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REVIEW 3 major objections 5 minor 42 references

Forward Target Propagation: A Forward-Only Approach to Global Error Credit Assignment via Local Losses

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper proposes Forward Target Propagation, a training rule that replaces the backward pass with a second forward pass and reports accuracy competitive with backpropagation on image-classification and time-series benchmarks, with…

desk verdict A clean incremental forward-only training rule that works on shallow benchmarks; the theory only covers a linearized case that the experiments never run. read the letter →

arxiv 2506.11030 v1 pith:43MNXPSK submitted 2025-05-20 cs.LG cs.AI

Nazmus Saadat As-Saquib , A N M Nafiz Abeer , Hung-Ta Chien , Byung-Jun Yoon , Suhas Kumar , Su-in Yi This is my paper · ORCID
classification cs.LGcs.AI
keywords backpropagation-freelearningforward-onlytargetpropagationlocalcreditassignmentbiologicallyplausibleneuromorphiccomputinglow-precisiontrainingtime-seriesforecasting
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Forward Target Propagation (FTP) is a proposed replacement for backpropagation that trains a neural network using only forward computations. The paper's idea is to run a second forward pass whose input is a target signal built by contrasting the label with the current output through a fixed random projection, so that each hidden layer receives a local target to move toward. On fully connected, convolutional, and recurrent networks, FTP reports accuracy competitive with backpropagation on MNIST, CIFAR-10, CIFAR-100, and three multivariate time-series benchmarks, while using only slightly more multiply-accumulate operations than backpropagation and far fewer than other forward-only methods. The paper also reports that FTP keeps higher accuracy than backpropagation when forward and backward weight symmetry is broken by programming noise at 8-bit and 4-bit precision. If these results hold, FTP offers a path to local, energy-efficient, hardware-compatible learning without symmetric weight transport.

What carries the argument

The central object is the fixed random projection matrix $G$, which maps both the label $y$ and the current output $h_L$ into a common space; the difference $\sigma(Gy) - \sigma(Gh_L)$ acts as a global error surrogate, and adding it to the first hidden layer's activation produces the target $\tau_1$. That target is then propagated forward through the network's own weights to generate targets for all deeper layers, so every layer can be updated with the locally computable squared loss between its activation and its target. This machinery is what removes the backward pass, the symmetric weight transport, and the non-local error signal from the training loop.

What would settle it

Train a network with five or more nonlinear hidden layers (for example, tanh or ReLU with standard initialization) on CIFAR-100 using FTP and compare test accuracy to backpropagation, while recording the cosine alignment between FTP and backpropagation updates for the first hidden layer; if that alignment stays near 90 degrees throughout training or accuracy collapses toward chance, the central claim that forward-propagated targets remain useful credit signals is falsified.

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Extended reading notes

Core claim

The central claim is that global error can be assigned to every hidden layer with two forward passes instead of a backward pass. The first forward pass computes activations $h_i$; the paper then forms a first-layer target $\tau_1 = \sigma(Gy) - \sigma(Gh_L) + h_1$ using a fixed random matrix $G$, propagates it forward through the same feedforward weights as $\tau_i = \sigma(W_i \tau_{i-1})$, and updates each weight by gradient descent on the local squared loss $\|h_i - \tau_i\|^2$, with the final layer trained by the global loss. The paper reports that this local rule produces update directions that increasingly align with backpropagation gradients during training and that accuracy tracks backpropagation on the tested benchmarks while exceeding it when hardware noise breaks forward-backward weight symmetry. The alignment theorem in the appendix is proved only for a linear two-hidden-layer network with $W_1$ and $W_3$ initialized to zero, so the argument's strength for deep nonlinear networks rests on the empirical results rather than on that proof.

Load-bearing premise

The method's usefulness rests on the unproven premise that a target formed by projecting the label and the current output through a fixed random matrix, then pushed forward through the network's own weights, still carries meaningful learning signal after passing through many nonlinear layers; the paper's proof of this only holds for linear activations with the first and last weight matrices initialized to zero.

Editorial extensions

If this is right

  • FTP removes the need for symmetric forward and backward weights: the only feedback connection is a fixed random matrix $G$, so no transposed weight copies need to be written or verified during training.
  • Because each layer minimizes a local squared loss, weight updates can in principle be computed asynchronously and in parallel once the target signal has passed through the layer.
  • Reported multiply-accumulate counts are within about 1-3% of backpropagation on the tested networks, versus 30-60% more for the target-propagation and forward-only baselines, which matters for edge and TinyML deployment.
  • Under simulated programming errors at 8-bit and 4-bit precision, FTP maintains higher accuracy than backpropagation, suggesting that hardware-induced forward-backward asymmetry is less damaging when no backward weight symmetry is required.
  • The scaling parameter $\gamma$ in $\tau_1 = \gamma(\sigma(Gy) - \sigma(Gh_L)) + h_1$ controls alignment with backpropagation, with smaller values producing stronger directional alignment on the tested two-hidden-layer network.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The proof of FTP's alignment with backpropagation assumes linear activations and zero-initialized outer weight matrices, while the experiments use nonlinear activations and standard initialization; a sympathetic reading is that the method's practical success is an empirical phenomenon whose theoretical guarantee remains open for deep nonlinear networks.
  • The fixed matrix $G$ is never adapted during training, so one testable extension is whether learning $G$ (while staying forward-only) improves both alignment and accuracy on deeper networks.
  • The target signal is formed by contrasting $\sigma(Gy)$ with $\sigma(Gh_L)$, which suggests a capacity threshold: if the random projection $G$ has too few rows to separate many classes, the credit signal could degrade, making the dimensionality of $G$ a natural hyperparameter to study.
  • Reported accuracy on CIFAR-100 (about 26.8% for FTP versus 33.8% for backpropagation) indicates that the method's competitiveness is architecture- and dataset-dependent, so scaling to much deeper models or larger image datasets would be a direct stress test of the credit-assignment claim.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper introduces Forward Target Propagation (FTP), a training algorithm that replaces backpropagation's backward pass with a second forward pass. Targets for hidden layers are computed by projecting the label and output through a fixed random matrix G and then propagating the resulting first-layer target through the network's own feedforward weights. Each layer's weights are updated by a local squared loss against these targets. The authors evaluate FTP on fully connected, convolutional, and recurrent networks for image classification and time-series forecasting, and report accuracy close to backpropagation on MNIST, Fashion-MNIST, and CIFAR-10, with larger gaps on CIFAR-100, along with lower MAC counts than other biologically plausible methods and robustness to low-precision hardware noise.

Significance. The forward-only formulation is a distinctive synthesis of target propagation and fixed random feedback, and the empirical results on shallow networks are promising. The paper is commendable for providing a complete algorithm description, implementation details, and a self-contained theoretical appendix. If the credit-assignment mechanism were shown to hold in nonlinear, deeper networks, FTP could be a useful contribution to biologically plausible and hardware-efficient learning. However, the theoretical justification currently applies only to a degenerate linear case that is not the experimental setting, and the claimed competitiveness with backpropagation is not fully supported by the reported accuracy gaps.

major comments (3)
  1. [Appendix A.1; Appendix B] Lemma 1 and Theorem 1 assume linear activations and initialize W1 and W3 to zero, whereas the experiments (Appendix B) use tanh activations and He initialization for all networks. This mismatch is load-bearing: the proof's rank-one structure, which forces e ∝ y (Eq. 35), is exactly what prevents the hidden layers from collapsing the propagated target signal. Consequently, Theorem 1 does not justify the claim that FTP gradients align with BP gradients in the evaluated setting. The paper should either extend the proof to more general conditions, report alignment measurements on the actual nonlinear architectures used in Tables 1 and 3, or explicitly restrict the theoretical claims to the linear case.
  2. [Section 3.2, Eqs. (4)-(5)] For layers i≥2, the local loss is L_i = ||σ(W_i h_{i-1}) − σ(W_i τ_{i-1})||^2 because τ_i is generated with the same W_i in Eq. (4). This loss can be minimized by making W_i map both h_{i-1} and τ_{i-1} to the same output, for instance by aligning W_i with the left nullspace of (h_{i-1}−τ_{i-1}); in the linear case the update is (W_i(h_{i-1}−τ_{i-1}))(h_{i-1}−τ_{i-1})^T, which vanishes when the error is in the nullspace. The theoretical analysis in Appendix A rules out this degeneracy only through the special initialization of Lemma 1; under He initialization and tanh nonlinearities no such guarantee is provided. The central claim that FTP performs meaningful global credit assignment to hidden layers beyond the first is therefore not established for the architectures actually tested. Additional experiments with deeper networks or an analysis of the nullspace behavior are needed to support the claim.
  3. [Table 1; Abstract] The abstract and Section 5 state that FTP achieves accuracies competitive with BP, but the reported CNN results show a 8.56-point gap on CIFAR-10 (56.32% vs. 64.88%) and a 6.99-point gap on CIFAR-100 (26.84% vs. 33.83%). These are substantial differences, not the within-noise margins seen on MNIST. The accuracy claim should be qualified to reflect the architectures and datasets where the result actually holds, or the comparison should be rerun with a more competitive architecture.
minor comments (5)
  1. [Algorithm 1] The pseudocode shows the loop for computing τ_i and ∆W_i but does not spell out how ∆W_i is derived from L_i at the boundaries i=1 and i=L; writing the explicit local-loss expressions would make the algorithm reproducible without referring to Eq. (5).
  2. [Eq. (7)] The formula for RRSE is corrupted in the rendering (the square root appears as 's'), and the CORR expression is split across lines; please re-typeset these equations.
  3. [Section 3.2] The phrase 'This also eliminates the need to compute the first hidden layer's activation from the input in a second forward pass' is confusing: in FTP the second pass starts from τ1, not from the input, so 'from the input' seems inaccurate; consider rewording.
  4. [Table 4] The MAC-count estimates would be easier to trust if the text explained how the dataset size, epochs, and per-layer operation counts were combined; as written the reader cannot independently verify the percentages.
  5. [Appendix A.2] The derivation of the pseudo-inverse in the proof of Theorem 2 uses rank-one identities that are not stated; please add a brief justification for each equality.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: FTP's target/loss construction, the gradient-alignment theorem, and the empirical comparisons are self-contained; the proof-assumption mismatch is a soundness concern, not a circularity.

full rationale

FTP's derivation chain is: define the first target tau1 via Eq. (3); propagate targets through the same feedforward weights via Eq. (4); define layer-local losses via Eq. (5); and take gradient steps via Eq. (6). This is an algorithmic definition, not a circular prediction. The claimed global-credit property is an empirical and theoretical assertion about that update rule, not an input to the rule. The alignment results in Appendix A.1 (Lemma 1, Theorem 1) provide a genuine invariant proof: starting from the update rules, they prove that, under the stated assumptions (linear activations, W1 = W3 = 0), the FTP updates stay in a rank-one family and the inner products in Eqs. (36)-(37) are positive. The proof does not assume the conclusion; it derives it. The fact that the experimental setup uses tanh and He initialization while the proof assumes linear activations and zero initialization is a real limitation of the theoretical justification, but it is a soundness/transfer gap, not a circular reduction. Similarly, the accuracy results in Tables 1 and 3 compare FTP against BP on held-out test sets; no test-set-derived parameter is relabeled as a prediction. The gamma sweep in Eq. (8) is a sensitivity analysis of a fixed hyperparameter, and the main reported results use gamma = 1. Figure 2b is a measured diagnostic of the weight/G correlation; because G enters FTP's own update rule, some correlation is plausible, but the reported angle trajectory is measured, not derived by definition from G. References to the authors' earlier hardware work are motivational and are not load-bearing for the FTP derivation; no uniqueness theorem or ansatz is imported from those citations. I therefore find no step in the paper where a claimed result is equivalent to its own input by construction.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The central method rests on the target-propagation assumption and on a proof that only covers linear networks with a special initialization. No fitted constants are needed for the main accuracy numbers, but gamma is introduced as a scaling hyperparameter in the alignment study and the missing initial learning rate makes exact replication impossible.

free parameters (1)
  • gamma (target scaling) = 1.0 in main results; 0.5 and 1.5 explored in alignment study
    Section 4.4 introduces gamma to scale the difference target tau1. The paper does not use gamma in the accuracy tables, so it is an auxiliary hyperparameter rather than a fitted constant in the central claim, but it does affect gradient alignment.
assumptions (4)
  • domain assumption A fixed random projection matrix G, uncorrelated with forward weights, yields target directions that improve hidden layer representations.
    This is the core design choice of FTP. The paper provides only a heuristic argument and a proof that holds under linear activations and zero initialization of W1 and W3 (Appendix A.1).
  • domain assumption Targets tau_i = sigma(W_i tau_{i-1}) propagated forward through the same feedforward weights remain meaningful credit signals for hidden layers.
    Section 3.2, Eq. (4). This is the load-bearing premise of the method; it is tested only on networks with at most two hidden layers.
  • ad hoc to paper The alignment theorems assume linear activations and W1 = W3 = 0 at initialization.
    Appendix A.1, Lemma 1 and Theorem 1 state these conditions explicitly. The experiments use tanh activations and He initialization, so the theorems do not apply to the reported results.
  • domain assumption MAC counts computed with the methodology of Pau et al. are a valid proxy for computational cost of training on edge hardware.
    Section 4.6 reports only estimated MACs based on dataset sizes and epochs, with no direct runtime or energy measurements on actual hardware.

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Cite this review

Pith. "Pith review of Forward Target Propagation: A Forward-Only Approach to Global Error Credit Assignment via Local Losses." pith.science (2026). https://pith.science/paper/43MNXPSK

@misc{pith2026250611030,
  author       = {Pith},
  title        = {Pith review of: Forward Target Propagation: A Forward-Only Approach to Global Error Credit Assignment via Local Losses},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/43MNXPSK}},
  note         = {Machine review of arXiv:2506.11030}
}
read the original abstract

Training neural networks has traditionally relied on backpropagation (BP), a gradient-based algorithm that, despite its widespread success, suffers from key limitations in both biological and hardware perspectives. These include backward error propagation by symmetric weights, non-local credit assignment, and frozen activity during backward passes. We propose Forward Target Propagation (FTP), a biologically plausible and computationally efficient alternative that replaces the backward pass with a second forward pass. FTP estimates layerwise targets using only feedforward computations, eliminating the need for symmetric feedback weights or learnable inverse functions, hence enabling modular and local learning. We evaluate FTP on fully connected networks, CNNs, and RNNs, demonstrating accuracies competitive with BP on MNIST, CIFAR10, and CIFAR100, as well as effective modeling of long-term dependencies in sequential tasks. Moreover, FTP outperforms BP under quantized low-precision and emerging hardware constraints while also demonstrating substantial efficiency gains over other biologically inspired methods such as target propagation variants and forward-only learning algorithms. With its minimal computational overhead, forward-only nature, and hardware compatibility, FTP provides a promising direction for energy-efficient on-device learning and neuromorphic computing.

Figures

Figures reproduced from arXiv: 2506.11030 by the authors.

Figure 1
Figure 1. Configuration of various learning methods: a) BP, b) DTP, c) PEPITA, and d) FTP. Black [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Evolution of alignment during training: (a) Alignment between gradient directions from [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Performance of FTP and BP when programming errors are considered for (a) 8-bit and (b) [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: Performance of BP when asymmetry is introduced in backward matrices due to read noise. [PITH_FULL_IMAGE:figures/full_fig_p017_4.png]

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