Gradient-adjusted Incremental Target Propagation Provides Effective Credit Assignment in Deep Neural Networks
Authors: Sander Dalm
Nasir Ahmad
Luca Ambrogioni
Marcel van Gerven
Venue: Accepted by TMLR ()

Abstract: Many of the recent advances in the field of artificial intelligence have been fueled by the
highly successful backpropagation of error (BP) algorithm, which efficiently solves the
credit assignment problem in artificial neural networks. However, it is unlikely that BP is
implemented in its usual form within biological neural networks, because of its reliance on
non-local information in propagating error gradients. Since biological neural networks are
capable of highly efficient learning and responses from BP trained models can be related
to neural responses, it seems reasonable that a biologically viable approximation of BP
underlies synaptic plasticity in the brain. Gradient-adjusted incremental target propagation
(GAIT-prop or GP for short) has recently been derived directly from BP and has been
shown to successfully train networks in a more biologically plausible manner. However,
so far, GP has only been shown to work on relatively low-dimensional problems, such as
handwritten-digit recognition. This work addresses some of the scaling issues in GP and
shows it to perform effective multi-layer credit assignment in deeper networks and on the
much more challenging ImageNet dataset.

Published in Transactions on Machine Learning Research (01/2023)
Gradient-adjusted Incremental Target Propagation Provides Effective Credit Assignment in Deep Neural Networks
Sander Dalm Nasir Ahmad Luca Ambrogioni Marcel van Gerven Department of Artificial Intelligence Donders Institute for Brain, Cognition and Behaviour Radboud University, Nijmegen, the Netherlands
Reviewed on OpenReview:
Abstract
Many of the recent advances in the field artificial intelligence have been fueled by the highly successful backpropagation of error (BP) algorithm, which efficiently solves the credit assignment problem in artificial neural networks. However, it is unlikely that BP is implemented in its usual form within biological neural networks, because of its reliance on non-local information in propagating error gradients. Since biological neural networks are capable of highly efficient learning and responses from BP trained models can be related to neural responses, it seems reasonable that biologically viable approximation of BP underlies synaptic plasticity in the brain. Gradient-adjusted incremental target propagation (GApT-prop or GP for short) has recently been derived directly from BP and has been shown to successfully train networks a more biologically plausible manner. However, far, GP has only been shown to work relatively low-dimensional problems, such as handwritten-digit rocognition. This work addresses some of the scaling issues in GP and shows it to perform effective multi-layer credit assignment in deeper networks and on the much more challenging ImageNet dataset.
1 Introduction
Backpropagation of error (BP) is learning algorithm that solves the credit assignment problem in deep neural networks, allowing for the formation of task-relevant internal representations and high performance in applications (Rumelhart, Hinton, and RJ 1986 LeCun, Bengio, and Hinton 2015 Schmidhuber 2014). Despite its efficacy, the default construction of BP does not appear a likely candidate for the computational steps involved in the learning algorithm of real neural systems. Early criticisms of BP's biological plausibility have already been put forward by Grossberg (1977 and Crick (1989) in the late 1980's. A recent review by Lillicrp et al. (2020) provides modern summary of the mechanisms that make BP biologically implausible. These include issues of symmetric synaptic weight matrices, error propagation machinery and more. In particular, these issues all pertain to the use of non-local information for the propagation and computation of error signals to update individual synaptic connections deep within a network. Therefore, biologically plausible learning algorithms are required to provide sensible methods for the assignment of credit deep within neural network model using local-only information.
Several lines of previous work have attempted to address the biological implausibilities inherent to BP. Lillicap et al. (2016) report the counter-intuitive result that a fixed random matrix of feedback weights can support learning in neural networks. This learning is presumed to take place as a result of the forward weight matrices adapting themselves to act as local pseudoinverses of the feedback matrices around the error manifold. This approach is called feedback alignment (FA) and it does away with the necessity of symmetric weight matrices. Direct foedback alignment (DFA) was proposed by Nokland (2016) and works by connecting random leedback weights from the output layer directly to each hidden layer. This approach has been used to

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train modern deep learning architectures (Launay et al. 2020). but its performance compared to regular FA on challenging datasets actually seems to be slightly worse (Bartunov et al. 2018 This may have to do with the relatively simplistic way of performing credit assignment, which does not fully exploit the expressiveness
of multi-layer architectures.
Equilibrium Propagation (EP) (Scellier & Bengio, 2017 is a two phase energy-based model. In the first phase, the network's input units are clamped to a particular input and the rest of the network settles to certain equilibrium state. In the second phase, the network's output units are weakly clamped, meaning that they are nudged toward desired network output using a small constant clamping factor. This perturbation then propagates backward throughout the network and is used to inform weight updates, following a contrastive Hebbian learning rule. These updates are shown to follow the error gradient as they would have been computed by BP. An extension of this work applied EP, with some improvements, to convolutional neural networks to show performance comparable to BP on CIFAR10 (Laborieux et al.) 2021). Though theoretically appealing, one downside appears to be that EP's energy dynamics come with significant computational overhead, with CIFAR10 run reportedly taking two days on a GPU.
Burstprop Payeur et al. 2021 is a learning rule for artificial pyramidal neurons that induces Long-Term Potentiation (LTP) if presynaptic event is followed by a postsynaptic burst and Long- Term Depression (LTDD) otherwise, where the bursting is controlled by topdown feedback connections. This method is shown Lo effectively learn on the ImageNet dataset and to approximate gradient descent on the error landscape (Greedy et al. 2022). A drawback of the method is that it simulates ensembles of neurons rather than individual neurons and demands a rather complex network topology with learned feedback connections and multi- conttitoomartmeent neurons.
Predictive coding [Whittington & Bogacz 2017 alternatively, handles error signals by assuming the existence of separate error neurons for each value neuron. These neurons compute differences between predicted and observed activations and these prediction errors can be used to inform local weight updates. The presence of an error neuron for each value neuron, however, is a rather strong assumption without any neurobiological evidence supporting it. Though most predictive coding research has focused on smaller datasets (Millidee let al.) 2022), Han et al. (2018) report success in learning CIFAR10 and ImageNet using a bidirectional predictive coding network.
Ideally, a biologically plausible algorithm would learn directly from changes in activities of neurons. An algorithm which aims to do just that is target propagation (TP) (Bengio 2014 Lee, Zhang, Fischer, and Bengio 2015). The key principle is to combinc a desired network output with an (approximate) inverse model of network in order to produce desired outputs for each layer of network. These computed layer-wise target activities can then be taken as local supervised labels and used for learning. TP addresses the weight transport problem, because it uses only local activity targets to achieve multi-layer learning. However, there is no explicit theoretical relationship between TP and BP. Furthermore, it has been shown that the efficacy of TP in deep networks that are trained to solve difficult tasks is questionable (Bartunov et al. 2018).
Bartunov et al. (2018) show that biologically plausible algorithms like TP, FA and DFA perform reasonably well, though still worse than BP, on MNIST (Deng) 2012) and CIFAR10 (Krizhevsky 2012 when applied to fully connected architectures. However, in locally connected architectures and on the ImageNet (Russakovsky ct al. 2014) dataset, a significant performance gap appears between all biologically inspired algorithms and BP. Particularly, TP and its variations fail to learn almost completely on ImageNet. FA learns the task to an extent, but its final top-1 test accuracy is about one fourth that of BP.
A common shortcoming among the above approaches is that no theoretical guarantees are made that their updates closely approximate the error gradient as computed by BP. Biologically plausible approaches tend to use local learning rules to avoid the weight transport problem. These local targets often provide useful leedback information in relatively shallow networks, but when multi-layer credit assignment is required, compounding errors drive the updates away from those computed by BP. This failure to accurately track the error gradient likely explains why no activity based (e.g. TP) learning algorithm has so far managed to scale to ImageNet and remain competitive with BP in terms of performance. Scaling biologically plausible learning to real world problems calls for a learning rule that works based on local information but somehow still guarantees a close correspondence to BP.
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Though TP in its original formulation does not track BP's weight updates very accurately, Ahmad al. (2020) recently found that there exists direct correspondence between larget-based learning and BP, though this relationship only exists locally and under specific network constraints. The correspondence between target-based learning and BP can be arrived at through a method called gradient-adjusted incremental target propagation (GP). GP was derived specifically to maintain exact correspondence to BP in the presence of orthogonal weight matrices and exact inverses, which may themselves not be biologically plausible. However, the main contribution of GP is that it allows for a local, activity-based learning rule that still closely approximates BP. This is accomplished by computing those layer-wise targets which would produce weight updates equivalent to BP. In practice, GP was shown to match the performance of BP on the MNIST, FMNIST and KMNIST datasets for relatively shallow networks. However, deeper networks and more challenging tasks were not explored.
The current paper shows that GP its standard form can diverge from BP due to issues of precision and proposes a solution to these issues. In particular, we propose layer-wise target normalisation procedure that stabilizes learning with GP in deeper works. We also describe an easy to implement procedure for inverting convolutional neural network layers in order to efficiently extend GP's utility to problems like the ImageNet classification task (Russakovsky et al. 2014
2 Methods
2.1 Target propagation and GAIT propagation
The key idea is that layer-wise activity targets, rather than error gradients, are propagated backwards through the network. The difference between the current activity and target activity acts as a local error signal to compute weight updates. An inverse mapping from layer to layer 1- can be used to determine which activation vector in layer /-1 would have produced this more desirable activation in layer I, which then becomes the target for layer / 1, and so on. As discussed, though elegant, this method does not produce weight updates that are similar to those produced by BP. The first reason why TP yields different weight updates from BP, is simply because TP uses either learned or exact matrix inverses to propagate its activity targets, while BP uses the transpose of the forward matrix to propagate its errors backwards. For non-orthogonal weight matrices, the inverse differs from the transpose. Furthermore, TP does not account for local gradients when perturbing the current activities in the direction of target activities.
GP is based on the idea of target propagation [Bengio 2014 Lee et al. 2015)
GP seeks to address both of these problems. In order ensure that matrix inverses produce the same transformation as matrix transposes, it constrains weight matrices to be orthogonal. This is achieved by orthogonal initialization of weights, followed by regularization during each weight update. GP also accounts for local gradients by adjusting its activity perturbations, multiplying them by the square of the local gradient. Under the constraint of square orthogonal weight matrices, GP follows the gradient as computed by BP arbitrarily closely and performs competitively with BP on datasets on the scale of MNIST. In this paper, we find that when scaling up GP to more complex datasets, and thus larger networks, numerical instability issues occur. The resolution of this scaling issue the first major subject of this paper.
Let us define the function F as the forward-pass mapping between layers in a neural network such that for layer [ 1+,,
a++1 = (a:) = f (Wiar + by)
(1)
where at is the activation vector, W2 is the weight matrix and by is a bias vector for layer L. The function f()) represents an activation function. In TP, layer-wise targets, ttP are propagated backwards from layer to layer I- 1, by taking the targets of layer I and applying a (learned) inverse function to them:
t G (
where G2 the (learned) inverse mapping from layer layer 1- so that with a perfect inverse mapping, (F)()))) aj. Exact inverses require an equal number of neurons in each layer, however, this constraint can be relaxed by using auxiliary units which be interpreted as form of perceptual memory Ahmad
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Normalized Target
Nearby Target
Distant Target
Target
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A
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Figure 1: A depiction of the activation and target vectors. A) Targets and activations are normalized to be at specific distance. B) Targets and activations become too close such that floating point precision issues occur. C) Targets and activations diverge, weakening the correspondence between GP and BP.
et al. 2020 In this work (and in the original GP work), the inverse function G2 is defined exactly such that a = FT (a1+1) Gi (a++1) = W/ W (a1+1) bt)
(3)
In order to enable this exact inversion, weight matrices are square and initialized orthogonal and the Leaky-ReLU activation function (invertible for all real-valued outputs) was used.
GP proposes a modification to the targets being inverted. The inverted targets were modified such that they conslitute a small perturbation from the forward-pass activily toward the target activity. The perturbation is also multiplied by the square of the activation function derivative at the forward-pass activity. Given this definition, the GP targets, 1EP, are computed after transformation such that:
(4) where DD is perturbation parameter with "n constant with very small magnitude and Dr diagonal square matrix with the derivatives dft of the forward mapping Fi on its main diagonal. This target day adjustment is constructed in order to obtain updates equivalent to BP under the condition of orthogonal weight matrices Ahmad et al. (2020)
T = G ((I -4) 4t) at + 4 tPP)
2.2 Scaling GP to ImageNet
Scaling GP to problems like ImageNet requires us to address the numerical instability that arises in GP when it is applied to deeper networks, as described in the next section. In addition, it requires the implementation of local connectivity to make parameter estimation feasible. Recall that it was the application of deep convolutional neural networks (CNNs) to the ImageNet challenge in 2012 that sparked the revival of neural networks for image classification (Krizheskyy et al. 2012). To our knowledge, there are no practical methods for achieving high performance on ImageNet without using local connectivity. To address these challenges, we propose a method of normalizing target-activation distances and a method to utilize an invertible convolution operation using GP, which sidesteps the need to invert large matrices in memory.
2.2.1 Normalized targets
We find empirically that when inverting targets deep in a network, precision errors in the inversion can occur when the target versus activity difference becomes very small. Repeated inversion with fixed layer-wise incremental factors can result in net increase or decrease (explosion vanishing) of the target-activation distance, which is the key error term used for learning.
To alleviate the problem target- actevvticti differences becoming susceptible to precision errors (or increasing beyond the limits of our linear approximation), we propose a local normalization of the incremental factors " based upon the aagget-cctiattin difference (Figure D. For each layer independently, we divide the incremental factor by the 1.-norm of the target-activale intationtivation difference 71 This ensures that our 'distance to target' is fixed
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a layer-wise basis, ensuring robust error propagation. Specifically, the value of the incremental factor for layer 1 is computed as
"
(5)
||: 1/p11
where is normalization constant determining the desired Euclidean distance between targets and activations. Note that when parge-acctiation distances are very small, y may be bigger than 1. This amounts to inflating the target-activation distance rather than decreasing it. See Appendix B for a pseudocode explanation of GP's backward pass which includes normalized targets.
2.2.2 Application of GP to CNN architectures
GP roquires network layers to be invertible. Because inverting large sparse matricos in memory is prohibitively expensive, we introduce way to invert convolutional layers in a patch by patch manner, greatly reducing computational cost. The output of convolutional layer is inverted by running a transposed convolution that takes as its input the layer's output and as its weight matrix the same kernel as used in the forward pass, but inverted. For these four-dinensional kernels to be inverted, they first need to be reshaped to be square, then transposed, then inverted, then reshaped back to their original format. See Appendix [ for a pseudocode explanation. Note that if these kernels were always perfectly orthogonal, simply taking the transpose would be equivalent to inversion. However, since our regularizer does not ensure perfect orthogonality, we take the exact inverse.
The above method correctly inverts a convolutional layer's output only in the case where receptive fields do not overlap. When receptive fields do overlap, those parts of the input image that are in the scope of multiple receptive fields will be reconstructed multiple times, requiring renormalization. An easy way accomplish this, to divide each pixel in the reconstruction by the number of receptive fields it a part of. This will correctly reconstruct any input image.
However, this process will also divide error information flowing through the pixel by the same mumber, causing reduction in the error signal. To remedy this in the context of GP, we use a different method: the input reconstruction is corrected by removing from it the forward signal (rather than feedback signal) equivalent to the number of reconstructions (minus one). This allows the layer to reconstruct its input exactly once. Importantly, any error information coming from higher layers will be preserved this way such that error information is fed back and integrated from multiple overlapping receptive fields. See Appendix for a pseudocode explanation.
2.2.3 Additional constraints
Exact inverses can only be computed for full-rank square matrices in which there is no loss of dimensionality. This means that, at minimum, the number of output channels of a filter must be equal to the imput dimensionality, that is, the number of output channels must be equivalent to the kernel height times kernel width times number of input channels (Cout = Ci H W).
In order to achieve change in network width, auxiliary output channels can be used akin to the auxiliary units used by Ahmad et al. (202)). This amounts to slicing off a number of channels from the output of a convolutional layer, before feeding the result into the next layer. When inverting the convolutional layer, its entire output, including the auxiliary units, should be used as input to the inversion operation. The auxiliary outputs should therefore not be discarded after the forward pass, but kept in memory.
2.3 Network Models and Training
To investigate the above two proposed improvements, together referred to as normalized GP, we trained and lested several deep neural network models with the CIFAR10 and ImageNet datasets. We trained two fully connected networks on CIFAR10 using either BP, FA, GP or normalized GP; one shallow (three hidden layers) and one deep (six hidden layers). We further trained a convolutional network on ImageNet using only normalized GP, FA or BP. FA was chosen as an additional bio-plausible benchmark because performed the best of all bio-plausible algorithms in Bartunov et al. 2018), though it still lagged significantly behind BP.
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Figure 2: Architectures used to train CIFAR10 shallow (left), CIFAR10 deep (middle) and ImageNet (right). Fully connected layers are indicated with FC, convolutional layers with Conv. For FC layers, the number indicates the number of units. For Conv layers, the numbers represent the kernel size, stride and output channels, respectively.
The convolutional architecture and hyper-parameters were also adapted from Bartunov et al. 2018 Due to memory constraints, we reduced the number of filters by half when training ImageNet. This likely impacts performance, however desired to investigate whether GP can perform competitively with BP on large dataset in deep convolutional network, rather than to maximize peak performance.
During training, parameters were updated for all methods using the Adam optimizer (Kingma & Ba 2014). In addition, when using GP, a regularizer was used to encourage orthogonality of weight matrices. The strength of the regularization component is a scalar hyper-parameterr that indicates by what number the regularization component is multiplied before adding it. to the weight update. See Ahmad et al. (2020) for details. In orthogonal,
addition to being regularized to be orthogonal, all weight matrices were also initialized ensure efficient training from the start of the experiment. Other performance enhancing techniques like batch normalization, 12-regularization and dropout not used. All network layers implemented the Peaky-ReLU activation function and categorical cross-entropy was used as loss function.
3 Results
In the following, consider networks trained on CIFAR10 and ImageNet using different learning algorithms. Appendix A provides overview of the train and Lest accuracies at the end of training. We will use vGP for the vanilla GP formulation as used in Ahmad et al. (2020) and reserve GP for the formulation which uses normalized targets as proposed in this paper.
3.1 CIFAR10 results for shallow network
The correspondence between weight updates from different algorithms was mcasured by treating each update as a one dimensional vector and measuring the angle between them. Figure 3 and show the angles between (v)PP's and FA's weight updates with respect to those of BP when training a relatively shallow fully connected network on CIFARI0. The angle between the updates fluctuates around one degree for Gv)GP, indicating a very high degree of correspondence between the algorithms. FA's updates show angles between and 80 for all non-output layers. Note that random update would produce 90 degree angle in expectation.

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Figure 3: Performance of BP, vGP, GP and FA CIFARIO for fully connected network with 3 hidden layers. For GP and FA, correspondence to BP's updates are displayed. A) Angles between GP's and FA's weight update vectors and BP's per layer. B) Angles in degrees wetween GP's and FA's updates and BP's weight updates per train step. C) Train accuracy BP, GP and FA per epoch. D) Test accuracy for BP, GP and FA per epoch.
Figure 3A shows that the angle is zero for the output layer for all algorithms. This is expected, as the updates are by definition identical for that layer. Updates in earlier layers in the network, which are arrived at after more iterations of (v)GP or FA, diverge very slightly the case of GP and significantly in the case of FA. They are also almost identical in their test performance, as can be seen in Figure 3D, though BP stays ahead in train performance, almost perfectly, though both are present in the figure. Though there is gap in test performance, BP does seem to converge much more quickly on the train set. This likely reflects two things. (v)GP works on the asumption of perfectly orthogonal weight matrices, which are not present in practice, despite the regularizer being used. This means GP can only approximate BP's performance. Second, the presonce of the ort ort hogonaliyty regularizer itself, dilutes (v)GP's lask-related updates with orthogonality-relttel updates, slowing down convergence.
shown in Figure 3C. Note that the lines for GP (orange) and vGP (pink) overlap
3.2 CIFAR10 results for deep network
Figure 4 shows results for the same experimental setup, except with deeper network, consisting of six hidden layers. As can be seen in Figure A and B, updates for vGP quickly diverge from BP's updates, with updates for GP maintaining their correspondence to BP for all layers. In terms of performance, vGP shows poor performance this setting, while GP remains competitive with BP on the test FA outperforms vGP this deeper network, but lags behind both BP and GP.
3.3 Deep convolutional networks
On the ImageNet dataset, like on CIFAR10, GP performs competitively with BP in terms test performance, see Figure 5 FA almost completely fails to learn this experiment, showing some learning only the carliest. epochs of training, followed by a decrease in performance training continues. vGP was not included in the ImageNet experiment, it failed to learn well even the much simpler CIFAR10 task in deep network.

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Figure 4: Performance of BP, vGP, GP and FA CIFARIO for fully connected network with 6 hidden layers. For GP and FA, correspondence to BP's updates are displayed. A) Angles between GP's and FA's weight update vectors and BP's per layer. B) Angles in degrees wetween GP's and FA's updates and BP's weight updates per train step. C) Train accuracy BP, GP and FA per epoch. D) Test accuracy for BP, GP and FA per epoch.
Angles between GP's and BP's updates are again consistently low, though higher than in the CIFAR10 experiments. Angles between FA's 's and BP's updates fluctuate around 90 degrees, indicating little to no correspondence. Even for GP can be seen that the updates slightly diverge from BP's in deeper layers, with the average angle at around 10 degrees, but reaching slightly over 20 degrees for the first hidden layer. It can also be seen Figure that GP converges slower than BP on the train set, though it eventually reaches parity in test accuracy. As in the CIFAR10 experiments, we attribute the gap in train performance to weight matrices not being perfectly orthogonal in practice and the orthogonality regularizer slowing down convergence. There is thus tradeoff when using GP, where regularizing more strongly gives better correspondence to BP at the cost of slower convergence.
The above interpretation was confirmed by running the network without performing any task-rclated weight updates, in which case the angles stay near zero throughout the network.
4 Discussion
This paper set out to investigate how to scale GP to more challenging problems. We found that in principle, GP scales to larger and more complex architectures than the relatively shallow fully connected networks used previous research. This was achieved by tackling its numerical instability issues and adapting the algorithm CNN architectures. Other biologically inspired learning rules either fail to scale to problems the
size of ImageNet [Bartunov et al. 2018) require complex network topologies (Whittington & Bogacz 2017
Millidge et al. 2022 Han et al. 2018 Payeur et al.] 2021 Greedy et al. 2022
require a slow iterative
process to learn Scellier & Bengio (2017); Laboricux et al. 2021).
The main limitation of GP in terms of its performance is its reliance on invertible network architectures. This makes it incompatible with certain activation functions, as well as non-invertible operations like max-poole noolino.ing.
Additionally, GP, in its current implementation, runs significantly slower than BP. This is due the need to invert weight matrices, include auxiliary units, apply the orthogonality regularization and perform
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Figure 5: Performance and correspondence with BP for GP and FA on ImageNet in convolutional neural network with convolutional layers. A) Angles in degrees between BP's updates and those by GP and FA per layer. B) Angles in degrees between BP's updates and those by GP and FA per train step. C) Top train accuracy for BP, GP and FA per epoch. D) Top test accuracy for BP, GP and FA per epoch.
customized, and thus less optimized, backward pass. This slowdown relative to BP stems from the specifies of modern computer architectures. biologically inspired algorithm such as GP would be expected to run much faster on neuromorphic hardware optimized for such computations compared to current hardware. Due to its lack of biological realism, BP likely would run on such hardware at all, if this hardware is subject to the same physical constraints the human brain, c.g. the locality of information.
Finally, the presence of orthogonality regularizer and the fact that GP's assumptions are not fully realized in practice appear to slow down convergence relative to BP.
As discussed above, some of the remaining limitations may be remodicd in various ways. The goal of the current. research, however,
demonstrate that, in principle, biologically plausible learning can scale real-world problems. This demonstrated by GP's competitive performance with BP on the complex ImageNet problem, given some constraints on the architecture. We argue that demonstrating the efficacy of any learning algorithm to large-scale problems is crucial since algorithms that seemingly work well on toy problems have tendency to break down in more complex settings.
A reader may also ask whether the issues addressed in this work are relevant for biology. For example, the issue overcome by normalization the target outputs is an issue precision of floating point numbers. However, we would argue that biology, any noise floor would also ensure that a sufficiently small target signal would be drowned out amidst network activity (Faisal et al. 2008). One might also ask whether orthogonality regularizer is biologically plausible, given that in order to orthogonalize weights, non-local information needs to be used. Here, would argue that lateral inhibition is well-established phenomenon in neuroscience that could produce similar decorrelating effect (Békésy) 1967). Therefore we propose that these improvements are also relevant to any biologically motivated analysis of these learning rules.
The presence of (approximately) inverse weights between layers and the need for separate forward and backward phases in the network seem be the most prominent remaining departures from biological plausibility, making the elimination of these mechanisms excellent targets for future research.

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Timothy P Lillicrap, Daniel Cownden, Douglas B Tweed, and Colin Akerman. Random synaptic feedback weights support error backpropagation for deep learning. Nature Communications, 7(1):1-10, 2016.
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Published in Transactions on Machine Learning Research (01/2023)
Beren Millidge, Tommaso Salvatori, Yuhang Song, Rafal Bogacz, and Thomas Lukasiewicz. Predictive coding: Towards future of deep learning beyond backpropagation? ArXio Preprinl ArXi::2202.09407, 2022.
Arild Nøkland. Direct feedback alignment provides learning in deep neural networks. Advances in Neural Information Processing Systems, 29, 09 2016.
Alexandre Payeur, Jordan Guerguiev, Friedemann Zenke, Blake A Richards, and Richard Naud. Burst- dependent synaptic plasticity can coordinate learning hierarchical circuits. Nature Neuroscience, 24(7): 1010-1019, 2021.
D. Rumelhart, G. Hinton, and Williams RJ. Learning internal representations by error propagation. Parallel Distribuled Processing: Explorations in the Microstruchure of Cognition, 1, 07 1986.
Olga Russakovsky, Jia Deng, Hao Su, Jonathan Krause, Sanjeev Satheesh, Scan Ma, Zhiheng Huang, Andrej Karpathy, Adilya Khosla, Michael Bernstein, Alexander Berg, and Li Fei-Fei. Imagenet large scale visual recognition challenge. International Journal of Computer Vision, 115, 09 2014.
Benjamin Scellier and Yoshua Bengio. Equilibrium propagation: Bridging the gap between energy-based models and backpropagation. Frontiers in Computational Neuroscience, 11:24, 2017.
Juergen Schmidhuber. Deep learning in neural networks: An overview. Neural Networks, 61, 04 2014.
James Whittington and Rafal Bogacz, An approximation of the error backpropagation algorithm in prodictive coding network with local Hebbian synaptic plasticity. Neural Computation, 29:1-34, 03 2017.
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Published in Transactions on Machine Learning Research (01/2023)
A Performance comparisons
Table 1: Final train and test accuracies for networks trained on CIFARIO (top-one accuracy) and ImageNet
(top-five accuracy) using backpropagation (BP), normalized and vanilla GP (GP) and feedback alignment
(FA).
CIFAR10 (shallow)
CIFAR10 (deep)
ImageNet (deep)
Algorithm
Train
Test
Train
Test
Train
Test
BP
92.4
53.2
99.2
52.5
52.7
26.4
GP (normalized)
64.9
53.7
61.4
52.0
46.6
26.5
GP (vanilla)
64.8
53.8
14.9
14.8
FA
66.2
51.3
40.8
38.2
0.8
0.8
B Pseudocode for normalized GP's backward pass
Algorithm Pseudocode for normalized GP's backward pass.
parameters: tL, the network's activation target at the output layer; a, the network's activation from the forward pass for each layer; G1, an inverse mapping for each network layer; Dı, the data dependent. derivatives for each layer; 7; small constant; a, learning rate for layer / from L to do
Wi Wi- (Di [
Compute weight update based on quadratic loss b Compute bias update > Set perturbation factor to normalize target-activation differences P Adjust perturbation factor for local gradients & Propagate targets back through network
by t by -aDi(aa - ft) 7
" ||a--1|l"
E = nDf
4-1 Gi ((I - Et) ai + el 41)
end for
return updated weights Wi, updated biases b
C Pseudocode for inversion of a convolutional layer
A Algorithm Inversion of convolutional layer
parameters: yt, the convolutional layer's output; wi, the convolutional layer's kernel; æ, the input of the convolutional layer's forward pass; n, mapping that contains for each element of how many overlapping filters it is in the receptive field of
f = Transposedd caanmppssd_Convvolttiooooo/// w/))
î + 2 - 3 (n= 1)
return reconstructed input &
D Code availability
Code used for the experiments can be found here:
_pr____rp______ pcaassiiggn
12


Reviews:
[{"review_id": "v4gq3O7vPM", "reviewer": "Reviewer_r9jh", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Overall, this is an important contribution to the field, which tries to scale up existing biologically plausible algorithms. But there are several points that should be ideally addressed/clarified as detailed below.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper aims to extend the previously introduced GAIT-prop learning algorithm to deeper networks and to convolutional layers. They argue that GAIT-prop is a more biologically plausible learning mechanism that has been theoretically shown to approximate backprop but that it is necessary to show that this approach scales. Specifically, they introduce a modification to the GAIT-prop targets to normalise the magnitude of their difference to the true activity. By normalising, this alleviates an issue they identified with standard GAIT-prop where small target differences are prevented from propagating due to numerical precision and large differences from deviating the updates far from backprop. To extend the approach to convolutional layers, they note that the convolution operation can be viewed as a simple matrix multiplication that can be inverted provided the same conditions of square and orthogonal weight/kernel matrices are enforced. They introduce a way to invert these matrices in a patch-by-patch manner to greatly improve computational efficiency. Finally, they show that their new model’s updates align to backprop and is competitive with backprop’s test accuracies on the CIFAR10 and ImageNet datasets.\n\n", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "Note: If some of the points below are not feasible, we will take this into account.\n\nMajor points:\n\n•\tIn the introduction it seems the authors are incorrect in their summarisation of the work of Guerguiev et al. (2017). They state that the separation of neurons into multiple compartments resolves the implausibility of requiring symmetric feedback weights which is not the case. This is evidenced by the fact that this work uses fixed random feedback weights (i.e. feedback alignment) for many of their experiments to avoid the implausibilities of this issue. Additionally, they claim that no separate feedback pathway is used despite the fact that separate feedback connections are used. Finally, they state that the weight transport problem is a separate problem that is also addressed however in this case it is the same problem as the symmetric feedback weights and, again, this work does not provide a new solution to this problem.\n\n•\tIt is stated that “Target Propagation addresses the weight transport problem, because it uses only local activity targets to achieve multi-layer learning.” I don’t believe this is true since TP requires knowledge (i.e. weight transport) of the feedforward weights to construct the feedback weight matrix as its inverse in order to generate the targets. It can be potentially argued that TP allows for an easier mechanism of learning the feedback weights to avoid weight transport, but this is a separate argument.\n\n•\tIn the introduction it says “no biologically plausible learning algorithm has so far managed to scale to ImageNet and remain competitive with BP”. While it is not competitive with BP it should be mentioned that Burstprop (Payeur et al. Nature Neurosci 2021) has been shown to scale to ImageNet. Moreover, a new model (BurstCCN, Greedy et al. NeurIPS 2022) is shown to align near-perfectly with backprop under certain conditions (see their CIFAR-10 task results). As BurstCCN builds on Burstprop it would imply that it would do well in Imagenet. Importantly, BurstCCN is currently the model that gets the closest to biology, capturing a wide range of experimental observations. These models should be discussed, as they are currently the best biologically plausible models of backprop-like credit assignment.\n\n•\tIn Section 2.2.2 it is not immediately clear or explained how the reshaping of the 4d kernel matrix to be square is being accomplished. It would be useful to explain more clearly what the shape of the reshaped kernel is after reshaping but before transposing and inversing. Also, in Figure 2, is “image patch” referring to for example a 3x3 patch (with kernel size 3x3)? If so I think it would be useful to replace the full image that is being shown there with just the small patch to avoid confusion.\n\n•\tThe plots in Figures 4-6 appear to be running with just a single seed even for the simpler CIFAR-10 task. It would be useful here to include the mean and variance of these quantities across multiple seeds.\n\n•\tDue to the large disparity between the training and test accuracy in all tasks for BP it would be important to discuss the use of regularisation in more detail. Firstly, does the orthogonality regulariser for used in GAIT-prop also impact generalisation in a similar way to an L1 or L2 regulariser? Secondly, why were other regularisation techniques not included (especially in BP) if it appears to be overfitting the training data significantly? Can they be successfully applied to GAIT-prop in combination with the orthogonality regulariser?\n\n•\tIt is stated multiple times that normalised GAIT-prop is competitive with BP (in terms of test performance) but the differences in training performance seem to indicate that BP is still significantly more powerful but simply failing to generalise, would be important to make this clear.\n\n•\tWhy are the models not trained until convergence in Figure 6?\n\nMinor points:\n•\tIn the introduction it is stated that “there is no explicit theoretical relationship between TP and BP” but also “there exists a direct correspondence between target-based learning and BP” which appear to contradict each other without a distinction being made more clear.\n\n•\tFor Figures 4-6 it would be more clear if these were imported in a vector format (svg/pdf) instead of a png. Alternatively, a higher resolution png could be used.\n\n•\tFor clarity, Fig. 6 should use the same colour for the “GAIT-prop normalised” as used in Fig. 5 and 4.\n\n•\tIn Section 3.3, Layer 0’s update angle is referred to as the angle for the input layer but this is actually the first hidden layer’s update angle.\n\n•\tIn Section 4, it is mentioned that normalised GAIT-prop does not use non-local information, but the layer-wide normalisation is arguable non-local information to the neuron. This point could be clarified that it is local to the layer.\n\n•\t“A biologically inspired algorithm such as GAIT-prop would be expected to run much faster on neuromorphic hardware optimized for such computations.”. Should clarify if this is in faster in reference to BP or just the currently used implementation.\n\n•\tIn Section 4, it is argued that BP would likely not run on neuromorphic hardware at all. It would be useful to provide a reference for this claim.\n\n•\tIn Section 4, it would be useful to include more insight into the link between the decorrelating effect of lateral inhibition and the orthogonality regularisation. The plausibility of this mechanism seems vital to the plausibility of the whole system.\n\n•\tTypos:\no\tIn Section 2.3 “hyper parameters”\no\tSection 4: “GAIT-prop scales to larger and more complex architectures than the relatively shallow fully connected networks used in previous research”\no\tSection 4: “algorithm that seemingly work”\n", "broader_impact_concerns": "No concerns.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "eY58IJaAKp", "reviewer": "Reviewer_iHJb", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strenghts\n\n- The scaling of bio-plausible algorithms to big networks/data is an important problem that is often ignored. \n- The manuscript is readable and highlights the problems that are to be solved without getting too bogged down in details (but see weaknesses below).\n\nWeaknessses\n\n- A lot of the details are missing. \n   -  To be self-contained, section 2 should be expanded to summarize the following (for detail the reader can be referred to the respective papers but a one or two sentence summary is good): \n      - What is the local gradients problem?\n      - how is the orthogonality constraint implemented in a bio plausible way. \n      - how is $D(a_l)$ computed in a bio-plausible way? Doesn't $d F_l/ d a_l$ have a weight transport problem? It needs $W^\\top.$\n- In section 2.1 it is not fully clear in the text why local connectivity is a problem (it is only later that it is mentioned that the inversion is the issue).\n- The exact methodology is hard to understand. It would be useful to have algorithm boxes such that the *exact* steps for both normalization and inversion can be understood.\n- The plots can use error bars. Running the each algorithm a few times so that the reader understands the run to run variability of the different methods is very important. \n- How does predictive coding scale with depth? The authors dismiss PC on realism grounds (error neurons) but this argument of what is more/less realistic is rather subjective. (I can see readers also doubting many of the choices made in GAIT-prop as plausible but unrealistic.) In this light, it would be useful for the authors to at least comment on the scaling properties of PC. (and ideally provide comparable experiments.) This would make the paper a lot more appealing as a work on how bio-plausible algorithms scale with depth. (Even if modified GAIT-prop ends up losing to PC in scaling, this would still be a very valuable contribution.)\n", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The submission considers modification of GAIT-prop, a biologically plausible alternative to backprop based on Target Propagation, that help GAIT-prop scale to larger networks and datasets. The main contributions are improvements to numerical stability and the introduction of an inverse convolution.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "Some kind of plot that shows the scaling of the different methods with the full depth of the network would be very useful. For example, how does the final test/train accuracy scale with the depth of the network. (i.e. train with a sequence of deeper networks not just 2, and summarize the results in one or two plots)\n\nI believe this is a good work in addressing the scaling properties of bio-plausible learning algorithms but with the current plots, it is demonstrated that the submission's proposal performs better than original GAIT-prop but the *scaling* itself has not become clear. \n\nFor other changes see weaknesses.", "broader_impact_concerns": "I don't foresee any broader impact concerns.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "xlkxxTwwqA", "reviewer": "Reviewer_wyuX", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "er is the shared weights (transposed) between the forward and the backward path (a.k.a the weight transport problem) and between the derivatives of the forward and backward \"activation\", however these joint computations would become \"local\" in the proposals where pyramidal neurons host both the forward and the backward computation.\n* insert \"test\" in the first line of page 7 (before \"performance\"), to be honest, and change the statement \"In terms of train accuracy, BP does seem to converge more quickly than the other algorithms\" to something like \"In terms of train accuracy, BP achieved much better performance much faster and the gap continued to grow by the end of training\". Similarly, remove the disingenous statement in the last sentence of sec 3.\n* the non-plausibility for biological networks of inverting matrices should not be only discussed in last paragraph of the paper but should be stated upfront in the introduction and section 2.1, e.g. around Eq. 3.\n* The cartoon explanation in Fig 2 is very confusing because it looks like applying an inverse matrix to a small vector (the output vector) could recover a rich high-dimensional image (the image patch, 2nd row of the figure). This is of course not plausible unless the output dimension is huge (corresponding to H x W x C_in). Hence the figure looks wrong. It is only when reading the text that we understand that indeed C_out is chosen in this way. This should be clarified in the caption or the whole figure should be redesigned. In any case it did not really make it clear to me what the procedure was and why it worked.\n\nMissing references:\nThere is a nice review of the litterature but it is missing what I consider some important papers, especially in the context of GAIT-prop:\n  * Equilibrium Propagation (Scellier et al, Frontiers Comp Neuro 2017)\n  * Sacramento et al NeurIPS 2018\n  * several follow-up on EqProp including Laborieux et al (also Front. Comp Neuro, but 2021)\n  and Scellier et al 2019 (Neural Comp)\n  * Meulemans et al NeurIPS 2020, ICML 2022 and NeurIPS 2022 (accepted, but it is on arXiv)\n", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "No", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper extends and improves on a previously proposed biologically motivated alternative to backprop, called GAIT-prop, which is a variant on target propagation (that worked better than previous variants). The authors propose to normalize the coefficient used to mix the fed back targets and the feedforward activations when computing the new targets, inversely proportional to the distance between feedforward activations and the fed back targets.  They also propose a way to deal with weight matrices that are not full, such as the implicit matrices corresponding to convolutions. On the chosen architectures, results are better than the compared methods (feedback alignment and the previous GAIT-prop version, unnormalized) and results comparable to backprop with a particular architecture (and no deformations?) are obtained by the authors on ImageNet on the test set but still substantially worse on the training set and MUCH worse than even AlexNet or the current state-of-the-art in test performance.", "claims_and_evidence_yn": "No", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "* as it stands, the title is overclaiming, since the actual results are MUCH worse than even the 2012 ImageNet state-of-the-art: either change the title (and the corresponding claims in the abstract, intro, discussions and conclusion) or obtain more credible results\n* provide actual numbers of final accuracy in a table\n* add parameter count comparisons and discuss the very weak results obtained on ImageNet, or better train larger models that can at least compete with backprop circa 2012\n* add and discuss the missing references\n* explain Eq. 4\n* justify Eq. 5 better and discuss why there are no issues with epsilon_l > 1 (or if there are, discuss them)\n* provide a clear and formal description of the method to deal with convolutions and demonstrate why it is the (or a) correct way to handle covolutional operators\n* add pseudo-code for normalized GAIT-prop with convolutions\n* clarify where requested (see above)\n* use honest statements when discussing the figures, not blaming the inability of GAIT-prop to match training performance of backprop on the overfitting of backprop, and not claiming that GAIT-prop would eventually catch up when the evidence shown in the figures 4 and 5 suggests exactly the opposite.\n", "broader_impact_concerns": "None", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: The reviewers brought up several points to be addressed in the first review round. All reviewers were fully satisfied with the revision. The reviewers noted that the manuscript provides an important contribution to the field. On the negative side, it was noted that the performance is still much worse than standard optimization with backprop. However, this is a general characteristic of any more biologically plausible algorithm available to date.
  Another point was that the biological plausibility of exact inverses is doubtful, which reduces the significance from a biological point of view. Nevertheless, the reported progress it is still worth publishing.
Decision: Accept (poster)