron. We agree that further clarification of this criterion is necessary and will add this discussion into Section 2.1.

3. In the second line of Eq. 15, the derivatives with respect to i+2, …, n are omitted

Thank you for pointing this out. There should indeed be an ellipsis when expanding $\mathcal{L}$ in the second line of Eq.15. The rest of Eq. 15 remains correct.

4. Implementation details of previous algorithms are missing.

As stated in Line 325 (left column), to ensure fairness, we use the same model architecture across all learning algorithms for MLPs, CNNs, and RNNs under a certain dataset. Detailed model specifications and training configurations can be found in Appendix C.2.

5. The authors did not discuss works using apical dendrites for credit assignments in the related work section.

Thanks for your suggestion. While apical dendrites have been discussed in previous literature for credit assignment and have been utilized for various purposes, our study takes advantage of their properties to design and implement more biologically plausible learning algorithms, which differ significantly from existing approaches. J Guerguiev et al. (eLife, 2017) primarily aimed to explain how deep learning can be achieved using segregated dendritic compartments, but it did not propose a specific learning algorithm. As for Sacramento et al. (NeurIPS 2018), we have acknowledged in Line 200 (right column) that we followed their division strategy for pyramidal neurons. A Payeur et al. (Nat Neurosci 2021) investigated burst-dependent synaptic plasticity. While their work shares conceptual similarities with ours in terms of apical dendritic processing, the primary objective of their study differs from ours. Our proposed method not only satisfies all three criteria but also achieves higher performance compared to existing biologically plausible learning methods. That said, we will add the discussion in the related work section. Thank you!

6. If $\Theta$ and $\mathbf{W}$ are initialized to be the same and $ξ_i$ in Eq.9 is replaced with $u_i$, the proposed rule becomes equivalent to backprop. I am not sure if this particular approximation was implemented before.

First, even if $\Theta^T$ and $\mathbf{W}$ are initialized identically, their update directions will differ because $\xi_i = u_i - x_i$ and $x_i$ from $\text{layer}_{i+1}$ cannot be exactly $2u_i$ or $0$ since it comes from lable, so $\xi_i$ and $u_i$ are different in both magnitude and direction. As a result, $\Theta^T$ and $\mathbf{W}$ will update in different directions according to Eq.8 and Eq.9. Furthermore, our formula is derived to minimize the loss, leading to the update formula for $\Theta^T$ in Eq.9. Therefore, replacing $\xi_i$ in Eq.9 with $u_i$ would result in an incorrect update direction. In conclusion, our method is not equivalent to BP.

Second, to the best of our knowledge, there are no existing algorithms that simultaneously satisfy all three criteria while also achieving competitive performance with BP. If you could kindly provide the relevant references, we would be pleased to discuss them in our related work section. Thank you!

7. L088: 'Pyramidal neurons consist of 70-85% of neurons': This statement is only true for the cortex.

Thank you for pointing it out. We will clarify this in our revised manuscript.

Thank you for your comments and suggestions, which are valuable for enhancing our paper. We are pleased that our contributions are well recognized.

Responses to your concerns and questions are hereby presented:

1. Why are the three criteria for biological plausibility important? More ablation studies are needed to prove that the three criteria contribute to improving the model's performance.

Thanks for your suggestion. The three criteria outlined in our paper were summarized from existing biologically plausible learning algorithms, rather than introduced as a means to enhance model performance. Our primary objective is to investigate the biological plausibility of learning algorithms. Based on this investigation and the biological limitations of existing methods, we propose a novel approach that satisfies all three criteria—making it more biologically plausible—while achieving performance comparable to backpropagation. We have conducted an ablation study on the backward weight $\Theta$ in Table 3. Could you please provide any suggestions on how to design further ablation experiments?

2. Gradient descent is still employed for training. Could this be the main reason why DLL outperforms previous work?

Thank you for your insightful comment. We agree that STDP is one of the most biologically plausible learning algorithms, as it not only avoids gradient descent but also satisfies the three criteria we summarized.

From the perspective of update rules, the high performance of our method stems from the local neuronal plasticity update rules we derived (Equations 8 and 9), rather than relying on a global gradient backpropagation process. We believe that local updates to neuronal plasticity are consistent with the form of computing gradients on local losses, which is a plausible mechanism that could have emerged through long-term evolution. Similar approaches have been employed in other biologically plausible learning methods, such as predictive coding, target propagation, and local losses. Although gradient descent is not used in STDP, it does not necessarily mean the neuronal plasticity rules can not be derived locally by using gradients.

1. Performance of BP on CIFAR10 with CNN (75%) is low.

Firstly, as stated in Line 325 (left column), to ensure fair comparisons between different algorithms, we used the same architecture for all methods on a given dataset. Specifically, for CIFAR-10, we employed a CNN with three convolutional layers (see Appendix C.2.2) without incorporating deep learning techniques such as batch normalization, residual connections, or dropout. This design choice was made to maintain full biological plausibility and prevent potential biases that could arise from the selective application of these techniques across different algorithms. Under these conditions, the accuracy achieved by our BP-CNN on CIFAR-10 is reasonable.

Secondly, previous studies like [1][2][3], which also did not incorporate additional training techniques, reported test accuracies of only achieving 60%-70% on CIFAR10 using similar CNN architectures. Our primary objective was to establish a fair and unbiased evaluation framework, ensuring that comparisons reflect the intrinsic differences between algorithms rather than the influence of external training enhancements.

2. Effect of common regularizations like weight decay or batch normalization. Does DLL work for modern architectures like ResNets?

As mentioned in the last question, we did not incorporate additional training techniques. If we were to use them, special design considerations would be necessary, as they may violate certain criteria. For example, batch normalization and dropout behave differently during training and testing, and directly applying them to our DLL framework would break Criterion 3 (non-two-stage training). We acknowledge that incorporating such techniques, particularly ResNets, could be beneficial for future optimizations and will consider them in subsequent work. However, our primary objective in this paper was to explore and evaluate the biological feasibility of the algorithms without relying on external enhancements.

3. How does DLL perform on more complex datasets like CIFAR100 or Tiny-ImageNet?

We followed your suggestions and trained CNNs with BP and DLL on CIFAR-100 and TinyImageNet. Consistent with previous work [1], we report test accuracy for CIFAR-100 and test error rate for TinyImageNet.

Note that we do not use any additional training techniques such as batch normalization or residual connections. Our CNN architecture is similar to that in [1]. For CIFAR-100, the CNN consists of four convolutional layers with channel configurations of 3-64-64-128-64, followed by two fully connected layers. For TinyImageNet, the CNN consists of five convolutional layers with filter configurations of 3-64-64-128-128-64, followed by two fully connected layers.

4. Discussion on convergence properties and guarantees would make the manuscript much stronger.

Thanks for your advice. We will add the following discussion on convergence properties and guarantees: The loss function (Eq. 3) is designed to minimize the discrepancy between the top-down predictions and bottom-up outputs of each pyramidal neuron in the network. To achieve this, we employ local gradient descent–based learning rules and neural plasticity mechanisms to update both forward and backward weights. During each iteration, the differences between the network’s predictions and the ground truth propagate back through localized errors, effectively coordinating all neurons in an orchestrated manner. As a result, neural responses collectively refine predictions over successive iterations, gradually reducing local errors and driving the network toward convergence. While providing formal convergence proofs remains challenging due to the network’s nonlinear operations, our empirical results consistently demonstrate a steady decrease in loss throughout training, supporting the stability and effectiveness of our approach.

5. How can the performance gap between BP and DLL be bridged?

Currently, most biologically plausible learning algorithms exhibit a significant performance gap compared to BP. In this study, our primary goal is to move closer to BP while preserving biological plausibility. If the sole objective were performance improvement, specialized training techniques such as normalization and dropout—adapted to our proposed DLL algorithm—could be explored in future work.

Reference

[1] Bartunov S, Santoro A, Richards B, et al. Assessing the scalability of biologically-motivated deep learning algorithms and architectures. Advances in neural information processing systems, 2018, 31.

[2] Millidge B, Tschantz A, Buckley C L. Predictive coding approximates backprop along arbitrary computation graphs. Neural Computation, 2022, 34(6): 1329-1368.

[3] Salvatori T, Song Y, Yordanov Y, et al. A Stable, Fast, and Fully Automatic Learning Algorithm for Predictive Coding Networks. ICLR, 2024

1. More experimental validation, especially real-time learning, to show simultaneous forward and backward.

Our approach enables real-time learning, as higher-layer neurons propagate signals backward only when a discrepancy between the output and the label is detected. Otherwise, no adjustments are made, and no backpropagation signals are generated. While this mechanism is theoretically sound, we are uncertain about the best way to empirically validate it. We would greatly appreciate any suggestions on how to design experiments to test and confirm this behavior.

2. Accuracy gap on complex datasets.

We primarily focus on comparing the biological plausibility of various learning algorithms rather than optimizing for high performance. Within these biological constraints, we propose the DLL algorithm, which further narrows the performance gap with BP. As shown in Table 1, DLL achieves the best performance among algorithms that satisfy all three criteria.

3. Theoretical analysis for convergence.

Please refer to the 4th question of Reviewer 2B1A.

4. Complex datasets like ImageNet or NLP tasks are ignored.

Given our current computational resources (four 2080 GPUs with 12GB each), we estimate that conducting experiments on the full ImageNet within 7 days would not be feasible. Therefore, we chose to evaluate our methods on Tiny-ImageNet. The results are presented in our response to the third question from reviewer 2B1A.

For NLP tasks, we have followed [1] to conduct experiments on the "next-character-prediction" task in Section 4.3. Additionally, we performed experiments on the text classification datasets Subj and Movie Review (MR), with results summarized below:

The architecture is the same as the original TextCNN (Kim, EMNLP2014).

5. Computational cost, memory efficiency, or sensitivity to hyperparameters

The time consumption and memory usage for these experiments are summarized below:

To fairly compare time consumption across architectures, we used the CPU instead of the GPU. DLL requires more training time and memory because both the forward weight $\mathbf{W}$ and backward weight $\Theta$ are updated simultaneously. Our design is not driven by computational or memory efficiency; rather, we prioritize biological plausibility.

As for sensitivity to hyperparameters, we have included a comparison of various learning rates and sequence lengths in Figure 3.

6. The authors do not establish whether DLL optimizes a well-defined loss function in a way that guarantees stable learning

Figure 2(c) shows that models trained with DLL exhibit a stable decrease in training loss. This figure is based on time-series forecasting experiments using RNNs on the Electricity dataset. Similarly, for MLPs and CNNs, the training losses also show a consistent downward trend. We will release all training logs upon acceptance.

7. No ablation study to determine which aspects of DLL are most responsible for its performance.

Our DLL algorithm is designed based on three criteria that we have summarized from current biologically plausible algorithms. It can be viewed as a faster (without iteration) and more biologically plausible implementation of predictive coding (PC), leveraging the unique properties of pyramidal neurons. Our design removes weight symmetry and separates forward and backward computations. As a result, the performance of DLL is similar to that of PC under certain conditions, and PC has been shown to achieve performance comparable to backpropagation [1][3]. We have conducted an ablation study on the backward weight $\Theta$ in Table 3. We would appreciate any suggestions on additional ablation studies.

8. Code lacks documentation. No details about hyperparameter selection.

Upon acceptance, we will release our code with detailed documentation for reproducibility. The selection of hyperparameters follows the similar choices made in [1][2].

9. Discussion of other learning methods.

We will add related papers on apical dendrites. Please refer to the 5th question of reviewer Crph.

10. Robustness and scalability?

Robustness is an interesting aspect, and we plan to explore it in future work.

We conduct scalability experiments:

All MLPs are trained fairly, and the results show the scalability of DLL.

[1] Millidge B, et al. Predictive coding approximates backprop along arbitrary computation graphs. Neural Computation, 2022.

[2] Bartunov S, et al. Assessing the scalability of biologically-motivated deep learning algorithms and architectures. NeurIPS, 2018.

[3] Salvatori T, et al. A Stable, Fast, and Fully Automatic Learning Algorithm for Predictive Coding Networks. ICLR, 2024.

1. Poorly supported claim of 3 criteria without any citation. There are many other criteria explored in prior work.

Firstly, in Section 2.2, we provide a detailed introduction and evaluation of all representative biologically plausible learning algorithms. These algorithms served as the basis for the three criteria, as mentioned in Line 33 (left column). However, we acknowledge that we should have included more explicit citations before discussing these criteria in Section 2.1, and we appreciate the reviewer’s suggestion in this regard.

Secondly, our work specifically focuses on biologically plausible supervised learning algorithms. For instance, when discussing Hebbian Learning and STDP, we highlighted their limitation in leveraging supervised learning signals (Line 188, left column). While we acknowledge that studies such as [4][5] have proposed alternative criteria—such as self-supervised learning and independence between layers—we emphasize that these approaches are often based on unsupervised learning. We fully recognize the value of these criteria in assessing biological plausibility; however, it is challenging to incorporate all possible criteria within a single paper.

To address this, we will add a discussion and cite relevant studies in our revised manuscript to clarify our motivation. We appreciate the reviewer’s insightful feedback.

2. Ignorance of recent related work [3][4] on local learning. Local learning methods satisfy C3 naturally as there is no end-to-end update.

Firstly, as mentioned in the last question, our work focuses on supervised learning algorithms. [3] proposed a layer-wise self-supervised pre-training algorithm based on random masking and image recovery, while [4] explored deepening model layers within unsupervised contrastive learning frameworks. Although these methods help scale local learning approaches to larger settings, their biological plausibility is limited, as they do not fully adhere to Criteria 1 and 3. In this study, our primary goal is to take a step closer to BP while maintaining biological plausibility. To achieve this, we designed the DLL algorithm based on three key criteria derived from existing biologically plausible learning algorithms. That said, we acknowledge the relevance of [3] and [4] and will incorporate a discussion of these works in Section 2.2 and the related work section.

Secondly, we define Criterion 3 as non-two-stage training, meaning there is no strict temporal segregation between forward and backward processes. We believe that not all local learning methods satisfy this criterion. For instance, difference target propagation is a local learning method but does not meet Criterion 3, as we discussed in Line 143 (right column). While the training methods in [3] and [4] do not involve end-to-end updates, they still exhibit a clear separation between forward and backward processes.

3. Writing suggestions: Line 263 and Line 318.

Thanks for pointing them out. We will correct them in our revised manuscript.

4. Why is the performance of the selected architectures poor on CIFAR-10?

Please refer to the first question of reviewer 2B1A.

5. How were the architectures selected? How were the hyperparameters tuned? Were they the same for all methods?

Yes, as stated in Line 325 (left column) and Appendix C.2, we ensured that all algorithms were evaluated using the same hyperparameters and architectures for a given dataset. Our choices for architecture and hyperparameter settings were based on the methodologies outlined in [1][2].

6. Can this method be adapted to residual networks?

We recognize that there is biological evidence supporting inter-regional and inter-layer neuronal connections. However, whether these connections function equivalently to residual connections in deep learning remains an open question. Residual connections were originally introduced to mitigate the gradient vanishing problem in deep learning models, which is less of a concern for local learning-based approaches like ours. As a result, their potential benefits may not be as relevant or impactful in our setting.

We acknowledge the importance of exploring such architectural enhancements. In future work, we will consider incorporating residual connections to further investigate their potential impact and applicability.

Reference

[1] Bartunov S, Santoro A, Richards B, et al. Assessing the scalability of biologically-motivated deep learning algorithms and architectures. NeurIPS 2018.

[2] Millidge B, Tschantz A, Buckley C L. Predictive coding approximates backprop along arbitrary computation graphs. Neural Computation, 2022, 34(6): 1329-1368.

[3] Siddiqui S, Krueger D, LeCun Y, et al. Blockwise Self-Supervised Learning at Scale. TMLP.

[4] Xiong Y, Ren M, Urtasun R. Loco: Local contrastive representation learning. NeurIPS, 2020