{"id":"b763b282-88c6-4fcd-802a-33d2a7121eb6","arxiv_id":"2501.09976","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"DLL trains MLPs, CNNs, and RNNs with local errors and trainable asymmetric feedback, achieving the best accuracy among algorithms satisfying the paper's three biological plausibility criteria.","lead":"This paper presents Dendritic Localized Learning (DLL), a neural-network training rule that computes per-neuron errors from local apical and basal inputs and uses trainable asymmetric feedback weights instead of weight transposes. The authors report that DLL approaches backpropagation accuracy on small image, text, and time-series benchmarks and beats other algorithms that meet their three biological plausibility criteria.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Eq. (5) contains an unjustified chain-rule term \\partial u_{i+1}/\\partial x_i: the forward model has u_{i+1}=f(W_i u_i), so DLL's error-propagation rule is not derived from the stated loss and the paper's theoretical framing is unsupported.","rationale":"The reader's weakest assumption correctly identifies that Eq. (5) uses a chain-rule term not present in the forward model. This is the most load-bearing concern because the paper's contribution is framed as a theoretically derived biologically plausible algorithm, and the derivation of Eqs. (6)-(9) rests on this term. However, the central empirical claim, that DLL satisfies all three stated criteria and performs better than other algorithms meeting them, is largely independent of the derivation: the algorithm is fully specified in Algorithm 1, the code is available, and the reported results are consistent across architectures and datasets. The flaw changes the nature of the contribution from a loss-derived method to an empirically validated heuristic, which is a significant caveat but not a refutation of the benchmark results. The reader's CONDITIONAL verdict already captures this state: the theory needs repair or reframing, while the experiments can stand pending independent reproduction. For these reasons, my stress-test does not change the reader's verdict.","tokens_in":20565,"tokens_out":13127,"duration_ms":123560,"concrete_test":"On a small two-layer MLP, implement the Algorithm 1 forward graph: u_2=f(W_1 u_1), with x_1 and x_2 as separate target variables and x_2 fixed to the label. Compute the numerical partial derivative \\partial L/\\partial x_1 of L=-1/2[(x_1-u_1)^2+(x_2-u_2)^2] by finite differences, holding u_1, W_1 and x_2 fixed. Compare it with the expression in Eq. (5). They will differ because the second term of Eq. (5) assumes \\partial u_2/\\partial x_1\\neq 0, whereas the forward graph gives \\partial u_2/\\partial x_1=0. This single numerical check settles whether the DLL error recurrence is actually the gradient of the stated loss.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The derivation of the core DLL update rules breaks at Eq. (5). The paper defines the forward model in Eq. (4) as u_{i+1}=f(W_i u_i), so the sensory variable u_{i+1} depends on u_i, not on the expected value x_i. Eq. (5) nevertheless computes \\partial L/\\partial x_i and includes \\partial u_{i+1}/\\partial x_i = W_i^T[\\xi_{i+1}\\odot f'(W_i u_i)], later replaced by \\Theta_i^T in Eq. (6). Under the actual dependency graph, \\partial u_{i+1}/\\partial x_i=0, so the chain-rule step collapses. Consequently, the recurrence \\xi_i=\\Theta_i^T[\\xi_{i+1}\\odot f'(W_i u_i)] in Eq. (7) and the \\Theta update in Eq. (9) are not consequences of minimizing L=-1/2\\sum_i\\xi_i^2. The problem is systematic: the RNN derivation, Eq. (15), similarly differentiates h^p_{i+1} with respect to h^p_i even though h^p_{i+1}=h^s_{i+1}=f(W_h h^s_i+W_x x_{i+1}) depends on h^s_i, not h^p_i. In addition, Algorithm 1 never updates x_i to the fixed point implied by Eq. (7); it directly sets \\xi_i by the recurrence, so the derivation in the text does not describe the implemented rule. Appendix C explicitly concedes that no convergence proof is provided. This does not refute the empirical results: Table 1 shows DLL performing well and the code is public. It does mean that DLL is presently an empirically specified heuristic, and the paper's claims of a loss-based derivation and 'theoretical rigor' are not supported.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":20984,"tokens_out":4449,"duration_ms":41825,"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":[{"comment":"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.","section":"Section 3.2, Eq. (5)"},{"comment":"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.","section":"Appendix A, Algorithm 1"},{"comment":"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.","section":"Appendix C"}],"minor_comments":[{"comment":"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.","section":"Section 3.2, text near Eq. (5)"},{"comment":"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).","section":"Appendix B, Eq. (15)"},{"comment":"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.","section":"Section 4.2, Table 1"},{"comment":"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.","section":"Table 2"},{"comment":"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.","section":"Appendix E, Table 4"},{"comment":"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.","section":"Section 5, Related Work"},{"comment":"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.","section":"Figure 3 and Appendix D"}],"recommendation":"major_revision","confidential_remarks":"The empirical contribution is real and the code availability is a strength. The main issue is that the theoretical derivation has a genuine gap in a load-bearing equation, and the implemented algorithm does not match the described fixed-point procedure. This is not a case of merely missing a proof; the chain-rule step as written is not a consequence of the forward model. I believe the paper can be repaired by honestly re-framing DLL as an empirically motivated local-learning heuristic, and by either removing the theoretical-rigor claims or clearly labeling the derivation as a heuristic analogy. The RNN derivation and Algorithm 1 need the same correction. Fit with the journal/venue is appropriate for this line of work."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should know two things about this paper. First, it is a legitimate empirical contribution: DLL trains MLPs, CNNs, and RNNs on standard benchmarks to within a few points of backprop, better than most local learning rules that meet the authors' three biological-plausibility criteria. Second, the theoretical derivation that supposedly grounds the algorithm is not sound. The chain rule in Eq. (5) uses \\partial u_{i+1}/\\partial x_i, but the forward model u_{i+1}=f(W_i u_i) has no dependency on x_i. That term is zero, so the recurrence in Eq. (7) and the Theta update in Eq. (9) do not follow from minimizing L. The stress-test note gets this right.\n\nWhat is genuinely new is the trainable feedback matrix Theta with a local error rule and simultaneous updates of W and Theta, plus a broad empirical comparison across architectures and datasets. The ablation against fixed random Theta (DLL-FA) is a good addition—it shows trainable feedback matters, especially for RNNs where fixed random feedback fails to converge. Code is public and main results have error bars across seeds. This is reproducible work.\n\nThe derivation problem is systematic. The RNN derivation in Appendix B repeats it by differentiating h^p_{i+1} with respect to h^p_i when the forward dependence runs through h^s_i. Also, Algorithm 1 never updates x_i to the fixed point claimed in Eq. (7); it directly sets xi from the recurrence. So the text describes a derivation that the code does not implement. The paper honestly concedes in Appendix C that there is no convergence proof, but the conclusion still claims 'theoretical rigor.' That is not supported. The algorithm stands as an empirically strong heuristic with a biologically plausible story.\n\nThe C3 claim is stretched. Algorithm 1 has a forward pass, a backward sweep for local errors, and then a weight update—temporally two-stage. The spatial-separation argument for simultaneity is a plausible future implementation, not what the code does. If C3 is a hard criterion, DLL fails it as implemented.\n\nMinor: a few result tables (CIFAR-100, Tiny-ImageNet, TextCNN, scalability) have no error bars, so treat those as indicative.\n\nWho should read this: people working on biologically plausible learning or neuromorphic algorithms. The benchmark is a useful reference and the method is simple to try. Anyone looking for rigorous theory should not rely on the derivation.\n\nI would send this to peer review. The empirical work deserves serious referee time, and the problems are fixable: re-frame DLL as a heuristic local rule, either provide a valid derivation or clearly separate the heuristic from the loss, and be honest about C3. The paper does not need to be rejected; it needs revision so the claims match the actual algorithm.","headline":"DLL is a solid empirical contribution with a broken loss-based derivation—the algorithm works, but the theory behind it does not.","tokens_in":21511,"tokens_out":4676,"would_cite":true,"duration_ms":41854,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Dendritic Localized Learning trains multilayer networks with asymmetric weights, local errors, and simultaneous forward/backward passes, reaching backpropagation-comparable accuracy.","keywords":["biologically plausible learning","dendritic computation","local error representation","pyramidal neuron model","feedback alignment","backpropagation alternatives","recurrent neural networks","localized learning"],"falsifier":"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.","tokens_in":1749,"feed_emoji":"🧠","tokens_out":2182,"duration_ms":74330,"temperature":0.7,"pith_summary":"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.","feed_headline":"Local-error rule rivals backprop on image and sequence tasks","feed_subtitle":"DLL meets all three biological-plausibility criteria and converges across MLP, CNN, and RNN architectures.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Defines backpropagation, the algorithm whose biological plausibility deficits motivate the three criteria and the performance baseline DLL is compared against.","marker":"(Rumelhart et al., 1986)"},{"why":"Supplies the weight-symmetry criticism that becomes criterion C1.","marker":"(Stork, 1989)"},{"why":"Supplies the global-error criticism that becomes criterion C2.","marker":"(Crick, 1989)"},{"why":"Provides the neurobiological evidence that apical dendrites receive top-down inputs and basal dendrites receive lower-layer inputs, grounding the three-compartment model.","marker":"(Spruston, 2008)"},{"why":"The prior three-compartment pyramidal neuron model that DLL explicitly follows for dividing the neuron into soma, apical dendrite, and basal dendrite.","marker":"(Sacramento et al., 2018)"},{"why":"Introduces feedback alignment, the idea of replacing transposed weights with alternative backward matrices that DLL adapts into trainable $\\Theta$ weights.","marker":"(Lillicrap et al., 2016)"},{"why":"Defines the forward-forward algorithm, the main all-criteria baseline whose CNN failures DLL improves upon.","marker":"(Hinton, 2022)"},{"why":"Supplies the scalability assessment and CNN architectures used for the CIFAR-100 and Tiny-ImageNet comparisons.","marker":"(Bartunov et al., 2018)"}],"fun_headline_variants":["Local error rule: bio-plausible, near-backprop accuracy","Dendritic learning: all three bio-plausibility boxes checked","No backprop needed: local error works on MLP, CNN, RNN","Bio-plausible learning that actually converges on real tasks","Local errors, global wins: DLL rivals backprop"],"cache_read_input_tokens":23552,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Local error rule: bio-plausible, near-backprop accuracy","Dendritic learning: all three bio-plausibility boxes checked","No backprop needed: local error works on MLP, CNN, RNN","Bio-plausible learning that actually converges on real tasks","Local errors, global wins: DLL rivals backprop"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000969,"raw_usage":{"total_tokens":4109,"prompt_tokens":923,"completion_tokens":3186,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":539,"completion_tokens_details":{"reasoning_tokens":3097}},"tokens_in":539,"tokens_out":3186,"duration_ms":26221,"temperature":1.0,"reasoning_tokens":3097,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T19:29:11.372008+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"E., Hinton, G","cited_arxiv_id":null,"evidence_quote":"Defines backpropagation, the algorithm whose biological plausibility deficits motivate the three criteria and the performance baseline DLL is compared against."},{"cited_title":"Is backpropagation biologically plausible? In International 1989 Joint Conference on Neural Networks, pp.\\ 241--246","cited_arxiv_id":null,"evidence_quote":"Supplies the weight-symmetry criticism that becomes criterion C1."},{"cited_title":"Pyramidal neurons: dendritic structure and synaptic integration","cited_arxiv_id":null,"evidence_quote":"Provides the neurobiological evidence that apical dendrites receive top-down inputs and basal dendrites receive lower-layer inputs, grounding the three-compartment model."},{"cited_title":"Dendritic cortical microcircuits approximate the backpropagation algorithm","cited_arxiv_id":null,"evidence_quote":"The prior three-compartment pyramidal neuron model that DLL explicitly follows for dividing the neuron into soma, apical dendrite, and basal dendrite."},{"cited_title":"P., Cownden, D., Tweed, D","cited_arxiv_id":null,"evidence_quote":"Introduces feedback alignment, the idea of replacing transposed weights with alternative backward matrices that DLL adapts into trainable $\\Theta$ weights."},{"cited_title":"A., Hinton, G","cited_arxiv_id":null,"evidence_quote":"Supplies the scalability assessment and CNN architectures used for the CIFAR-100 and Tiny-ImageNet comparisons."}],"review_version":1}