Counter-Current Learning: A Biologically Plausible Dual Network Approach for Deep Learning
Authors: Chia Hsiang Kao
Bharath Hariharan
Venue: NeurIPS 2024 poster (2024)
Keywords: biologically plausible algorithm
backward locking problem
biological inspired algorithm
target propagation
Primary area: deep_learning_architectures
TL;DR: We propose counter-current learning, a biologically inspired dual network architecture that facilitates local learning and addresses weight transport, non-local credit assignment, and backward locking issues in backpropagation.

Abstract: Despite its widespread use in neural networks, error backpropagation has faced criticism for its lack of biological plausibility, suffering from issues such as the backward locking problem and the weight transport problem. 
These limitations have motivated researchers to explore more biologically plausible learning algorithms that could potentially shed light on how biological neural systems adapt and learn. 
Inspired by the counter-current exchange mechanisms observed in biological systems, we propose counter-current learning (CCL), a biologically plausible framework for credit assignment in deep learning. 
This framework employs a feedforward network to process input data and a feedback network to process targets, with each network enhancing the other through anti-parallel signal propagation. 
By leveraging the more informative signals from the bottom layer of the feedback network to guide the updates of the top layer of the feedforward network and vice versa, CCL enables the simultaneous transformation of source inputs to target outputs and the dynamic mutual influence of these transformations.
Experimental results on MNIST, FashionMNIST, CIFAR10, CIFAR100, and STL-10 datasets using multi-layer perceptrons and convolutional neural networks demonstrate that CCL achieves comparable performance to other biological plausible algorithms while offering a more biologically realistic learning mechanism. 
Furthermore, we showcase the applicability of our approach to an autoencoder task, underscoring its potential for unsupervised representation learning.
Our work presents a promising direction for biologically inspired and plausible learning algorithms, offering insights into the mechanisms of learning and adaptation in neural networks.

Counter-Current Learning: A Biologically Plausible Dual Network Approach for Deep Learning
Chia-Hsiang Kao
Bharath Hariharan
Cornell University ck6960cornell.eda
Cornell University
bharathhecs. cornell......o.oo cornell edu
Abstract
Despite its widespread use in neural networks, error backpropagation has faced criticism for its lack of biological plausibility, suffering from issues such as the backward locking problem and the weight transport problem. These limitations have motivated researchers to explore more biologically plausible learning algo- rithms that could potentially shed light on how biological neural systems adapt and learn. Inspired by the counter-current exchange mechanisms observed in biologi- cal systems, we propose counter-current learning (CCL), a biologically plausible framework for credit assignment in neural networks. This framework employs a feedforward network to process input data and feedback network to process targets, with each network enhancing the other through anti-parallel signal propa- gation. By leveraging the more informative signals from the bottom layer of the feedback network to guide the updates of the top layer of the feedforward network and vice versa, CCL enables the simultancous transformation of source inputs to target outputs and the dynamic mutual influence of these transformations. Exper- imental results on MNIST, FashionMNIST, CIFARIO 10, and CIFAR 100 datasets using multi-layer perceptrons and convolutional neural networks demonstrate that CCL achieves comparable performance to other biologically plausible algorithms while offering more biologically realistic learning mechanism. Furthermore, we showcase the applicability of our approach to an autoencoder task, underscoring its potential for unsupervised representation learning. Our work presents direction for biologically inspired and plausible learning algorithms, offering an alternative mechanisms of learning and adaptation neural networks,
Introduction
In deep learning, biological plausibility refers to the properties that deep learning algorithms could respect avoid inconsistency with current understandings of neural circuitry violation of funda- mental physical constraints, such as the localized nature of synaptic plasticity [Grossberg 1987 Crick
1989.. Consequently, error backpropagation (BP), despite its wide application, has been frequently criticized for its lack of biological plausibility, particularly for the following three challenges: (a)
The weight transport problem, which arises because BP requires the feedback pathway to use the same set of weights as the feedforward process, a mechanism not observed in biological systems
[Burbank and Kreiman, [2012] Bengio et al.) 2015 Lillicrap al. 2016). (b) The non-local credit
assignment problem arises because backpropagation relies on the global error signal to update the
synaptic weights throughout the network, instead of depending on local errors derived from local loss
"Code available wtttt////ggtaaa..co/aa/d///////e
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

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computation (c) The backward locking problem occurs because, in BP, each data sample must await the completion of both forward and backward computations of the previous sample, impeding online learning capabilities [Jaderberg et al. 2017 Czarnecki et al. 2017]. These limitations have propelled
the development of alternative credit assignment methods that aim to better align with biological principles and address these significant issues [Lillicrap et al.) [016)] Crafton et al. 2019 Launay
ct al.
2020 Nøkland 2016, [engioo| 2014| Lee et al. 2015] Ororbia and Mali, 2019 Meulemans et al.
2020 2021
Dellaferrera and Kreiman 2022 Shibuya et al. 2023].
Reaching Biological Plausibility With Dual Network Structure. To address the weight transport problem, we leverage a dual network architecture for processing feedback signals, which uses a different set of weights from the forward network. To tackle the non-local credit assignment issue, we use pairwise local loss, computing the difference in layerwise activations between the feedforward and feedback networks, and ensuring the local loss only updates local weight parameters through gradient detaching. We approach the backward-locks -oocing problem by preventing the feedback network from reusing latent activations and output signals from the feedforward networks. Since the feedback network operates independently of the output (prediction), the forward and feedback processes can occur simultaneously. These enhancements make our approach not only more biologically plausible but also potentially more effective in complex scenarios.
Analogy to Biological Counter-Current Mechanism. Our scheme draws inspiration from nature's counter-current exchange mechanisms, observed in fish gills, animal vessels, and renal systems. These physiological mechanisms use an aati-prallll structure to optimize resource or energy exchange between two flows. Similarly, our dual network learning scheme allows the input signals in the forward network, flowing from input space to target space, to receive target domain information from the aaget-to-source signal flow (in the feedback network) and reciprocally share their source information. Therefore, we name our learning scheme "counter-current learning," as this reciprocal exchange mirrors the efficiency and optimization seen in biological systems.
Contributions. This paper aims introduce counter-current learning as novel, biologically plausible alternative. We validate our approach through experiments on MNIST, FashionMNIST. CIFARIO, and CIFARI00 using MLP or CNN architectures. Additionally, we demonstrate the effectiveness of our model in autoencoder- application of biologically plausible algorithms in this area. Our approach addresses key challenges. including weight transposition and non-local credit assignment, and partially mitigates the backward locking, providing a promising avenue for advancing biological plausibility.
tasks, which, to our knowledge, representn the first
2 Literature Review
Addressing Biological Plausibility. The target propagation (TP) family [Ben- gio 2014) and its variants (c.g.. local target representations) [Ororbia et al. 2018 2023 first explored in the late 1980s [Le Cun 1986 Le Cun and Fogelman-Soulic 1987.. have been developed
Target Propagation:
to optimize the neural networks by using locally generated error signals. TP explicitly constructs local targets for each layer using a separate feedback network. Take difference target propagation (DTP) [ee et al. (2013) for example, an idealized global target signal is computed based on the labels and the prediction error the output layer. Then, the local idealized targets are generated by (1) propagating the idealized global targets through the feedback network and (2) computing a linear correction using the activations from the forward network. Subsequently, local losses are computed by comparing the layer activations with their corresponding local targets. The weights of both the forward and feedback networks are updated based on these local losses. Notable variants such Direct Difference Target Propagation (DDTP) [Meulemans et al.) 2020). Local- Differrnce Reconstruction Loss (L-DRL) |Ernoult et al. 2022]. and Fixed-Weight Target Propagation Shibuya al. 2023 further refine this approach by introducing mechanisms to improve feedback weight training and enhance the accuracy of local error signals. Despite these advancements, TP methods still encounter the backward locking issue, since TP methods depend the forward network's outputs and intermediate activations to compute targets. Moreover, recent iterations of TP algorithms, such as DDTP and L-DRL, can be computationally expensive. They require additional feedback
In biological systems, synaptic plasticity is believed to be governed by local learning rules, such as Hebbian learning. where synaptic changes depend on the correlated activity of the pne post-synaptic neurons Dan and Pool 2004. Bartunov et al. 2018 Whittington and Bogacz 2019

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Network
Targety
Targuty
N2
Forward Network
stop gradient operstor Lgrndient dedach)
80
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Source
Output
Source
Outgut
Before Training
Counter-Cureets Learning
source
melsy
target
Info
Figure 1: Overview of the Counter-Current Learning Framework: (a) At initialization, the counter-current learning framework establishes dual network structure, with a forward network that maps the input to the target output, and a complementary feedback network that mirrors the forward network's 's architecture but propagates information in the opposite direction. The framework leverages the data processing inequality (DPI) from information theory, which states that information content cannot be increased through signal processing. Consequently, both networks, information content decreases from the lower to the upper layers. (b) During training, the losses are computed in layer-wise manner, i.e., by calculating the difference of activations from corresponding layer pairs between the forward and feedback networks, allowing the networks to learn from each other's s complementary information. Notably, the dependency of the gradient on earlier layer parameters interrupted using the gradient detachment operator.
weight update loop per data batch, leading to a three -old increase training time compared to traditional backpropagation [Meulemans et al. 2020 Ernoult et al.) 2022 Shibuya et al. [023]]
Other Efforts in Enhancing Biological Plausibility. In addition TP, several other methods have been proposed to overcome the biological implausibility of traditional backpropagation. The
feedback alignment (FA) [Lillicrap et al.) 2016 Nokland [2066] Crafton et al. 2019 Launay et al.) 2020 Refinetti et al. [021] family uses random feedback weights, instead of the transpose of the weight in the feedforward layer, to approximate the error gradient, thereby eliminating the need for precise synaptic symmetry. To resolve the backward locking problem, direct random target projection (DRTP) [Frenkel al. 2021] proposed to randomly project the target signals each layer as ideal targets. While achieving biological plausibility. DRTP encounters a significant performance drop concerning BP compared to FA algorithms Frenkel et al. [021], Dellaferrera and Kreiman [2022]. Block-local learning (BLL) Kappel et al.] [2023] explores block-wise target signal propagation; however, the algorithm requires backpropagation update layers in the same block. Please refer to Section 6.5 for more literature review.
3 Counter-Current Learning Framework
In this section, we present the counter-current learning (CCL) framework, as shown in Figure [ focusing on its formulation and key components.
Setup and Feedforward Network. Consider input space X' and output space V. each with dimensions do and di, respectively. The objective is to learn mapping F X' - y that minimizes the discrepancy between the predicted output and the target. We adopt an L-layered feed-forward neural network with activation function Let 9r(-) denote the operation layer I, and define Fjw = gL o -L-1 o... o 91- Each gi is parameterized by weights U,. The output of layer is a (a,-)) (Ujo--1), where an
Feedback Network. The proposed learning scheme introduces complementary backward function F that mirrors Fi in an anti-parallel manner. Fr.. comprises layers [/,,,... ..)), with each by parameterized by weights Vi. We define Frw = h1 o... hr. The output of layer by = hr(b)) = (Vibi), with by y. The dimensions of hidden layers align between Fyw and Fb..
3

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from torch. un import Linear, Module import torch.mn functional
1 1iro torchrision. datasets import MNIST
torch.utila.de import Dataloader 3 3fron mn functionnll import one_hot 4fron torch inport optin
clasa C2Model (Module):
def
iinitt... (self);
super (C2Nodel, self) -__ 1nit__0
train_cri_ sttp (model, inputa, labels):
self enc1 C2Linear 256)
7 Iv_actvv model.fv_pass fa.... (inputt,,, True)
self.enc2 C2Linear (258, 20)
8 bv_actve model. www.paupppaaaeeeeeeeee. (labels, True)
self C2Linear (20, 10)
loss
10
def
fv_pazs x, detach):
for a, xip(fv_actvs, bu_actvs):
s1 self encl (x, detach)
loss F.mse_losass,
12
=2 solf. enc2 fa.paan(al, detach)
12 return lass
self enc3.nc.ccc.... fac3.fa_paasaa2,, detach)
14
return Dx, at, a2, a3]
def train_ (rss___P_ttepmoddil. inpats, labels):
def bu_pass(sell) target, detach)::
model. (inputs, False)
16
b2 self, enc3. bv.pasa(target: detach)
16
return (.nee_lonuttvaacttt]] labels)
b1
aec2.bx_paaa(b2) detach)
18
bo self encl bw by.pass(bl, detach)
18 train_datasee HNIST(root= /data')
[bo, bl, target]
Datalosder dataset) 20 model C2Model(()
20
21
(Minnear(oouulc):
optimizer Adannnol p
22
def
__iint___eelle in in_dims out_dims): super (C2Linsar, solf) innft....nit__..)
22 for inputs, labels in dataleader:
23
inputs vinptss
24
self. iv_laer Linear (in_dii,,,,, out oan_dins)
24
Por CCL
self. bellyer
(out_dims, in_dims)
ona_hot(labels, fo)att)
26
daf
fu_paas (self, detach):
26 loss train_C0___ttopnnoddl. inputa, labels) For BackProp
27
detach: d)
28
return F. F.elu(selfffv. (x,]aaye(x))
loss trass_8P. BP..ste inpsta, labets)
29
def bv.pase x, detach):
For both and BackProp
30
if detach: xx ()tacch)
loss. backvard()
31
return F.elu(self. bw bx_layer(x))
optimizer. steep()
Figure 2: Code Snippet For Counter-Current Learning With Dual Network Architecture.
Stop Gradient Operation. To address the backward locking problem and ensure local synaptic learning, we use the SG() () operation to decouple activations from weights in previous layers, disrupting the long bacoooopppppotoiionn chain into local update segments. The SG can be implemented using Py gradient detach operation easily. In CCL, each layer's input is processed with the the hat symbol to denote the exact activations in the CCL
SG operation. To avoid confusion, paradigm. Specifically, for I L:
à (â;-1) = o (U)SG(((-1),
by = hx(bi) = o (Visa((ii),
where åg x and by = y-
Loss Objective Function. The objective of the counter-curren learning algorithm is to minimize the difference between activations of l' and Fmm across all layers but the output layer. To avoid trivial solutions where activations converge zero, first reshape the activations into vectors and normalize them along the feature dimension using norm). This normalization ensures meaningful alignment between forward and feedback passes:
min [ (n)mm()))oom(()))) /||,
(2)
where == (U1....,UL-- VI..... Vi} are learnable parameters, and is the identity matrix. For the output layer (1 = L), the cross-entropy loss between the network outputs â and the one-hot encoded labels bg. y.
different objective: the weight matrix Uy. is trained to minimize
Biological Plausibility. We examine the biological plausibility of the proposed counter current learning scheme. This framework mitigates the weight transport problem by using different weight parameterization for the feedback network. For the non-local credit assignment problem, the update of the parameters is driven by local loss, instead of the back pockppropggtted global error signals. Finally, we partially address the backward update problem by removing the dependency the backward network and the forward network with careful gradient detachment.
we present a code snippet for the CCL algorithm in PyTorch, tailored for MNIST classification. It shows the independence of the forward process and the feedback
Implementation. In Figure 2

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Before training
After 10 CCL updates
After 100 CCL updates
After 1000 CCL updates
Figure Dynamic Feature Alignment Between Forward and Backward Models During Counter- Current Learning. This series of t-SNE plots demonstrates the evolution of feature space alignment over different stages of training. Circular dots represent features from the forward network processing MNIST images, while squares depict features from the feedback network handling one-hot encoded labels. Each color represents a distinct class, with every subplot providing an independent t-SNE visualization. This emphasizes how distinct classes increasingly converge within and across the forward and backward models as training progresses, highlighting the dynamic and reciprocal nature of learning within the counter-current framework.
(backward) process from each other. The main C2Model module comprises three C2Lineat layers, each inherits from nn Module and consists of fw_pass and bw bw_pass for forward propagation and backward propagation, respectively. These functions accept an additional Boolean input, detach, allowing for the quick toggling between non-local (i.e., BP) and local learning (i.e., CCL) modes. For loss computation, the train_ Frrin_ccL_ste CCL_step function calculates the loss for counter-current learning. Conversely, the train_BP fian__P_step step function computes the loss for error backpropagation.
Experiments
4.1 Counter Current Learning Facilitates Dual Network Feature Alignment
To investigate the alignment of latent features within the counter-current learning framework, we visualized embeddings from the penultimate layer of the forward network and the corresponding second layer of the feedback network at various stages of training. We employ a six-layer neural net trained MNIST and analyze embeddings from both networks. t-SNE was applied independently to these embeddings at each training iteration to effectively visualize the evolution of feature spaces.
As illustrated in Figure 3 the embeddings from the forward model trained MNIST data represented by colored dots, while the embeddings from the backward model related to one-hot encoded labels denoted by outlined squares of the same color. Throughout the training process, embeddings from the same class progressively align between the forward and backward models, suggesting that the forward and backward models mutually guide each other's s feature representations toward a coherent and discriminative structure. Please refer to Appendix 6.3 for visualization of feature alignment more layers and Appendix for experiments on weight alignement between forward and backward layers.
4.2 Classification Performance
Task Setup. We evaluate the performance of our proposed method against several biologically plau- sible algorithms, including direct target propagation (DTP), DTP with difference reconstruction loss (DRL), local difference reconstruction loss (L-DRL), and fixed-weigh difference target propagation (FW-DTP). The evaluation is conducted on MNIST, FashionMNIST, CIFAR-10 10, and CIFAR- All experiments are performed using stochastic gradient descent optimization with 100 epochs. We use cross-validation over 5 different random seeds and report the testing performance. The models are implemented using the Py Torch deep learning framework, and the code available in the Supple- mentary Material. For the experiments on multi-layer perceptrons, we apply image normalization as a preprocessing step. For the convolutional neural network experiments, we use additional data augmentation techniques, including cropping and horizontal flipping. For the counter-current learning (CCL.) algorithm, we search across different learning rates and gradient norm clipping values find the optimal hyperparameters following cross validation, detailed in Appendix 6.1
5

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Table 1: Test performance MNIST, PashionMNIST, CIFAR1O, and CIFAR100, evaluated using multi-layer perceptrons. Performance metrics are reported for error backpropagation (BP), feedback alignment (FA), target propagation (DTP), DTP with difference reconstruction loss (DRL), local difference reconstruction loss (L-DRL), fixed-weight difference target propagation (FW-DTP), and cross-correlation loss (CCL). Best values per task are bolded, and second-best values are underlined.
MNIST
FASHIONMNIST
CIFARIO
CIFARI00
BP
RUMELHART ET
1986
98.19 + 0.10
89.58 0.25
50.03 + 0.31
22.55 + 0.19
FA
LILLICRAP ET AL.
2016
96.96 1 0.05
87.38 + 0.12
45.76 + 0.38
22.13 1 0.41
DFA
NØKLAND 2016
97.27 + 0.06
87.35 + 0.99
42.86 1 1.94
19.87 1 1.50
DRL
MEULEMANS ET AL.,
20220]
93.05 1 0.24
83.40 1 0.18
1.27
19.94 0.28
D-DRL ET AL.
2022
93.29 0.21
83.60 = 0.20
42.19
19.96 0.27
FW-DTP ET AL.
[2023
97.20 = 0.16
87.78 = 0.47
45.91
21.09 0.31
DRTP FRENKEL ET AL.
DRTP
[021
92.16 = 0.18
82.03 = 0.56
33.85
15.53 0.33
CCL (OURS)
98.13 + 0.10
88.58 0.29
52.73 + 0.59
21.76 + 0.22
Multi-Layer Perceptrons (MLP). We follow Shibuya et al. [2023 for experimental setup and hyperparameter selection. For MNIST and FashionMNIST, we employ a fully connected network with 6 layers, each having 256 units. For CIFAR-10 and CIFAR- 100, we use a fully connected network with layers, each containing 1,024 units. We use the hyperbolic tangent activation function for BP and TP variant algorithms, while CCL adopts an ELU activation function. The results for ML.Ps are shown Table which demonstrates that the CCL. obtains comparable results to error backpropagation and bears consistency with other biologically plausible algorithms.
Table Computational Complexity Comparison on MNIST and CIFARIO. The computational efficiency of various learning algorithms is evaluated by measuring the estimated floating-point operations (FLOPs) per sample for single training cycle. All measurements were conducted with batch size of 32 samples. Values are reported in millions (M) of FLOPs, with the best performers highlighted for each dataset.
MNIST 6 LAYERS
CIFARIO LAYERS
ARCHITECTURE
BP
2.39 M
25.23 M
DTP
48.69 M
485.59 M
DRL
125.73 M
833.81 M
L-DRL
82.45
770.36
FWDTP-BB BN
18.06 M
170.08 M
CCL (OURS)
2.55 M
27.89 M
Table Test Accuracy on CIFARIO and CIFAR100 Using Convolutional Neural Network. The metrics are reported for error backpropagation (BP) and cross-correlation loss (CCL).
CIFARIO
CIFAR100
BP
87.12 11.76
51.92 +0.48
CCL (OURS)
82.94 +0.53
56.29 +0.25
MLP Runtime Analysis. We analyze the computational efficiency of various learning algorithms by measuring doating-poin operations (FLOPs) per sample. The FLOPs estimation is performed using torch.profiler on the complete training cycle: forward (and feedback) propagation, loss computation, and backward propagation. Backpropagation (BP) serves the baseline for comparison. Feedback Alignment (FA) and Direct Random Target Propagation (DRTP) are excluded from the analysis due to their algorithmic similarities with BP. All measurements conducted with batch size of 32
"Codebase: http:://gggii.oo////aaaaaaaaaaaa Weight weiehee-ddeeegghg---fffeeeooe Difference- Target-Proppant tree/main

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Before training
After training with BP
After training with CCL
Figure 4: Visualization of the First Layer Convolutional Kernels of the Forward Model Trained with Error Backpropagation (BP) and Counter-Current Learning (CCL). Kernels from models trained with BP have more high-frequency components, as manifested as neighboring white (e.g.. weight with high values) and black pixels (e.g.. weight with low values). In comparison, those with CCL have more low-frequency components. We posit this might be because the enror signal can contain more high-fre high-frequency information than the ideal target signal.
samples. The results in Table 2 show that CCL consistently outperforms the TP family algorithms in terms training time. Note that the forward and feedback process for CCL performed scquentially in this experiment.
Figure 5: Qualitative Comparison of an Eight-Layers Convolutional Autoencoder Trained Using Error Backpropagation (BP) and Counter-Curren Learning (CCL). The network structure does not contain skip connections. Testing set reconstruction results highlight CCL's comparable reconstruction as BP while achieving biological plausibility.
Convolutional Neural Network. We also evaluate CCL convolutional neural networks (CNN) consisting of five convolutional layers (each with a kernel size of and a max-pooling operation with kernel size of 2) followed by a linear classifier, tested on CIFAR-10 and CIFAR-100. As shown in Table 3 our CCL-based model performs comparably to, or slightly inferiorly than, the BP-based model. Additionally, we visualize the kernels in the first convolutional layer inspect the learned representations in Figure demonstrating that CCL enables the model to learn meaningful convolutional kernels without using error backpropagation. Furthermore, we compare our CNN results on CIFAR-10 10 with those of L-DRL,Emooe et al. [2022) in Appendix 6.2 while FW-DTP Sh-byya al. 2023
does not include CNN implementations.
4.3
Auto-Encoder Performance
We explore the applicability of the counter-current learning (CCL) algorithm to autoencoders.
Auto-Encoder STL-10 Dataset. A convolutional autoencoder with four-layer encoder and a four- layer decoder is used. Different from the classification tasks, this architecture replaces the 2x2 kernel max-pooling with convolution layers with stride of 2. Batch normalization is applied following each linear projection and before activation functions to ensure both stability and optimal performance. The hidden layers of the network are structured with dimensions of [128, 256, 1024, 2048). Orthogonal weight initialization is used for training stability. Data augmentation techniques such as random incorporated. Hyperparameters including gradient clipping,
cropping and horizontal flipping learning rate, momentum, and weight decay are subjected to grid search, while cross- validaaaoo across five different seeds is employed to assess the reconstruction L.2 loss.
Results. The test set's reconstruction metric-mean mean square error-is quantified as 0.0059 + 0.0001 for BP and 0.0086 + 0.0001 for CCL. The outcomes, illustrated in Figure underscore the models'
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proficiency on the test set. While both BP and CCL adeptly capture the general image structure, occasional artifact introduction, such as blurring, is observed in CCL compared to BP. This suggests that while CCI. augments certain facets of autoencoder training, further refinement or architectural adjustments may imperative to minimize visual artifacts and enhance detail preservation.
4.4 Empirical Analysis of Learning Dynamics for Counter-Current Learning
In this section, we provide insights into the functioning of the proposed Counter-Current Learning (CCL) algorithm by examining the representation similarity between the forward and feedback networks. We start by analyzing the feature similarity between randomly initialized feedforward and feedback models. Following this, we focus on the feature alignment in the high-level feature regime. Our investigation reveals the emergence of a reciprocal learning structure, where the top layers of both networks benefit from the bottom layers of each other during training.
We trained model on the MNIST classification task using a configuration of five convolutional layers topped with a linear classification head. The training was conducted over 160 steps with a batch size of 32, achieving an average testing accuracy of 88.88%. To measure the cross-network representational similarity, we utilized Centered Kernel Alignment (CKA) Kornblith et al. (001)) metric known for its robustness to invertible linear transformations. This was applied evaluate layer and architecture similarity. The results, obtained using five different seeds, are presented as averaged CKA values on the test set. Figure 6 displays the horizontal axis marking the activations of
forward layers, ranging from the input MNIST images to the logits (i.e., ag), whereas the vertical axis denotes the activations of the feedback network starting from one-hot labels (i.e., target).
Observation 1: Initialization Shows Noisy Top Layers and Misalignment Between Networks. At initialization, our premise that the bottom layers contain more relevant information is validated. The initial CKA (t = O, top-left subplot in Figure(6) reveals low similarity between the ag column (i.e., the top layer of the forward network) and the feedback network. Similarly, the bo row (i.e., the top layer of the feedback network) shows low CKA values with the forward layers, confirming our hypothesis of feature misalignment at random initialization.
Observation 2: Alignment of High-Level Features During Training. As training progresses, high-level features (i.e., the top layers of the forward network and the bottom layers of the feedback network) begin to show increasing CKA values (Figure t = 20 to t = 160, top row). This trend suggests that the forward and feedback networks gradually align their high-level representations.
Observation 3: Emergence of a Counter-Curreds Reciprocal Structure. The dynamics of the counter-current learning algorithm reveal significant changes in CKA. particularly at higher layers (illustrated in the bottom subplots of Figure 6). Notably, the increases are concentrated in the d4, as, 4g columns of the forward network and the bo. b1. by rows of the feedback network. This pattern supports the counter-curren intuition that top layers benefit from the more informative bottom layers, fostering reciprocal learning structure that guides each network's 's optimization process using the most informative features available.
This empirical analysis underscores the hypothesis that the counter-current learning scheme effectively leverages complementary and reciprocal alignment of representations between the forward and feedback networks. By exploiting the informative features from the bottom layers of one network to refine the noisy features the top layers of the other, the counter-current signal propagation algorithm achieves biologically plausible and efficient learning mechanism.
4.5 Ablation on Asymmetric Learning Rates for Dual Network Optimization
We delve deeper into the effects of varying learning rates between forward and feedback networks in CCL. As illustrated in 7 our experiments confirm that asymmetric learning rates in forward and feedback networks facilitate effective and robust learning. Particularly noteworthy is our observation that robust learning outcomes are achievable even when the feedback network operates under fixed random setting sppeificclly,, with a zero learning rate (refer to the bottom rows of the figure). This suggests that the random projection of target labels by the feedback network conveys meaningful target domain information, echoing findings from the DRTP (Frenkel et al.) 2021 Moreover, our experiments indicate superior performance compared to the DRTP approach (refer to Table [T). possibly hinting that random, non-linear neural network projection (i.c., CCL with a feedback
8

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== 0
t = 20
40
t 80
t= 160
target
b5
b4
CKA
b1
bo
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Feedback net
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Figure 6: Chunter-Current Signal Propagation Enables Learning Through Reciprocal Repre- sentation Alignment. (Top) Centered Kernel Alignment (CKA) between the forward and feedback networks during training. At the initial training step (1 0), cross-network CKA minimal, suggest- ing low similarity between networks. As training progresses, CKA significantly increases, especially in the top layers of both networks, indicating high similarity in learned high-level representations. (Bottom) Changes in CKA between consecutive training steps (i.e., from step step t + reveal significant increases in the top layers of both networks, consistent with our counter-current learning insights. Notably, increases are concentrated in the ag. A5. aş columns of the forward network and in the bu. b1. by rows of the feedback network, as highlighted by the green dotted box. These changes align with the expected reciprocal and complementary learning dynamics.
MNIST Test Accuracy
CIFAR10 Test Accuracy
Feedback Network Learning Rate
0.9
05.77
96.81
97.47
97.87
98.07
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Forward Network Learnnng Rate
Forward Network Learinng Rate
Figure 7: Learning With Asymmetric Learning Rates in the Networks. We investigate the influence asymmetric learning rate in the forward and backward MLPs. This study demonstrates that effective learning can occur with asymmetric learning rates, even when the feedback network has a fixed random configuration (i.e., the learning rate for the feedback
zero).
learning rate of zero) is more beneficial than a mere random linear projection (i.e., DRTP) of labels. This could be due to the neural network's ability to maintain and leverage more inductive biases, which can be crucial for the hierarchical learning processes.
5 Conclusion
In this paper, we introduced the counter-current learning framework, a novel approach addressing the critical limitations of traditional error backpropagation. Our dual network architecture enables

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dynamic and reciprocal interaction between feedforward and feedback pathways, supporting local learning and effectively resolving the backward locking problem. Our learning framework is validated across a diverse set of datasets including MNIST, FashionMNIST, CIFAR10, CIFAR 100, and STL100, and autoencoder tasks, demonstrates comparable performance with existing learning methods without compromising learning speed. This underscores the potential of our model to efficiently handle complex neural tasks and highlights its suitability for broader applications.
Limitations and Future Directions. We acknowledge several limitations that can guide future research in this field. Firstly, there is need for further theoretical insights into the counter-current learning scheme, focusing on its learning dynamics, stability analysis, and inductive biases. Secondly, continuous exploration of this dual model architecture is essential, such as integrating residual connections or self-attention modules. Thirdly, hardware acceleration to streamline berward-feedbuck computation through parallelization could potentially reduce computation time and yield improved results within the same time frame. Lastly, exploring the integration of counter-current learning with other biologically plausible alternatives holds promise for advancing research in this domain.
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6 Appendix
6.1 Experimental Setup
Hyperparameter Search. For experiments with MLP architectures, we conduct hyperparameter searches for each algorithm. We run all the combinations of the hyperparameters with 5 different ran- dom seeds and then select the hyperparameter set with the highest accuracies (or lowest loss for auto- encoder task) evaluated on the validation set and test on the testing set. For DTP, DRI., L.DRI,, and FWDTP, we search across forward learning rates [0.3, 1, 3]. step sizes [0.001, 0.003, 0.01, 0.03, 0.1), and backward learning rates [0.0001, 0.0003, 0.001, 0.003, 0.01]. Note that FWDTP-BN does not have the backward learning rate hyperparameter. For DRTP, the search space includes learning rates 1.00..000.0.33 and mean ((0.0.05) and standard deviation (0.010.030.. 030.10.3) for random project matrix. For BP and FA, we use learning rates [0.4, 0.2, 0.1, 0.05, 0.02, 0.01] for hyperparameter search. and gradient clip values of 00.5, 1]. For CCL, the search space includes learning rates [0.2, 0.5, 1, 1.5, 2] and gradient clip values of [0.5, 1].
Implementation Details. Training MLP and CNN models with CCL can be unstable initially since both the forward and feedback networks are randomly initialized. We use learning rate warm-up for the initial 200 steps for CCL, which is adopted in Ernoult et al. [2022). Moreover, unlike algorithms in the target propagation family [Meulemans et al. 2020 Ernoult et al. 2022]. where the feedback network weights are trained using additional for- for-Ioops to tune the weights for each data batch, we introduce some techniques to stabilize the training course. We found that normalizing the activations during loss computation helps stabilize the training process. Additionally, can also suffer from feature collapse, where a trivial solution to all pairwise losses is to produce constant activations,
counter-current learning
introduce remedy. Inspired by layer-wist training, for each latent activation X E TR bg where b stands for batch size and d stands for feature dimension, we added an additional 1.2 loss to minimize the difference between norm( X) )n)rmm (X) and the identity matrix. Finally, in contrast to error backpropagation, where the error signals can be zero and the weight updates can be small at the end of training, the layer-wise losses in CCL are seldom zero, thus the weights can keep changing during the training. This can lead to worse minima. To accommodate this, we use gradient centralization Yong al. 2020 to centralize the gradient for parameters for both BP and CCL and flooding method [Ishida et al. 2020 to prevent some weights from updating if the sample-wist -ise difference between output and target smaller than 0.2 in CCL on CNN.
6.2 Comparison of Implementation
Comparison with Ernoult al. [002]. We compare the results on CIFAR-10 10 using a VGG- like network architecture. As shown in Table 4 the results indicate that L-DRL [Emoult et al.) 2022] achieved similar testing accuracies to BP, reaching 89%. While these results are promising, it's worth noting some aspects of their methodology that may affect direct comparisons: (1) Their models were trained on the full training set, which differs from our approach; (2) The authors do not provide details on the hyperparameter search space or the implementation of cross-vall validaaii.......
Table 4: Comparison Results on CIFAR10 Using VGG-like Convolutional Neural Network." indicates that we report the results from the literature.
OUR IMPLEMENTATION L-DRL ERNOULT BT AL. 2022
TRAIN VALIDATION SET
x
BP
87.12 11.76
89.07 -.22
L-DRL ERNOULT ET AL. 2022
89.38 =0.20*
CCL (OURS)
82.94 +0.53
Comparison with Shibuya et al. [2023] There are some quantitative differences in the results
between ours and (Shibuya et al. [2023 Here, we identify the sources that can potentially lead to different empirical results:
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We note that the original implementation normalizes FMNIST, CIFARIO, and CIFAR100 using the mean and standard deviation of MNIST. We fix this by using each dataset's mean and standard deviation;
We proceed with performing a hyperparameter grid search again using five random seeds because directly using the paper's best hyperparameters does not fit. We consider slightly smaller hyperparameter search set than the original onc, as shown in Section 6.1 due to the computation and time budget for the grid search of all the combinations.
6.3 Visualization of Representations in Other Layers
To show that the feature alignment can present in more than layer, we show the results of five-layered CNN model trained on CIFARIO. The model is trained for 3000 steps using CCL, and the t-SNE for layers 1, and 5 with different time steps shown in Figure g
eetrre train
ofter 10 CD. spoones
Afer 200 cc. spdates
Alter 1000 CCL Up33005
ARL 3000 COL upsates
CAND
Figure 8: Feature Alignments Across Layers. We show that similar feature alignment organiza- tion are observed in the third and fifth layer. While feature alignment in the first layer is inapparent.
6.4 Experiments on Weight Alignment
To better understand how CCL learns, we wonder if CCL learns by aligning the forward and backward weight. We compute the cosine similarity between the forward layer weights and the corresponding feedback layer weights for the model trained on MNIST using a five-layered MLP architecture.
As shown in Figure our findings reveal two distinct phases during training:
Phase one: Layers and show rapid alignment increases.
Phase two: Alignments layers and saturate. Alignments in the intermediate layers gradually increase until saturation.
Interestingly, intermediate layers showed bottom-up up convergence pattern, with layer 2 achieving highest similarity first, followed by layers 3 and 4. We found that high alignment isn' always achieved necessary for effective learning. This may be due to (1) Dimensional reduction: Information propagation through the network affects alignment, and (2) Non-linearity: Activation functions impact the relationship between forward and backward weights. These factors may contribute to observed alignment patterns without requiring perfect symmetry between forward and backward weights.
6.5 Literature Reviews More Biologically Plausible Algorithms
Algorithms with only forward passes. Forward-forward algorithm (FFA) Hinton 2022 only uses forward passes update the neural network. Specifically, in the supervised learning scheme,
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Cosine Similarity between Forward Feedback Weights
Layer
0.25
Layer
Layer
0.20
Layar
Similartty
Layer 5
0.15
0.10
0.05
3.00
1000
1500
2000
2500
3000
3500
4000
Fpoch
Figure 9: Weight Alignment Between the Forward and the Backward Network Trained With CCL. We find that the weight alignment of the first and the last quickly achieve plateau, while that of the intermediate layers increase gradually.
FFA sends the supervision signals with recurrent architecture in top-down manner, i..., propagating the signal one layer by one layer for 3 steps. In contrast, our CCL scheme seeks to propagate the signal directly, thus attaining more computational efficiency. Similarly, Error-driven Input Modulation (PEPITA) Dellaferrera and Kreiman| [022] adopts two forward passes for updating the
neural network, but the second forward pass requires perturbed input, obtained via adding the input with the global error signals. Thus, PEPITA causes the update locking problem, and the neurons have to track two forward activities for loss computation, making it less biologically plausible. SoftHebb improves upon several state-of-the-ard biologically plausible algorithms by achieving non-local property, mitigating time-locking of layer updates, and operating SobtHebb and CCL perform similarly on CIFAR- 10 and CIFAR-100 datasets, SoftHebb requires an extensive layer-wise hyperparameter search (24 hyperparameters). Additionally, SoftHebb's 's potential for auto-encoding tasks remains unexplored.
unsupervised manner. While
Algorithms for spiking neurons. BurstCNN Greedy al. 2022 and BurstProp [Payeur al. 2021] propose to learn spiking neural networks using a two-compartment neuronal model. This model incorporates unique dendritic properties, synaptic plasticity, and gighffregency bursting, enabling information multiplexing across hierarchical levels. While these approaches offer high biological plausibility and attain good performance, extensions auto-encoding tasks lacking, and the required computation times are not provided.
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Reviews:
[{"review_id": "XH2oj0rucb", "reviewer": "Reviewer_sTDo", "summary": "The article proposes a learning framework that addresses three major critiques of the backpropagation algorithm regarding its biological plausibility: (i) the weight transport problem, i.e., the error feedback weights being the transposes of the feedforward weights, (ii) the nonlocal update problem, i.e., the local weights being dependent on the global error signal, and (iii) the backward locking problem, i.e., the need to freeze layer activations until error feedback information becomes available.\n\nThe basic idea of the proposed framework is to construct a dual network that operates in the reverse direction to the original inference network. This dual-network has transpose dimensional weights (but not necessarily the transposes of the forward weights), with some similarity to the case of backpropagation. However, this dual network propagates the target value backward. Each layer of the dual network is compared to the corresponding layer (of the same dimension) in the original inference network. The loss is defined as the mean square error between the activations of the corresponding layers.  The numerical results demonstrate that the test performance in classification task are on par with backpropagation and the state-of-the-art models.", "questions": "- How does the proposed approach train the forward neural network to minimize the mean square error (MSE) at the final layer?\n\n- It would be useful to report the alignment (angle) between the forward and backward network weights, to check if the weight symmetry problem still exists.\n\n- The sentence \"Due to random weight initialization, the information content ... decreases ...\" in the caption of Figure 1(a) is imprecise. How do you measure information content and relative to what, and why is it decreasing? Furthermore, the next sentence does not really apply to the \"before training\" (a) part of Figure 1.\n\n- Figure 3: Do we see a similar organization of representations in other layers? Since the target label input for each class is fixed, we expect the backward network representations to be a single point for each class in any layer. Do the forward network layer activations appear as clouds around these points?\n\n- Section 4.2: MLP Experiments: Why do the models used for comparison employ tanh nonlinearity, while CCL uses ELU? Test results with the same model properties would be more meaningful.\n\n- Table 1: Is the DTP in the line below FA correct? As Noakland 16 is cited, is it actually DFA? Shibuya et al. [2023], which the authors claim to base their experiments on, report 52.17% accuracy for DTP/CIFAR10, which is not available or conflicts with Table 1. Similarly, Shibuya et al. [2023] report 51.33% for FA-CIFAR10, but Table 1 in the current article reports FA performance as 45.76%. The FW-DTP performances also do not match.\n\n- Figure 4: It would make more sense to compare the conv. kernels of CCL after training with the corresponding kernels of BP, as the initial random kernels shown in this figure are not informative.\n\n- Section 4.4: Do we need the CKA-based measure to compare the representations of the forward and backward networks? The proposed approach tries to minimize the mean square error (MSE) between them, so do we expect any linear transformation uncertainty? Isn't it more meaningful to look directly at the angle of alignment or the MSE normalized by the norm-square of the forward representations?\n\n- Figure 7 and Section 4.5: The best performance appears to occur toward the top right corner, suggesting that feedforward-feedback learning rates should be similar (symmetric) rather than asymmetric?", "limitations": "The authors sufficiently discuss the limitations of their work at the end of the article. Related points are raised in the weaknesses and questions part.", "rating": 6, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "- Novelty: The article offers a novel approach for the biologically plausible learning framework. The use of a dual network to propagate the target value backward and using the mean square error between the corresponding representations of the network as the loss function appears to be a new idea.\n\n- Numerical Experiments: The article presents several experiments demonstrating the performance of the proposed approach in comparison to existing methods. The performance appears to be close to that of backpropagation (and even significantly better in the case of CIFAR-10).", "weaknesses": "- Analytical Support/Discussion: There is no analytical justification provided for the proposed approach, making it unclear why it should work. This is the main weakness of the article. At the very least, there could be more discussion aimed at providing intuition about the underlying mechanism of the proposed approach that would enable effective training of neural networks. The authors cite \"counter-current\" mechanisms observed in nature as their primary motivation.\n\n- Presentation: The presentation is generally smooth and clear. However, it can be further improved as addressed by the questions below.\n\n- Reporting of Numerical Experiments: The results reported for the numerical experiments are confusing, especially as they conflict with those of Shibuya et al. [2023], whose experimental setup is adopted in the current article.\n\n- Scalability: It is uncertain whether this idea would work for large-scale inputs/models (e.g., ImageNet 1K/ResNets).", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "**[W1] We provide a theoretical extension for the existing framework**\n\nWe provide an analytical framework for understanding CCL in General Response 1 [GR1]. In short, we show that CCL can be considered as optimizing an ELBO for a hierarchical model. \n\n**[W4] Validation on large-scale dataset/models**\n\nWe acknowledge that the ImageNet dataset poses significant challenges due to its size and complexity. It typically requires batch normalization, whose biological plausibility is under-explored and often necessitates extensive architectural changes. Additionally, recent work on biologically plausible learning algorithms (e.g., FA, DTP, DRL, FW-DT, and DRTP) has not evaluated these methods on large datasets like ImageNet, as datasets such as CIFAR-10 and CIFAR-100 demonstrate sufficient complexity for current studies in this field. We will include this discussion in the revised manuscript, acknowledging the need for further research on scalability to more complex datasets like ImageNet.\nNonetheless, to showcase the capabilities of the CCL algorithm, we have applied it to an autoencoder task, which, to our knowledge, is the first time a biologically plausible algorithm has been scaled to an image autoencoding paradigm.\n\n**[Q1] What is the loss for the final layer?**\n\nWe thank the reviewer for reminding us of this. We adopt cross-entropy loss for the final layer, which is not explicitly mentioned in the paper. We will modify the content accordingly. \n\n**[Q2] Suggestion on reporting weight alignment**\n\nWe appreciate this insightful suggestion. Please refer to General Response 4 [GR4] for discussion. \n\n**[Q3] Suggestion on caption for Figure 1(a)**\n\nWe thank the reviewer for checking the details. The main idea is that signal processing cannot increase information, which corresponds to the data processing inequality (DPT) concept in information theory [1]. From a more intuitive perspective, due to dimensional reduction and the random initialization of the weight parameters, the information of the input signals can be lost during data processing. We will revise the sentence accordingly. Also, we agree that the sentence “Notably, … operator” should move to “(b) During training,” and we will also revise this accordingly.\n\n[1] Tishby, N., & Zaslavsky, N. (2015). Deep learning and the information bottleneck principle.\n\n**[Q4] Do we see a similar organization of representations in other layers?**\n\nYes, similar organization is presented in other layers, especially for deeper layers. In Figure (2) in the rebuttal document, we show the results of a five-layered CNN model trained on CIFAR10. The model is trained for 3000 steps using CCL, and the t-SNE for layers 1, 3, and 5 with different time steps are shown.\n\n**[Q5] Why does CCL utilize ELU?**\n\nWe find that ELU empirically gives better results than Tanh in CCL training when we perform exploratory analysis on MNIST in the early stage of research. We posit the main reason is that Tanh is a symmetric function and thus would induce feature collapse when trained with CCL. For example, at the beginning of training, say the first entries of $a_l$ (i.e., activation from forward network layer $l$) and $b_l$ (i.e., activation from feedback network layer $l$) are 1 and -1 respectively, then solving the local loss function, the weight would be updated such that the first entries of $a_l$ and $b_l$ becomes 0. This would lead to feature collapse for both the forward and feedback networks. We appreciate the reviewer for the insight and will add this to the Supplementary Section. \n\n**[W3, Q6] Result inconsistency from Shibuya, et al.**\n\nFirst of all, we thank the reviewer for noting the error, where the citation for DTP should be corrected. We will revise it in the camera-ready version. \n\nSecond, we would like to kindly remind the reviewer that the statistics Shibuya et al. [2023] reported are the test error (%), not the test accuracy (%) shown in Table 1. \n\nThird, we would like to identify the sources that can potentially lead to different empirical results:\n- (a) We note that the original implementation for the code is incorrect, where the original codebase from Shibuya, et al. (2023) normalizes FMNIST, CIFAR10, and CIFAR100 using the mean and standard deviation of MNIST. We fix this by using each dataset’s mean and standard deviation;\n- (b) We proceed with performing a hyperparameter grid search again using five random seeds because directly using the paper’s best hyperparameters does not fit. We consider a slightly smaller hyperparameter search set than the original one, as shown in Section 6.1, due to the computation and time budget for the grid search of all the combinations. \n\nThe above all can give rise to the difference between the results presented in our work and the original results. \n\n**[Q7] Suggestion on presenting BP and CCL convolutional kernels**\n\nWe show the result in Figure (3) in the rebuttal document, where kernels learned by BP and CCL are visually different. Kernels from models trained with BP have more high frequency components, as manifested as neighboring white (e.g., weight with high values) and black pixels (e.g., weight with low values). In comparison, those with CCL have more low frequency components. We posit this might be because the error signal can contain more high-frequency information than the ideal target signal. \n\n**[Q8] Suggestion on similarity measurement**\n\nWe thank the reviewer for pointing out this part. In Figure 6, we not only compare the similarities between pairwise features in the same layer (i.e., the diagonal entries) but also across different layers. We believe that using CKA can better capture relations across layers since it considers invertible linear transformation and orthogonal transformation.\n\n**[Q9] Does Figure 7 suggest feedforward-feedback learning rates should be symmetric?**\n\nYes, it suggests that symmetric learning rates perform better.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "ezhlQoEKNN", "reviewer": "Reviewer_EBaT", "summary": "In their paper ‘Counter-Current Learning: A Biologically Plausible Dual Network Approach for Deep Learning’ the authors introduce their new ‘counter-current learning (CCL) framework’ which they use to train neural networks of multiple network classes (MLPs, CNNs, Autoencoders) using a learning mechanism which solely relies on local calculation of learning signals, without the need to fully propagate gradients through the network. Through this, authors establish a link the brain’s counter-current exchange mechanism. The authors also show an analysis how feature representations are shaped over the course of learning in their learning framework. Whereas many biologically-plausible learning mechanisms have been proposed throughout the last couple of year, the author’s work particularly stands out due to its generality / easy of being applied to various different network classes used in machine learning.", "questions": "If the authors see their work primarily focused on an engineering solution, I am wondering whether their technique generalises to Transformers and / or State Space Models (like Mamba or Griffin). Otherwise, I wonder whether authors could make specific predictions based on the learning dynamics observed in their model which would allow the community to test for its plausibility using neural data.\n\nHow does your work relate to the bioplausible works referred to above?", "limitations": "I think authors address limitations on the machine learning / engineering side well but could potentially expand their discussion of limitations with regards to biological evidence for / against their proposed learning framework.", "rating": 6, "confidence": 3, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "As mentioned in the summary, I think many biologically-plausible learning mechanisms have been proposed throughout the last years and the author’s framework stands out to me due to it apparently generality as it seems to easily be applied to network classes which are, from my point of view, not commonly considered in bio-inspired learning mechanism papers, like autoencoders. This generality could allow for the authors framework to have an impact beyond theoretical neuroscience but also influence the more engineering focused side (e.g. neuromorphic computing). The new learning framework is supported by some additional empirical investigations, giving an intuitive understanding of the learning dynamics within the network.", "weaknesses": "While authors mention a potential biological link, not much space is given to discussing specific biological implementations and authors do not seem to make specific predictions which would allow for empirical validation of their proposed mechanism in biology, to validate whether the mechanism is actually utilised in the brain. As such, I see this paper as being more focused on the engineering / computing solutions side, where it does a good job, mentioned above. Given this focus on the computing side, I would have hoped that authors would also consider training a model using a current SOTA architecture (i.e. Transformers or State Space Models) using their technique, to show whether their algorithm can also handle these cases.\n\nIt would be important to discuss how the proposed work relates to a recent line of work in which two streams of information are assumed to exist in the brain: burstprop (Payeur et al. Nature Neuroscience 2021) and burstCCN (Greedy et al. NeurIPS 2022). Moreover, this work seems to have links with Sacramento et al. 2018 NeurIPS, in that both require specialised feedback neurons that encode the error signals. Both Sacramento et al. and Greedy et al. use an interneuron that learns to match the feedback, which conceptually seems to match the idea proposed by the authors. Perhaps even more relevant is the predictive coding of backprop by the group of Rafal Bogacz, which requires separate error neurons and both the forward and backward phases to align with each other. These should be discussed, and it may be worth making a more explicit link in future work.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "**[Q1] Generalization of models**\n\nA: While MLPs and CNNs have established biological plausibility, the biological relevance of Transformers and State Space Models remains unclear and under-explored. For vision tasks, CNNs have been shown to match Transformer performance when computational resources, datasets, and techniques are equalized [1,2]. Recognizing the need for generalization, we've extended our research to auto-encoding tasks. To our knowledge, this represents the first application of a biologically plausible algorithm to an auto-encoder architecture, broadening the scope of our approach beyond traditional supervised learning paradigms.\n\n[1] Liu, Z., Mao, H., Wu, C. Y., Feichtenhofer, C., Darrell, T., & Xie, S. (2022). A convnet for the 2020s. CVPR.\n\n[2] Liu, S., Chen, T., Chen, X., Chen, X., Xiao, Q., Wu, B., ... & Wang, Z. (2023). More convnets in the 2020s: Scaling up kernels beyond 51x51 using sparsity. ICLR.\n\n\n**[Q2] Literature review and comparison**\n\nWe appreciate the reviewer's suggestion to compare our Counter-current Learning (CCL) approach with recent work on dual-stream information processing in the brain. Here's a detailed comparison of CCL with BurstCNN/BurstProp, Sacramento, et al. (2018), and Predictive Coding:\n\nNetwork Structure:\n- CCL uses two separate networks: a feedforward network and a feedback network.\n- BurstCNN/BurstProp and Sacramento et al. (2018) use a single network with two firing modes.\n- Predictive Coding uses a hierarchical structure with prediction and error units at each layer.\n\nError Representation:\n- CCL doesn't use global error signals; differences between forward and feedback local activations guide the learning.\n- BurstCNN/BurstProp encodes errors implicitly in burst firing patterns.\n- Sacramento, et al. (2018) allow local error signals to be generated by comparing different sources of neural input activity; the axonal output is a corrected target signal.\n- Predictive Coding uses explicit error neurons.\n\nBiological Plausibility:\n- CCL mitigates the update locking problem, allowing for more flexible and potentially more biologically plausible learning dynamics.\n- BurstCNN/BurstProp is based on observed burst firing patterns in neurons, enhancing biological plausibility. However, it still suffers from update locking problems.\n- Sacramento, et al. (2018) does not resolve the update locking problem, nor does it elaborate on multi-layered structures, limiting its biological plausibility in complex networks.\n- Predictive Coding aligns with theories of how the brain processes information, but explicit error encoding and propagation remains debated in neuroscience.\n\nComputational Complexity:\n- CCL allows for parallel processing in forward and feedback networks, potentially offering computational efficiency.\n- BurstCNN/BurstProp and Sacramento, et al. (2018) require distinct forward and backward phases, which may result in slower processing.\n- Predictive Coding involves iterative refinement, which can be computationally intensive.\n\nOur CCL approach offers several advantages in this context. By using separate forward and feedback networks, we avoid the update locking problem that affects BurstCNN/BurstProp and Sacramento et al. (2018). This separation allows for more flexible learning dynamics that could better reflect the parallel processing capabilities of biological neural networks. Moreover, CCL's approach to error representation - using differences between forward and feedback activations rather than explicit error signals - offers a unique perspective on how learning might occur in biological systems without the need for specialized error neurons or complex temporal coding schemes.\n\nWe will add the above discussion over comparison in the camera-ready version.\n\n**[Q3] Testable prediction**\n\nWe appreciate the reviewer's thought-provoking question. Based on our observations of the learning dynamics in our model, we propose the following testable predictions:\n\n- Organized increase in cross-layer neural activity correlation: As illustrated in Figure 6 of our manuscript, we observe an organized increase in cross-layer neuron activity correlation (highlighted by red and green boxes). This suggests a reciprocal and complementary learning dynamic among neurons. We predict that this phenomenon would be more pronounced when the network is exposed to novel, unseen patterns. This could be tested by measuring cross-layer neural correlations in biological networks before and after exposure to new stimuli.\n\n- Low similarity in local error signals across layers: Unlike backpropagation or related algorithms where error signals are globally coordinated, CCL's architecture suggests that error signals at different layers should exhibit low correlation or similarity. This prediction could be tested by comparing the similarity of local error signals across layers in biological neural networks, particularly during learning tasks.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "h0e6EHQLDf", "reviewer": "Reviewer_ijwx", "summary": "In this work, the authors proposed the counter-current learning algorithm, a novel algorithm for biologically plausible training of feedforward neural networks. The learning rule is built upon the target-propagation algorithm and its variants. In these algorithms, the backward pathway is typically trained in a way that the target information propagated to each hidden layer provides a layer-wise target that nudges the forward pathway's output toward the true label. Instead, the authors jointly trained both feedforward and feedback weights projected to each layer, in a way that forward and backward pathways exhibit roughly the same activity at each layer. The authors applied this algorithm to various image recognition tasks (MNIST/Fashion-MNIST/CIFAR10/CIFAR100) and an autoencoder task and demonstrated that the proposed algorithm outperforms other variants of target-prop algorithms. The authors empirically analyzed the learning process of the algorithm by examining the development of hidden layer representations and through an ablation study.", "questions": "Considering that the algorithm is minimizing the difference between $a_l$ and $b_l$, I expect to see large diagonal components in Figure 6 after learning. However, there is no diagonal trace in the figure. Why is that the case? \n\nIn addition, at t = 0, $b_1$ to target are already weakly aligned with the input, but $b_0$ is not aligned with the input. What is the origin of initial alignment and the absence of it in $b_0$? \n\nOn a related point, I wonder if the authors checked whether the stop gradient is implemented correctly, for instance, by calculating the gradients by hand and implementing them manually into the network.", "limitations": "The authors addressed the limitations, especially the lack of theoretical justification, nicely.", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The proposed learning algorithm is simple yet novel. The key novelty is the training algorithm for the backward pathway, where, unlike previous target-prop algorithms, activity matching between the forward and feedback pathway at each layer is used as the objective. I was positively surprised by the algorithm's strong performance because this learning algorithm clearly deviates from the backprop of the global loss. This training approach also makes the algorithm distinct from previous related approaches where the backward weights were fixed instead (Frenkel et al., Frontiers 2021; Shibuya et al., AAAI 2023).", "weaknesses": "The learning rules for hidden weights have trivial solutions, hence susceptible to representation collapse. The authors discussed their techniques for avoiding the collapse briefly in the supplement, but the implementation is not discussed in detail. Biological plausibility of the regularization mechanisms should also be discussed.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "**[W1] Discussion of biological plausibility of the regularization mechanisms.**\n\nWe thank the reviewer for pointing out and allowing us to elaborate on this. \n\n- Normalization: Our use of activation normalization during loss computation aligns with divisive normalization observed in biological neural circuits [1]. This process helps maintain neural activity within a functional range, similar to the divisive normalization phenomenon observed in animals in the early 1990s in the primary visual cortex.\n\n- Modulation. The additional loss provides signals for surround modulation [2], a foundational property of visual neurons at many levels of the visual system (e.g., retina, visual cortex), auditory system, and so forth. Surround modulation describes that dissimilar stimuli between inside and outside neuron's receptive field would evoke stronger responses for a neuron, compared to similar stimuli. This is similar to the regularization methods used to prevent the model from representation collapse.\n\n- Flooding method: This approach, preventing weight updates when the sample-wise difference is small, can be likened to the concept of activation thresholds in biological neurons, where small input changes do not trigger action potentials.\n\nImplementation details:\n-  Modulation. We implemented a technique to prevent feature collapse for each latent activation $X ∈ R^{b×d}$, where $b$ represents the batch size and $d$ represents the feature dimension. We added an L2 loss term to minimize the difference between norm($X$)norm($X$)$^\\top$ and the identity matrix. The loss is computed as: L_feature = $||norm(X)norm(X)⊤ - I||_F$\n- Avoid large gradient norms. We adopted gradient centralization [3] to mitigate the issue of large gradient norms. This technique centralizes the gradient vectors by removing their mean values before updating the model parameters: $g_{c} = g - mean(g)$, where $g$ is the original gradient and $g_c$ is the centralized gradient.\n- Focus on major differences. We implemented the flooding method [4] to focus the model on cases where predictions significantly differ from the ground truth. The modified loss function is expressed as $L_{flooded} = max(L, b)$, where $b$ is set to 0.2 as the original paper suggested.\n\n**[Q1] Why is there no diagonal trace in the figure 6.**\n\nThe observed pattern in Figure 6 arises from the functional asymmetry when comparing forward and backward networks because their input signals are different. If we were to compare two independently trained forward networks using CCL, we would expect to see diagonal traces in the CCA analysis because both networks perform similar compression and learning functions. However, forward and backward networks exhibit functional asymmetry (e.g., one for feature extraction and another one for reconstruction), leading to rank disparity (e.g., the backward activations have a lower rank). This reduced detail and compressed information in the feedback activations result in lower pair-wise CCA values between forward and backward layers, manifesting as a non-diagonal trace in Figure 6.\n\n**[Q2] What is the origin of initial alignment?**\n\nSince MNIST is a relatively small and simple dataset, digits of the same class are relatively similar and are more likely to be projected to neighboring features in the latent space, forming several loose clusters. This would cause a relatively positive alignment in terms of CCA.\n\n**[Q3] Checking stop gradient implementation.**\n\nWe followed your advice and checked the gradient implementation. We checked that the weight detachment is implemented correctly such that optimization over a local loss only involves the nearest layers.\n\n[1] Carandini, M., & Heeger, D. J. (2012). Normalization as a canonical neural computation. Nature reviews neuroscience.\n\n[2] Angelucci, A., Bijanzadeh, M., Nurminen, L., Federer, F., Merlin, S., & Bressloff, P. C. (2017). Circuits and mechanisms for surround modulation in visual cortex. Annual review of neuroscience.\n\n[3] Yong, H., Huang, J., Hua, X., & Zhang, L. (2020). Gradient centralization: A new optimization technique for deep neural networks. ECCV.\n\n[4] Ishida, T., Yamane, I., Sakai, T., Niu, G., & Sugiyama, M. (2022). Do We Need Zero Training Loss After Achieving Zero Training Error?. ICML.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "itpQQh7FRH", "reviewer": "Reviewer_kHhR", "summary": "In this paper, the authors propose counter-current learning (CCL), a biologically plausible framework for credit assignment in neural networks.", "questions": "1. Could the author discuss the biological plausibility of their approach from weight transport, locality, freezing of neural activity, and update locking points of view?\n2. Could the author provide some insight how their approach compares against or relates to the state of the art in the domain: FF, PEPITA, and CFSE?\n[1] Hinton, G. (2022). The Forward-Forward Algorithm: Some Preliminary Investigations. Technical report. \n[2] Dellaferrera, G., Kreiman, G., and Kreiman, G. (2022). Error-driven Input Modulation: Solving\nthe Credit Assignment Problem without a Backward Pass. In Chaudhuri, K., Jegelka, S., Song, L.,\nSzepesvari, C., Niu, G., and Sabato, S., editors, Proceedings of the 39th International Conference\non Machine Learning, pages 4937–4955. PMLR.\n[3] Andreas Papachristodoulou, Christos Kyrkou, Stelios Timotheou, and Theocharis Theocharides.\nConvolutional channel-wise competitive learning for the forward-forward algorithm. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 38, pages 14536–14544, 2024.\n3. Why is CCL working considerably better than BP in Table 3 for CIFAR100?", "limitations": "The limitation of the work has been briefly discussed.", "rating": 4, "confidence": 4, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "The research direction of having new learning algorithms inspired biologically seems relevant and interesting.", "weaknesses": "- The authors discuss biological plausibility as a motivation, but do not discuss thoroughly how the proposed approach is biological plausibility. For instance, update-locking and non-frozen activities should be discussed in detail.\n\n- The comparison with the state of the art in the domain is missing:\n[1] Hinton, G. (2022). The Forward-Forward Algorithm: Some Preliminary Investigations. Technical report. \n[2] Dellaferrera, G., Kreiman, G., and Kreiman, G. (2022). Error-driven Input Modulation: Solving\nthe Credit Assignment Problem without a Backward Pass. In Chaudhuri, K., Jegelka, S., Song, L.,\nSzepesvari, C., Niu, G., and Sabato, S., editors, Proceedings of the 39th International Conference\non Machine Learning, pages 4937–4955. PMLR.\n[3] Andreas Papachristodoulou, Christos Kyrkou, Stelios Timotheou, and Theocharis Theocharides.\nConvolutional channel-wise competitive learning for the forward-forward algorithm. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 38, pages 14536–14544, 2024.\n\n- The results are sometimes counter intuitive. For example, the results of CCL vs BP for CIFAR100 in Table 3. Why is CCL working considerably better than BP?\n\n- The presentation and organization of the paper can be improved. For instance, Figure 2 can be presented at a higher abstraction level as pseudo-code", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "**[Q1] In-depth discussion of bIological plausibility for counter-current learning**\n\nWe thank the reviewer for the opportunity to further elaborate on the biological plausibility of our method from weight-transport, locality, freezing of neural activity, and update locking perspectives. Although the explanation has been provided in the manuscript Line 41-50 and Line 124-129 on how the model architecture and algorithm design mitigates these problems, we discuss more explicitly below\n- Weight-transport. We leverage a feedback network architecture for processing feedback signals, which uses a different parameter set from the forward network. The weights for feedback and forward networks are updated independently, mitigating the problem of weight transport by construction.\n- Locality. We leverage local layer-wise loss functions and gradient detach (i.e., stop gradient operation) to ensure that the parameters are updated by a local loss function, not a global error signal that propagates through the network.\n- Freezing of neural activity. The back propagation method requires keeping the neural activations for the global error back propagation since the process involves element-wise multiplication of the error signal with the derivative of the neural activations and thus it has to remain static. Our CCL algorithm, like all the algorithms in the Target Propagation Family, also requires freezing the neural activity for the loss computations. However, the duration of freezing time is halved with our CCL scheme since the two networks can process the signals simultaneously. \n- Update locking. In back-propagation, The latent activations of the forward network are kept and re-used for the backward phase. \nMoreover, the backward phase can only begin after the forward phase is finished, causing the update locking problem. In CCL, the backward process is performed through the feedback network, which is independent of the forward phase, and thus the forward and feedback processes occur simultaneously.\n\n**[Q2] Literature review and algorithm comparison**\n\nQuantitative comparisons are provided in the general response [GR2]. We discuss algorithmic differences here:\n- Forward-forward algorithm, FFA [1]\n  - Compared to Section 3.3 in [1], CCL does not embed the label in the image. CCL adopts a feedback network to process and project the one-hot label, while in FFA the label is embedded in the input data, e.g., for MNIST training the first 10 pixels of an image is replaced with a one of N representation of the label.\n  - Compared to Section 3.4 in [1], CCL does not need recurrent architecture to propagate the supervised signal backwardly: FFA adopts a recurrent network architecture to send the supervision signals in a top-down manner, i.e., propagate the signal one-layer by one-layer for 3 - 5 steps. In contrast, our CCL scheme seeks to propagate the signal directly, thus attaining more computational efficiency. Rolling out the signal transmission step-by-step would make the algorithm hard to scale to deep networks.\n\n- Error-driven Input Modulation, PEPITA [2]\n  - PEPITA requires two forward passes. In PEPITA, the parameter update function requires computing the difference between the two forward passes. This causes the update locking problem\n  - PEPITA requires the neurons to track two states. As PEPITA requires two forward passes,it requires every neuron to keep track of two neural activities, which is more biologically implausible.\n  - PEPITA does not follow locality. The global error signal is directly transformed and added to the latent feature; this does not follow the locality learning concept where synaptic changes depend on the correlated activity of the pre-and postsynaptic neurons.\n- Channel-wise Feature Separator and Extractor, CFSE [3]\n  - CFSE is a layer-wise training method that contains structural inductive bias. CFSE groups input channels in the forward stage, which can potentially provide inductive bias preferable for training.\n  - CFSE works with two different normalization methods. CFSE adopts batch normalization and group normalization methods to work, while CCL attains similar results without group normalization.\n  - CCL is validated on auto-encoder experiments, which CFSE does not.\n\t\n**[Q3] Explanation for the result on CIFAR-100.**\n\nFor a detailed discussion, please see General Response 3 [GR3] and Figure 1.a in the rebuttal document.\n\n[1] Hinton, G. (2022). The Forward-Forward Algorithm: Some Preliminary Investigations. Technical report.\n\n[2] Dellaferrera, G., Kreiman, G., and Kreiman, G. (2022). Error-driven Input Modulation: Solving the Credit Assignment Problem without a Backward Pass. ICML. \n\n[3] Andreas Papachristodoulou, Christos Kyrkou, Stelios Timotheou, and Theocharis Theocharides. (2024). Convolutional channel-wise competitive learning for the forward-forward algorithm. AAAI.\n\n[4] Shaw, N. P., Jackson, T., & Orchard, J. (2020). Biological batch normalisation: How intrinsic plasticity improves learning in deep neural networks. Plos one, 15(9).", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "qWoGDoHGeH", "reviewer": "Reviewer_myCb", "summary": "Inspired by the counter-current phenomenon observed in nature, the authors propose a counter-current learning (CCL) framework that decouples the input and backpropagation information in different circuits, overcoming many biological implausibility issues of backpropagation learning (BP).", "questions": "-\tI am surprised that the performance of BP is worse than the proposed local learning method. Can the authors explain more about this gap?\n-\tIn forward-forward [1], Hinton proposes a recurrent network architecture that inputs the label at the top layer. This architecture seems similar to what is used in this work. Can the authors elaborate on the differences?\n\n\nreference\n[1]. Hinton G. The forward-forward algorithm: Some preliminary investigations[J]. arXiv preprint arXiv:2212.13345, 2022.", "limitations": "I think the authors have discussed it well.", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "I am not an expert in target propagation, but the general idea presented in this work sounds novel and interesting to me. I believe it also offers some insights into nature-inspired learning. I appreciate the constructive figures that help me quickly grasp the core idea of this work.", "weaknesses": "-\tIn Table 1, the authors should compare their results with other biologically plausible methods, like forward-forward and deep softhebb, or justify their criteria in model selection.\n-\tA minor drawback is the lack of validation in complicated dataset like Imagenet.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "**[W1] Algorithm selection.**\n\t\nWe appreciate the reviewer's question regarding concurrent work. A performance comparison is provided in general response 2 [GR2], demonstrating that our CCL algorithm outperforms other methods. While SoftHebb achieves similar results, it requires 24 hyperparameters compared to our 3, making our approach more efficient. We acknowledge the importance of comparing with other emerging biologically plausible approaches and will include more rigorous comparisons to methods like forward-forward and deep SoftHebb, as suggested.\n\n**[W2] More validation.**\n\nWe appreciate the reviewer's questions regarding dataset and model scalability. However, due to the size and complexity of ImageNet, one typically requires batch normalization and other tricks in model architectures, which may not be biologically plausible. Additionally, recent work on biologically plausible learning algorithms (e.g., FA, DTP, DRL, FW-DT, and DRTP) has not evaluated these methods on large datasets like ImageNet, as datasets such as CIFAR-10 and CIFAR-100 demonstrate sufficient complexity for current studies in this field. We will include this discussion in the revised manuscript, acknowledging the need for further research on scalability to more complex datasets like ImageNet. Nonetheless, to showcase the capabilities of the CCL algorithm, we have applied it to an autoencoder task, which, to our knowledge, is the first time a biologically plausible algorithm has been scaled to an image autoencoding paradigm.\n\n**[Q1] Explanation for the result.**\n\nFor a detailed discussion, please see General Response 3 [GR3] and Figure 1.a in the rebuttal document.\n\n**[Q2] Comparison to the forward-forward algorithm.**\n\nQuantitative comparisons are provided in the general response [GR2]. We discuss algorithmic differences here:\n- Compared to Section 3.3 in [4], CCL does not embed the label in the image. CCL adopts a feedback network to process and project the one-hot label, while in FFA the label is embedded in the input data, e.g., for MNIST training the first 10 pixels of an image is replaced with a one of N representation of the label.\n- Compared to Section 3.4 in [4], CCL does not need recurrent architecture to propagate the supervised signal backwardly: FFA adopts a recurrent network architecture to send the supervision signals in a top-down manner, i.e., propagate the signal one-layer by one-layer for 3 - 5 steps. In contrast, our CCL scheme seeks to propagate the signal directly, thus attaining more computational efficiency. Rolling out the signal transmission step-by-step would make the algorithm hard to scale to deep networks.\n\n[1] Lee, D. H., Zhang, S., Fischer, A., & Bengio, Y. (2015). Difference target propagation. ECML PKDD.\n\n[2] Meulemans, A., Carzaniga, F., Suykens, J., Sacramento, J., & Grewe, B. F. (2020). A theoretical framework for target propagation. NeurIPS.\n\n[3] Shibuya, T., Inoue, N., Kawakami, R., & Sato, I. (2023). Fixed-weight difference target propagation. AAAI.\n\n[4] Hinton, G. (2022). The forward-forward algorithm: Some preliminary investigations. arXiv preprint arXiv:2212.13345.\n\n[5] Nøkland, A. (2016). Direct feedback alignment provides learning in deep neural networks. NeurIPS.\n\n[6] Launay, J., Poli, I., Boniface, F., & Krzakala, F. (2020). Direct feedback alignment scales to modern deep learning tasks and architectures. NeurIPS.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:
We thank the reviewers for their insightful feedback. We're grateful for their assessment of our work as novel (sTDo, ijwx, myCb), distinctive (ijwx), and empirically supported (sTDo, EBaT), with EBaT noting its 'apparent generality'. We address the main critiques below, focusing on theoretical foundation, literature review, result clarification, and intuition explanation. We will add the corresponding results and discussions to the next version of the paper.

**[GR1] A theoretical explanation and interpretation of CCL**

We appreciate reviewer sTDo for requesting a more analytical approach to explain and support the proposed counter-current learning (CCL) algorithm. We offer the following analysis through the lens of variational inference and hierarchical latent variable models.

CCL fundamentally aims to learn a conditional distribution p(y|x), where x represents input data and y represents target outputs. The algorithm employs a hierarchical structure, for example, with latent variables z1 and z2, utilizing both forward (encoding) and backward (decoding) pathways. The key insight is that CCL uses two different graphical models with which CCL can be interpreted as optimizing the Evidence Lower Bound (ELBO) on the log-likelihood of y given x:
- For the generative model p: x → z1 → z2 → y
- For the inference model q: y → z2 → z1 → x

### ELBO Derivation
Starting from the goal of maximizing log p(y), we derive the ELBO as follows with simplifications based on the conditional independence from graphical models:

$\log p(y|x) = \log E_{q(z1,z2|y)}[p(y,z1,z2|x) / q(z1,z2|y)]$

$≥E_{q(z1,z2|y)}[\log (p(y,z1,z2|x) / q(z1,z2|y))]$ (by Jensen's inequality)

$=E_{q(z1,z2|y)}[\log p(y|z2) + \log p(z2|z1) + \log p(z1|x) - \log q(z2|y) - \log q(z1|z2)]$

$=E_{q(z2|y)}[log p(y|z2)] - KL(q(z2|y) | p(z2|z1)) - KL(q(z1|z2) | p(z1|x))$

The CCL algorithm implements this variational framework in a simplified and computationally tractable form. The implemented losses connect to the ELBO terms as follows:
- Target reconstruction: This first term represents the reconstruction of y given the latent variable z2. In CCL, it is implemented as a loss between the output of the forward pathway and the target y.
- Latent consistency: The two KL divergence terms encourage alignment between the distributions of z2 (z1) inferred from y (z2) and predicted from z1 (x). In CCL, this is approximated by L2 losses between latent features from forward and feedback networks. While this simplification assumes zero variance in the distributions, it captures the essence of aligning the forward and backward pathways. Using Monte Carlo sampling by adding noise to latent features may be viewed as an approximation.

**[GR2] A comparison with previous SoTA**

We have compiled results from relevant papers and conducted additional experiments to provide a comprehensive comparison. Please note that hyperparameters and architectures may vary across studies. To account for this, we have added the results of our CCL approach with an architecture similar to the one used in PEPITA.

| Algorithm | CIFAR10 | CIFAR100 | 
|--|--|--|
|FFA [1]| 59 (*) |-|
|CFSE [3]|78.11|51.23|
|SoftHebb [4]|80.30| 56.00|
|CCL (original)| 82.94 |56.29|
|PEPITA [2]|56.33|27.56|
|CCL (matched with PEPITA)|62.19|34.89|

Key observations:
- CCL outperforms other methods on both CIFAR10 and CIFAR100. Even with the simplified architecture used by PEPITA, CCL yields higher accuracy.
- SoftHebb [4] performs extensive layer-wise hyperparameter search (24 hyperparams), while CCL searches for only 3 parameters.

We will revise our paper to include this comprehensive comparison, providing a clearer context for our algorithm's performance relative to other recent work in the field.

**[GR3] Result clarification**

We thank the reviewers for carefully reading the results. We show the training loss during training in Figure (1.a) in the rebuttal document. BP actually optimizes the training objective better and yields much lower training loss than CCL but a high testing error, indicating that BP may be overfitting. We believe that more aggressive model regularization techniques may be helpful. 

**[GR 4] Suggestion on reporting weight alignment**

We appreciate this insightful suggestion from reviewer sTDo and show the weight alignment between the forward and backward network in Figure (1.b) in the rebuttal document. Cosine similarity between the forward layer weights and the corresponding feedback layer weights are computed for the model trained on MNIST using a five-layered MLP architecture. Our findings reveal two distinct phases during training:
- Phase one: Layers 1 and 5 show rapid alignment increases.
- Phase two: Alignments in layers 1 and 5 saturate. Alignments in the intermediate layers gradually increase until saturation.

Interestingly, intermediate layers showed a bottom-up convergence pattern, with layer 2 achieving highest similarity first, followed by layers 3 and 4. We found that high alignment isn't always achieved or necessary for effective learning. This may be due to (1) Dimensional reduction: Information propagation through the network affects alignment, and (2) Non-linearity: Activation functions impact the relationship between forward and backward weights. These factors may contribute to observed alignment patterns without requiring perfect symmetry between forward and backward weights

(*) uses CNN with local receptive fields

[1] Hinton, G. (2022). The forward-forward algorithm: Some preliminary investigations.

[2] Dellaferrera, G., Kreiman, G., and Kreiman, G. (2022). Error-driven Input Modulation: Solving the Credit Assignment Problem without a Backward Pass.

[3] Andreas Papachristodoulou, Christos Kyrkou, Stelios Timotheou, and Theocharis Theocharides. (2024). Convolutional channel-wise competitive learning for the forward-forward algorithm.

[4] Journé, A., Rodriguez, H. G., Guo, Q., & Moraitis, T. (2023). Hebbian deep learning without feedback.

Meta-review: The article presents a new biologically-plausible learning network that addresses the weight transport problem, the nonlocal update problem, and the backward locking problems of backpropagation. A key idea is to use a feedback network with trained weights. Reviewers agree to the novelty of the ideas in the paper, and note strong experimental results.
Decision: Accept (poster)