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Learning without feedback: Fixed random learning signals allow for feedforward training of deep neural networks

T0 review · 2 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Fixed random label projections train deep networks without backpropagation or feedback pathways.

desk verdict Simple, useful training rule with honest empirical work; the proof of 90-degree alignment has a fixable gap that should be requested before publication. read the letter →

arxiv 1909.01311 v2 pith:J4EPQ3A3 submitted 2019-09-03 stat.ML cs.LGcs.NE

Charlotte Frenkel , Martin Lefebvre , David Bol This is my paper · ORCID
classification stat.MLcs.LGcs.NE
keywords directrandomtargetprojectionfeedbackalignmentweighttransportupdatelockingbiologicallyplausiblelearningedgecomputingfeedforwardtraining
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

This paper proposes that supervised classification can train deep feedforward networks using only the one-hot labels and fixed random projection matrices, eliminating the backpropagation feedback pathway entirely. The authors introduce the direct random target projection (DRTP) algorithm, which replaces propagated error signals with fixed random projections of the target label. They aim to show this works because, for classification outputs bounded between 0 and 1, the error sign is known in advance from the label, serving as a proxy for the true gradient direction. If true, this would make neural network training cheaper, faster, and more biologically plausible, enabling low-cost adaptive edge devices.

What carries the argument

The central object is the direct random target projection (DRTP) update rule, which sets each hidden layer's estimated gradient to \(\delta y_k = B_k^T y^*\), where \(B_k\) is a fixed random connectivity matrix and \(y^*\) is the one-hot target. The insight is that replacing the error vector with the target is equivalent to a shift-and-rescale of the error sign, so no feedback pathway is needed. The proof of 90-degree alignment relies on a lemma showing that with zero-initialized weights and a single training example, each weight matrix and hidden activation becomes a scalar multiple of a random target projection, which then lets the DRTP modulatory signal be expressed as a negative scalar multiple of a Moore-Penrose pseudo-inverse times the error.

What would settle it

Train a deeper fully-connected tanh network on a classification dataset with DRTP while measuring the angle between DRTP modulatory signals and backpropagation gradients; if the angle exceeds 90 degrees for any hidden layer while accuracy is still improving, or if the network fails to learn despite the angle claim, the central claim is falsified. More directly, evaluate the alignment identity \(e^T(\prod W_i)(\prod W_i)^+e\) in a single-example linear-hidden-layer setup; if it is non-positive at any update step, the proof's conclusion fails.

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Extended reading notes

Core claim

The central claim is that for classification networks with sigmoid or softmax outputs and cross-entropy loss, the one-hot target encoding y* can replace the error signal in feedback alignment-based training, and the resulting modulatory signals remain aligned within 90 degrees of the backpropagation gradient directions. This makes it possible to train hidden layers of feedforward networks using only layer-local updates computed from a label-dependent random vector selection, with no dedicated feedback pathway. The paper proves this alignment for networks with linear hidden layers trained on a single example, and demonstrates empirically that DRTP trains fully-connected tanh networks on MNIST and CIFAR-10, reaching accuracies between backpropagation and shallow learning.

Load-bearing premise

The mathematical guarantee that DRTP updates stay within 90 degrees of backpropagation is proven only for a network with linear hidden layers, weights initialized to zero, and a single training example, and even that proof contains an unproven step where a matrix times its pseudo-inverse is treated as the identity; the paper assumes this result transfers to tanh networks trained on minibatches, supported only by measured angles.

Editorial extensions

If this is right

  • Layers can be updated immediately during the forward pass, eliminating update locking and reducing memory buffering overhead.
  • The weight transport problem disappears because no symmetric feedback weights are needed at all.
  • DRTP is computationally cheaper than direct feedback alignment, since projecting the one-hot target is just selecting a precomputed random vector per layer per class.
  • On MNIST and CIFAR-10 fully-connected networks, DRTP outperforms shallow learning and approaches feedback alignment accuracy, making it a candidate for adaptive edge computing.
  • The purely feedforward update rule can be formulated as a three-factor synaptic plasticity rule, connecting to biologically plausible learning.

Reading between the lines

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

  • The proof relies on the alignment term \(e^T(\prod W_i)(\prod W_i)^+e\), which the paper labels as \(e^Te\), but the product of a matrix and its pseudo-inverse is an orthogonal projection, not the identity. Strict positivity of the alignment is therefore not established by the algebra as written, though the conclusion may still hold for other reasons.
  • The empirical success on tanh networks suggests the 90-degree alignment may hold more broadly than the linear-hidden-layer proof, but the proof itself does not cover nonlinear activations or minibatch training.
  • The authors note that DRTP drives tanh activations toward saturation, which stops learning; this could be exploited deliberately to produce networks with binarized activations for efficient inference.
  • Since the algorithmic benefit is clearest for fully-connected layers and the paper observes a bottleneck effect in convolutional layers, an interesting extension would be combining DRTP for fully-connected layers with local-error methods for convolutional layers.
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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

2 major / 3 minor

Summary. The manuscript proposes Direct Random Target Projection (DRTP), a learning rule for feedforward classifiers that replaces backpropagated error signals with fixed random projections of the one-hot target vector. Because the target vector is known at the start of the forward pass, DRTP removes the feedback pathway and the update-locking constraint of backpropagation. The paper gives a theoretical alignment argument in Supplementary Note 1 for networks with linear hidden layers, zero-initialized weights, and a single training example, and it supports the method with experiments on synthetic regression and classification tasks, MNIST, and CIFAR-10, comparing BP, FA, DFA, sDFA, and DRTP over 10 runs with reported standard deviations. Public code is provided.

Significance. If the central claim holds, DRTP is a significant contribution: it is a simple, purely feedforward alternative to backpropagation that addresses both weight transport and update locking, with demonstrated accuracy on MNIST and CIFAR-10 and a plausible hardware-cost argument supported by an independent implementation in [33]. The empirical evaluation is careful by the standards of the field: fixed train/test protocols, 10 trials, standard deviations, grid-selected learning rates, and publicly available code. The theoretical guarantee is not yet presented in a fully valid form as printed, but the gap is localized and correctable, and the empirical angle measurements provide substantial supporting evidence for the transfer of the idealized result to the actual tanh/minibatch experiments.

major comments (2)
  1. [Supplementary Note 1, Alignment step] The displayed derivation in the Alignment step is not valid as printed. The product of the forward matrices and its Moore-Penrose pseudo-inverse is labeled as the identity, but it is an orthogonal projection, not the identity. Strict positivity of the dot product can still be recovered from the earlier parts of the proof: for the rank-one matrix P = α s_WK^t (B_k^T y*)^T, we have e^T P P^+ e = (s_WK^{tT} e)^2 / ||s_WK^t||^2, and the positivity of s_WK^{tT} e follows from the previously shown sign agreement between s_WK^t and the error vector. This argument is absent, so the proof as written does not establish the claimed strict positivity; the gap is correctable but must be repaired in the final version.
  2. [Section 2.3 and Section 4] The theorem establishing the within-90-degree alignment is explicitly restricted to linear hidden layers, zero-initialized weights, and a single training example, while the headline classification experiments use He-initialized tanh networks trained with minibatches (Section 4, Tables 1 and 2). The paper should state clearly that the formal guarantee does not cover the evaluated setting and that the transfer to tanh/minibatch training rests on the empirical angle measurements in Figures 2, 3, S1, and S2. This is a scope-transfer gap rather than a contradiction of the empirical results, but it affects the strength of the central claim as worded.
minor comments (3)
  1. [Section 4, synthetic regression and classification] The sentence "the forward weights are drawn from a He uniform distribution and are zero-initialized for feedback-alignment-based algorithms" is ambiguous; please clarify which weight matrices are zero-initialized in the FA/DFA/sDFA/DRTP experiments.
  2. [Supplementary Note 1, network dynamics] The product notation ∏_{i=k+1}^K W_i^t is used for noncommuting matrices without specifying the multiplication order; please define it explicitly (for example, as W_K^t W_{K-1}^t ... W_{k+1}^t or the reverse).
  3. [Figure 4 and Supplementary Figure 3 captions] The captions could be clearer about which curves correspond to which algorithm; in particular, Supplementary Figure 3 plots three algorithms but the caption refers to them collectively as feedback-alignment-based.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the DRTP derivation is self-contained, the theory is proven from stated dynamics, and the empirical claims are external benchmarks.

full rationale

The derivation chain is not circular. DRTP is defined in Algorithm 1 as an update rule using fixed random projections B_k^T y* of the one-hot label, and the claim that targets are a proxy for the error sign is an exact algebraic identity for sigmoid/softmax outputs: y*_c = (1 + sign(e_c))/2, stated in Section 2.3. The theoretical alignment result in Supplementary Note 1 is a self-contained proof under explicit assumptions (linear hidden layers, zero-initialized weights, a single training example); it proves the lemma, derives the theorem from that lemma, and then derives the positive dot product. No step of that proof assumes the conclusion as an input, and no fitted parameter is renamed as a prediction. The empirical sections compare DRTP against BP, FA, DFA, sDFA and shallow learning on MNIST and CIFAR-10, with learning rates grid-searched per algorithm in the standard benchmarking sense; the test errors are measured outcomes, not quantities determined by construction. The only self-citation, [33], is an independent chip implementation measuring power and area overhead, which is real external evidence for the hardware-efficiency claim rather than a load-bearing self-citation chain. The paper also explicitly acknowledges limitations, including the convolutional bottleneck, tanh saturation, and the classification-specific derivation of DRTP, which further indicates that the claims are not manufactured. No circular step can be exhibited from the paper's equations or citation structure.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

DRTP introduces no new physical or ontological entities; its random connectivity matrices are inherited from DFA/FA. The load-bearing assumptions are the error-sign/label equivalence and the transfer of the idealized proof to practical settings. Learning rates are the main fitted quantities.

free parameters (2)
  • Per-algorithm learning rates = 5e-6 to 1.5e-2 depending on algorithm, dataset, network (Table 3)
    Selected by grid search separately for BP, FA, DFA, sDFA, DRTP and shallow learning; the accuracy comparisons depend on these tuned rates.
  • Synthetic data generation parameters = n=256, ninf=128, 10 classes, 5 clusters/class, class_sep=4.5
    Used for the synthetic classification experiment in Section 2.1.2; not part of the DRTP algorithm itself.
assumptions (4)
  • domain assumption For sigmoid/softmax output units, outputs are strictly between 0 and 1, so sign(e_c) = +1 for c=c* and -1 otherwise
    Core premise of DRTP, stated in Section 2.2.
  • ad hoc to paper The alignment theorem assumes zero-initialized weights, linear hidden layers, and a single training example
    Supplementary Note 1; the main experiments use He-initialized tanh networks and minibatches, so transfer of the guarantee is assumed.
  • standard math Moore-Penrose pseudo-inverse and matrix algebra identities
    Used in the theorem and alignment proof in Supplementary Note 1.
  • domain assumption Saturating activation functions (tanh) are required for DRTP to work; ReLU is not compatible
    Discussed in Section 3; DRTP relies on the vanishing derivative of tanh to stop learning.

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Cite this review

Pith. "Pith review of Learning without feedback: Fixed random learning signals allow for feedforward training of deep neural networks." pith.science (2026). https://pith.science/paper/J4EPQ3A3

@misc{pith2026190901311,
  author       = {Pith},
  title        = {Pith review of: Learning without feedback: Fixed random learning signals allow for feedforward training of deep neural networks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/J4EPQ3A3}},
  note         = {Machine review of arXiv:1909.01311}
}
read the original abstract

While the backpropagation of error algorithm enables deep neural network training, it implies (i) bidirectional synaptic weight transport and (ii) update locking until the forward and backward passes are completed. Not only do these constraints preclude biological plausibility, but they also hinder the development of low-cost adaptive smart sensors at the edge, as they severely constrain memory accesses and entail buffering overhead. In this work, we show that the one-hot-encoded labels provided in supervised classification problems, denoted as targets, can be viewed as a proxy for the error sign. Therefore, their fixed random projections enable a layerwise feedforward training of the hidden layers, thus solving the weight transport and update locking problems while relaxing the computational and memory requirements. Based on these observations, we propose the direct random target projection (DRTP) algorithm and demonstrate that it provides a tradeoff between accuracy and computational cost that is suitable for adaptive edge computing devices.

Figures

Figures reproduced from arXiv: 1909.01311 by the authors.

Figure 1
Figure 1. The proposed direct random target projection algo [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Error-sign-based direct feedback alignment (sDFA [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Error-sign-based direct feedback alignment (sDFA [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: Network of DRTP-updated linear hidden layers cons [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]

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Forward citations

Cited by 1 Pith paper

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    A spiking predictive-coding network with a local, spike-triggered alignment rule achieves 4.72% error on MNIST in a single online pass and forgets less on split continual-learning benchmarks.

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