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REVIEW 3 major objections 7 minor 1 cited by

Dendritic Localized Learning: Toward Biologically Plausible Algorithm

T0 review · 3 major / 7 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read Dendritic Localized Learning trains multilayer networks with asymmetric weights, local errors, and simultaneous forward/backward passes, reaching backpropagation-comparable accuracy.

desk verdict DLL is a solid empirical contribution with a broken loss-based derivation—the algorithm works, but the theory behind it does not. read the letter →

arxiv 2501.09976 v2 pith:ZTJTXKPW submitted 2025-01-17 cs.NE

Changze Lv , Jingwen Xu , Yiyang Lu , Xiaohua Wang , Zhenghua Wang , Zhibo Xu , Di Yu , Xin Du
show 2 more authors
Xiaoqing Zheng Xuanjing Huang
This is my paper · ORCID
classification cs.NE
keywords biologicallyplausiblelearningdendriticcomputationlocalerrorrepresentationpyramidalneuronmodelfeedbackalignmentbackpropagationalternativesrecurrentneuralnetworkslocalized
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

The paper proposes Dendritic Localized Learning (DLL), a training rule inspired by pyramidal neurons, and claims it satisfies three criteria that backpropagation violates: asymmetric forward and backward weights, local error signals, and no strict separation of inference and training phases. On image benchmarks spanning MNIST to CIFAR-10, DLL-trained MLPs and CNNs converge on every dataset and reach accuracy close to backpropagation, which the authors say is the best among algorithms meeting all three criteria. DLL also trains recurrent networks for next-character prediction and time-series forecasting, where it converges where several biologically motivated baselines fail. The point of the claim is that biological plausibility and competitive performance need not be traded off.

What carries the argument

The three-compartment pyramidal neuron model: the basal dendrite receives sensory input $u_i$, the apical dendrite receives the expected value $x_i$, and the soma computes the local error $\xi_i = x_i - u_i$. The backward pass uses trainable matrices $\Theta_i$ in place of transposed forward weights, with updates $\Delta x_i = -\xi_i + \Theta_i^T[\xi_{i+1} \odot f'(W_i u_i)]$, and both $W_i$ and $\Theta_i$ are updated from the same local errors. This spatial separation is what lets forward and backward signals coexist in time, implementing criterion C3, while $\Theta$'s asymmetry implements C1 and the soma-local computation implements C2.

What would settle it

Compute the exact gradient $\partial L / \partial x_i$ on the computational graph the paper defines, where $u_{i+1}$ depends only on $u_i$ and $W_i$, and compare it with the paper's update $\Delta x_i = -\xi_i + \Theta_i^T[\xi_{i+1} \odot f'(W_i u_i)]$; the two will disagree whenever $\xi_{i+1} \neq 0$. A second check: freeze $x_i$ at its initial value for all hidden layers and train only $W$ and $\Theta$; if accuracy collapses, the $x$-update is load-bearing, whereas if it stays high, DLL's effective learning rule is simpler than its derivation suggests.

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Extended reading notes

Core claim

The central claim is that a network can be trained by replacing the global backpropagation error with a locally computed error $\xi_i = x_i - u_i$ at every layer, where $u_i$ is the layer's sensory input and $x_i$ is an expected value delivered along trainable backward weights $\Theta_i$. Because each soma computes its own error and both $W$ and $\Theta$ are updated simultaneously, the algorithm satisfies the paper's three criteria C1-C3. Empirically, DLL achieves, for example, 70.89% on CIFAR-10 with CNNs compared with 75.10% for backpropagation, and it is the only method satisfying all three criteria that converges across all tested datasets and architectures; the authors conclude that reconciling biological plausibility with high performance is possible.

Load-bearing premise

The derivation assumes that changing a layer's expected value $x_i$ changes the next layer's input $u_{i+1}$, even though the forward pass defines $u_{i+1}$ as a function of $u_i$ only; if that hidden dependence does not exist, DLL's cross-layer error propagation does not follow from the stated loss.

Editorial extensions

If this is right

  • Among algorithms satisfying all three plausibility criteria, DLL is the only one in the benchmark that converges across MNIST, FashionMNIST, SVHN, and CIFAR-10 for both MLPs and CNNs.
  • DLL-trained CNNs average 86.61% across the four image benchmarks, close to backpropagation's 90.67%, and the CIFAR-10 gap is about four points.
  • On sequential tasks, DLL is the only all-criteria method that converges, with competitive MSE and MAE on electricity and traffic forecasting datasets.
  • Updating the backward weights $\Theta$ matters: freezing $\Theta$ at random values ('DLL-FA') degrades results and causes RNNs on Harry Potter to fail to converge.
  • TextCNNs trained with DLL reach within a few points of backpropagation on text classification, and DLL scales with depth on MNIST.

Reading between the lines

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

  • A consequence the paper leaves implicit: because DLL's per-layer errors depend only on adjacent layers, the learned backward matrices $\Theta$ could be transferred from one task to another as a generic error-transport mechanism; this is testable by pretraining $\Theta$ on one dataset and freezing it on another.
  • The derivation of the $x$-update suggests a cleaner interpretation: treat the expected values $x_i$ as auxiliary variables of an inner loop rather than as backpropagated gradients; under that reading DLL resembles target propagation with learned targets, and convergence might be provable by standard fixed-point arguments.
  • The spatially separated apical and basal pathways suggest an immediate neuromorphic implementation: a chip with separate pathways for input and expected value could run inference and learning in the same clock cycle, something the temporal two-phase requirement forbids.
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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 / 7 minor

Summary. The paper proposes Dendritic Localized Learning (DLL), a biologically plausible alternative to backpropagation in which each layer computes a local error ξ_i = x_i − u_i between an apical "expected value" x_i and a basal "sensory input" u_i, and in which error propagation uses trainable backward weights Θ instead of transposed forward weights. The authors claim DLL satisfies three criteria of biological plausibility: asymmetric forward/backward weights (C1), local error representation (C2), and non-two-stage training (C3). They benchmark DLL against a wide range of existing biologically plausible algorithms on MLPs, CNNs, and RNNs across image classification, text character prediction, and time-series forecasting, reporting that DLL achieves the best accuracy among algorithms satisfying all three criteria. The paper also provides an RNN extension, an ablation of the trainable Θ, sensitivity analyses, scalability experiments, and a public implementation.

Significance. The empirical contribution is substantial: the paper provides a broad, externally benchmarked comparison of biologically plausible learning algorithms on standard datasets, with code publicly available. DLL is one of the few methods that demonstrably converge on all tested architectures while satisfying the authors' three stated criteria, and its accuracy on MNIST, FashionMNIST, SVHN, and CIFAR-10 is meaningfully better than other three-criteria-satisfying baselines. If the algorithm is taken as an empirically specified heuristic, the practical claim is credible. However, the paper also claims a loss-based derivation and "theoretical rigor"; that part is not supported, because a load-bearing chain-rule step in the derivation is unjustified and the implemented algorithm does not follow the derivation's fixed-point procedure. The central theoretical framing therefore needs substantial revision, while the empirical findings can stand.

major comments (3)
  1. [Section 3.2, Eq. (5)] The derivation of the error-propagation rule is not justified by the forward model. In Eq. (4), u_{i+1} = f(W_i u_i), so u_{i+1} depends on u_i and not on x_i. Eq. (5) nevertheless computes ∂L/∂x_i and includes the term ∂u_{i+1}/∂x_i = W_i^T[ξ_{i+1} ⊙ f'(W_i u_i)]. Under the stated dependency graph, ∂u_{i+1}/∂x_i = 0, so the chain-rule step collapses, and Eqs. (6)–(9) do not follow from minimizing L = −(1/2)Σ ξ_i^2. The same problem appears in Appendix B, Eq. (15), where ∆h_i^p includes ∂h_{i+1}^p/∂h_i^p, while h_{i+1}^p = h_{i+1}^s = f(W_h h_i^s + W_x x_{i+1}) depends on h_i^s, not h_i^p. Unless the authors introduce and justify an explicit dependence of u_{i+1} on x_i (or of h_{i+1}^p on h_i^p), the theoretical derivation should be withdrawn or explicitly labeled as a heuristic inspiration rather than a derivation from the stated loss.
  2. [Appendix A, Algorithm 1] The text surrounding Eqs. (5)–(7) describes a fixed-point procedure: x_i is updated by x_i ← x_i + η_x Δx_i, and at stability Δx_i = 0 gives Eq. (7), ξ_i = Θ_i^T[ξ_{i+1} ⊙ f'(W_i u_i)]. However, Algorithm 1 never updates x_i at all; it directly sets ξ_i using the recurrence in the "Compute Local Errors" loop. Thus the implemented rule is not the equilibrium of the stated dynamics, and the derivation in Section 3.2 does not describe the algorithm that produces Tables 1 and 2. This gap is load-bearing because the paper's theoretical narrative is that DLL minimizes Eq. (3); the actual algorithm is better described as an ad hoc local-error recurrence. The authors should either modify Algorithm 1 to implement the fixed-point updates (and analyze convergence), or revise the text to present the recurrence as the definition of DLL.
  3. [Appendix C] Appendix C concedes that "providing formal convergence proofs remains challenging" and offers only empirical loss curves in support of convergence. This is consistent with the two problems above, but it directly contradicts the paper's concluding claim of "theoretical rigor". The claim is also load-bearing because the Introduction and Section 3 present DLL as derived from a well-defined loss. A heuristic algorithm with strong empirical results is acceptable, but the text must not claim a theoretical derivation that is not present. The authors should either supply a rigorous statement (e.g., conditions under which the Δx_i = 0 fixed point exists and the recurrence is equivalent to gradient descent on Eq. (3)) or remove the theoretical-rigor language.
minor comments (7)
  1. [Section 3.2, text near Eq. (5)] The sentence "x_i depends solely on ξ_i and ξ_{i+1}" is misleading because the loss in Eq. (3) couples all layers through the forward model; the claim is only true under the local-recurrence ansatz that the paper is trying to derive.
  2. [Appendix B, Eq. (15)] In Eq. (15), the derivative ∂y_i/∂h_i^p is written with g'(W_y h_n^s), using index n where i is intended, and y_i = g(W_y h_i^s) depends on h_i^s, not h_i^p; this appears to be a typographical manifestation of the same dependency confusion as in Eq. (5).
  3. [Section 4.2, Table 1] The claim that DLL achieves performance "comparable to backpropagation" is overstated for CNNs: DLL's average is 86.61% versus 90.67% for backpropagation, a gap of about four points. I recommend saying "competitive" or quantifying the gap.
  4. [Table 2] On Harry Potter, DLL reaches 33.7% versus 51.9% for backpropagation; on Metr-la and Pems-bay, DLL is worse than backpropagation on most metrics and comparable to predictive coding. The phrase "on par with or surpassing those of backpropagation in several metrics" should be checked against the table; it is true only for specific metrics such as Electricity MSE/MAE.
  5. [Appendix E, Table 4] CIFAR-100 and Tiny-ImageNet results are reported without standard deviations or number of seeds, while Tables 1 and 2 include such statistics; please add them for consistency.
  6. [Section 5, Related Work] The citation of Weed and Hursting (1998) in the related-work paragraph about reviews of learning algorithms appears out of place; that reference concerns causal inference methodology and is not a review of biologically plausible learning.
  7. [Figure 3 and Appendix D] Figure 3(a/b) shows that the best learning rate is around 1e−4 for both MLPs and CNNs, but Appendix D reports widely varying per-dataset rates (e.g., 1e−3 for MNIST MLPs and 5e−5 for CNNs). Please clarify whether Figure 3 uses a fixed architecture and a separate learning-rate sweep, and how the chosen rates in Table 1 relate to the observed optimum.

Circularity Check

2 steps flagged · score 6.0 of 10

DLL's local-error propagation rule is assumed in Algorithm 1 and then re-derived from a loss whose chain rule is not supported by the forward model; the theoretical derivation reduces to the algorithm's own definition.

  1. self definitional [Section 3.2, Eqs. (4)-(7); Algorithm 1 (Appendix A)]
    "∆xi = ∂L/∂xi = ... = −ξi + ∂ui+1/∂xi ξi+1 = −ξi + W T i [ξi+1 ⊙ f ′(Wiui)] ... Ultimately, when xi approaches stability, ∆xi = 0, leading to the expression ξi = ΘT i [ξi+1 ⊙ f ′(Wiui)]."

    Under the paper's own forward model, Eq. (4), u_{i+1}=f(W_i u_i) depends on u_i, not on the expected value x_i; hence ∂u_{i+1}/∂x_i in Eq. (5) is zero and the chain-rule step does not follow from L. The text then replaces W_i^T by Θ_i^T by assumption (Eq. 6) and calls the resulting ξ_i=Θ_i^T[ξ_{i+1}⊙f'(W_i u_i)] a fixed point (Eq. 7). Algorithm 1 never iterates x_i to that fixed point; it directly computes ξ_i with exactly this recurrence. The 'derived' error-propagation rule is therefore the algorithm's own definition, not a consequence of minimizing the stated loss.

  2. self definitional [Appendix B, Eqs. (10) and (15)-(20)]
    "hp i = hs i = f (Whhs i−1 + Wxxi) ... ∆hp i = ... = −ξh i + WT y ξy i + WT h [ξh i+1 ⊙ f ′(Whhp i + Wxxi+1)] ... we can directly assign the value of ξh i as: ξh i = ΘT y ξy i + ΘT h [ξh i+1 ⊙ f ′(Whhs i + Wxxi+1)]."

    Eq. (10) makes h^p_{i+1}=h^s_{i+1}=f(W_h h^s_i + W_x x_{i+1}), a function of h^s_i, not of the backpropagated activity h^p_i. Yet Eq. (15) differentiates h^p_{i+1} with respect to h^p_i to introduce the W_h^T (later Θ_h^T) recurrence, and Eqs. (19)-(20) then 'directly assign' ξ^h_i by that same recurrence. As in the MLP case, the temporal error-propagation rule is inserted as a definition and re-derived from the loss, so the RNN derivation is circular in the same way.

full rationale

The paper's empirical results are benchmarked on external datasets (MNIST, FashionMNIST, SVHN, CIFAR-10, Harry Potter, Electricity, Metr-la, Pems-bay), so the headline performance comparisons are not circular and this is not a case of a fitted parameter being renamed a prediction. The circularity is confined to the theoretical derivation: the local-error propagation rule that DLL implements is assumed in Eq. (6)/Algorithm 1 and then presented as derived from the loss in Eqs. (5)-(7); the chain-rule term ∂u_{i+1}/∂x_i is not justified by the forward model Eq. (4), and no x_i fixed-point iteration is performed. Appendix C effectively concedes that no convergence proof is supplied ('While providing formal convergence proofs remains challenging...'). The RNN extension repeats the same self-definitional move. Thus the algorithm itself is an empirically specified heuristic with independent benchmark success, but its claimed first-principles derivation reduces to its own definition, giving partial circularity (score 6).

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

The central algorithm rests on three nonstandard assumptions: the three-compartment neuron model, the equilibrium relation Eq. (7) that defines local errors without actual convergence, and the loose interpretation of criterion C3. In addition, the gradient derivation in Eq. (5) uses a chain-rule term ∂u_{i+1}/∂x_i that is not supported by the forward model, which is a significant unstated dependency.

free parameters (1)
  • Learning rates ηW, ηΘ, ηx and hidden sizes = Varies per dataset: e.g., 1e-3 for MNIST MLP, 5e-5 for CNNs, grid searched for RNNs
    Performance is sensitive to these hyperparameters (Section 4.5); they are chosen by grid search or by hand, not derived from the algorithm.
assumptions (4)
  • domain assumption Three-compartment pyramidal neuron model (basal/soma/apical) is an appropriate substrate for local error computation.
    Invoked in Section 3.1 following Sacramento et al. (2018); no independent neurobiological validation is provided in this paper.
  • ad hoc to paper Local error ξ = x − u plus the equilibrium condition Δx_i = 0 (Eq. 7) defines the correct training signal.
    The pseudocode uses Eq. (7) directly without iterating x_i to equilibrium; the equilibrium is assumed rather than proven (Appendix C admits no formal convergence proof).
  • domain assumption The three criteria C1-C3 are sufficient and accurate definitions of biological plausibility.
    Section 2.1 defines the criteria from literature, but the non-two-stage criterion is interpreted loosely enough that DLL's sequential pseudocode can be said to satisfy it.
  • domain assumption Trainable feedback matrix Θ can stand in for the transpose of the forward weights without destabilizing learning.
    This is the core algorithmic premise (Section 3.2); supported only by the paper's experiments, not by theory.
invented entities (1)
  • Trainable backward weight matrix Θ
    purpose: Replaces W^T for propagating top-down expectations and errors, satisfying criterion C1.
    A new parameterized matrix with update rule Eq. (9); no external falsifiable prediction outside this paper's experiments.

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Cite this review

Pith. "Pith review of Dendritic Localized Learning: Toward Biologically Plausible Algorithm." pith.science (2026). https://pith.science/paper/ZTJTXKPW

@misc{pith2026250109976,
  author       = {Pith},
  title        = {Pith review of: Dendritic Localized Learning: Toward Biologically Plausible Algorithm},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZTJTXKPW}},
  note         = {Machine review of arXiv:2501.09976}
}
read the original abstract

Backpropagation is the foundational algorithm for training neural networks and a key driver of deep learning's success. However, its biological plausibility has been challenged due to three primary limitations: weight symmetry, reliance on global error signals, and the dual-phase nature of training, as highlighted by the existing literature. Although various alternative learning approaches have been proposed to address these issues, most either fail to satisfy all three criteria simultaneously or yield suboptimal results. Inspired by the dynamics and plasticity of pyramidal neurons, we propose Dendritic Localized Learning (DLL), a novel learning algorithm designed to overcome these challenges. Extensive empirical experiments demonstrate that DLL satisfies all three criteria of biological plausibility while achieving state-of-the-art performance among algorithms that meet these requirements. Furthermore, DLL exhibits strong generalization across a range of architectures, including MLPs, CNNs, and RNNs. These results, benchmarked against existing biologically plausible learning algorithms, offer valuable empirical insights for future research. We hope this study can inspire the development of new biologically plausible algorithms for training multilayer networks and advancing progress in both neuroscience and machine learning. Our code is available at https://github.com/Lvchangze/Dendritic-Localized-Learning.

Figures

Figures reproduced from arXiv: 2501.09976 by the authors.

Figure 1
Figure 1. Illustrations of biologically plausible learning algorithms. (a) Backpropagation; (b) In feedback alignment, the weight matrix W is replaced with a random matrix during backpropagation; (c) In local losses, classic backpropagation is applied layer by layer; (d) In predictive coding, the transposed weights WT are used iteratively for local convergence; (e) In perturbation learning, the weights W are randomly perturbe… view at source ↗
Figure 2
Figure 2. (a) Overview of Dendritic Localized Learning. (b) Our DLL algorithm satisfied all 3 criteria. (c) Models trained by DLL successfully converge and achieve comparable performance to those trained by backpropagation. networks. The weight update in Hebbian learning is given by ∆wij = ηxixj , where xi and xj are the activations of neurons i and j. In STDP, the update depends on the spike timing difference ∆wij = ( A +e −… view at source ↗
Figure 3
Figure 3. (a) MLPs trained with DLL by various learning rates. (b) CNNs trained with DLL by various learning rates. (c) Loss curves of RNNs trained with DLL across different sequence lengths. (d) Loss curves of RNNs trained with DLL by different learning rates. Notably, in sequence modeling tasks, such as RNNs-based predictions on Harry Potter, DLL outperforms DLL-FA by a significant margin, with DLL-FA failing to converge in… view at source ↗

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.