{"id":"8a07f167-885e-4446-845e-f3bd2d412a2b","arxiv_id":"2501.15403","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"Training time to a fixed accuracy for perturbative gradient methods grows far slower than linearly with network size, challenging a long-standing scaling objection.","lead":"This paper studies multiplexed gradient descent, a training method that probes a neural network with small random perturbations to estimate gradients without backpropagation, and finds that the time to reach a fixed accuracy does not grow linearly with network size. It matters because it suggests analog neuromorphic chips, which cannot easily run backpropagation, could be trained in place on large networks.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The flat time-to-train scaling is measured against an 80% threshold with no backpropagation baseline at the million-parameter scale, so the headline 'solves the problem' and 'accuracy comparable to backpropagation' are not yet supported.","rationale":"The reader's weakest_assumption focused on the unstated hardware model that makes one MGD iteration O(1) in wall-clock time. That is a real and important bridge from iteration counts to the paper's time-based claims. My concern is complementary but distinct: even before translating iterations to wall-clock, the iteration-count scaling itself is only demonstrated for an 80% accuracy threshold, with no BP baseline at the million-parameter scale. The abstract and Section 3.2 make the stronger claim that MGD matches backpropagation accuracy on networks with over one million parameters, but the only side-by-side accuracy curve with BP is for N = 2.55e5 (Fig. 4a). If the flat scaling disappears at higher thresholds, the headline 'time to solve the problem does not scale with N' would fail on its own terms, independent of hardware timing. The concrete test is purely computational and uses the paper's own protocol, so it would settle the concern without new hardware assumptions. If the test passes, the central claim is strengthened and the current conditional verdict is appropriate; if it fails, the verdict should move toward rejection or at least a substantially narrowed claim. I therefore keep 'UNCHANGED' relative to the reader's CONDITIONAL verdict, while adding the threshold/BP-baseline condition as the most load-bearing element to verify.","tokens_in":17756,"tokens_out":16520,"duration_ms":165114,"concrete_test":"Rerun the Fig. 4b protocol for the same network sizes (d = 1 through 64) and record the number of iterations to reach 85%, 90%, and the backpropagation-comparable final accuracy (e.g., 91.6% or the accuracy BP reaches at each size), with BP baselines reported at every N. Also show full accuracy-versus-iterations curves for the largest network (N ~ 1,014,874). If iterations-to-threshold grow appreciably with N at 90% or above, or if MGD never reaches BP's final accuracy at 1M parameters, then the flat-scaling result is specific to the 80% threshold and the central claim should be weakened.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim (Abstract, Section 3.2) is that time to target accuracy does not scale with network size and that a >10^6-parameter network can be trained to the same testing accuracy as backpropagation. The only scaling evidence is Fig. 4b, which counts iterations to 80% test accuracy. However, 80% on FashionMNIST is substantially below the 91.6% accuracy backpropagation reaches in Fig. 4a, and no backpropagation curve is reported for networks larger than 2.55e5 parameters. Thus the observed flatness may be specific to an undemanding threshold: in wider networks, reaching 80% may become easier even if the time to reach a BP-comparable accuracy still grows with N. The phrase 'actually solve the problem of interest' and the claim of matching backpropagation accuracy require evidence at the accuracy BP actually attains, not at an arbitrary threshold. The estimator mathematics in Appendix A appears correct, so the gap is between the simulation metric and the headline conclusion, not in the unbiasedness derivation.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies multiplexed gradient descent (MGD), a perturbative zeroth-order training method intended for in-situ training of analog neuromorphic hardware. It extends the MGD framework from weight perturbation to node perturbation, proves that the MGD estimator is unbiased and computes its covariance in Appendix A, and reports FashionMNIST simulations on convolutional networks whose size ranges from roughly 2.5e4 to more than 1e6 parameters. The central claims are that the time to estimate the gradient grows with network size in the expected way, but the time to reach a target accuracy does not follow this scaling and in fact often decreases with network size, and that MGD can train a network with more than one million parameters to the same testing accuracy as backpropagation. The paper also demonstrates that MGD can drive the Adam optimizer and discusses hardware-specific tuning of the MGD time constants.","tokens_in":17985,"tokens_out":12086,"duration_ms":108300,"significance":"If the central scaling claim holds on real hardware, it would materially change the practicality of perturbative, model-free training for large analog neuromorphic systems, because the standard objection that gradient-estimation time grows linearly with parameter count would not translate into wall-clock training time. The paper has genuine strengths: the unbiasedness and covariance derivation in Appendix A is careful and appears correct; the distinction between gradient-estimation time and time-to-accuracy is an important and testable framing; the pseudocode is complete; and the code is publicly available. However, the empirical support for the strongest form of the claim is incomplete, as detailed in the major comments, so the paper needs revision before the headline conclusions are fully supported.","major_comments":[{"comment":"The claim that 'a network with more than 1 million parameters can be trained to the same testing accuracy as backpropagation' is not supported by the reported experiments. Fig. 4a shows a backpropagation baseline only for N=2.55e5, reaching 91.6%, whereas Fig. 4b measures time to 80% accuracy for larger networks and shows no backpropagation curve at all. Since 80% is substantially below the accuracy backpropagation attains, the observed flat scaling could be an artifact of an undemanding threshold; the authors should report the time to reach the accuracy that backpropagation actually reaches for each network size, or otherwise demonstrate that the flatness persists at BP-comparable accuracy.","section":"Section 3.2, Fig. 4"},{"comment":"The paper converts iteration counts into wall-clock time through an implicit hardware model. The statement that '10^9 MGD time iterations would take 16 minutes on hardware with a modest speed of 1 MHz for perturbations, inference and updates' assumes that one iteration costs O(1) time independent of N, i.e., that perturbations, loss readouts, and weight updates are fully parallel. Fig. 4b/c report iterations, not time; if the per-iteration cost on the target hardware grows with network size, the headline scaling of 'time to train' with N would be different. The authors should state this hardware model explicitly and, ideally, plot the projected wall-clock time under that model.","section":"Section 3.2, Conclusion"},{"comment":"There is a mismatch between the theoretical estimator and the pseudocode. Appendix A analyzes G defined in Eq. (11) for simultaneous perturbations of all K parameters, with one cost change Delta C per timestep. Algorithm 1, however, loops over layers and computes a separate output, cost, and cost change Delta C_l for each layer l using only that layer's perturbation theta_l. This is a layer-wise perturbative estimator, not the simultaneous estimator of Eqs. (10)-(12). The normalization Gamma in Table 2 uses the total K, but the per-layer cost change involves only K_l parameters. The authors should clarify which algorithm was actually simulated, and either correct the pseudocode to match the theory or provide the corresponding moments for the layer-wise estimator.","section":"Appendix A, Algorithm 1"},{"comment":"The central scaling result is reported without error bars or a statement of how many random initializations were used. Fig. 3 reports quartile bands over 10 initializations, but the caption for Fig. 4 does not; the claim that time to 80% accuracy 'varies by less than one order of magnitude' needs variance information to be interpretable, especially because the conclusion contradicts earlier scaling arguments.","section":"Fig. 4b/c"}],"minor_comments":[{"comment":"The text says 'node perturbation was reduced by approximately a factor of 2 over node perturbation'; this should read 'over weight perturbation.'","section":"Section 3.2"},{"comment":"There is a typo in 'weight nad node perturbation,' and the caption should state whether the dashed lines are theoretical predictions or fits to the data.","section":"Fig. 3b caption"},{"comment":"The perturbations are called Bernoulli but they are symmetric +-delta; the term 'symmetric Bernoulli' or 'Rademacher' would be more precise.","section":"Section 2.2"},{"comment":"The layer index l is used but the perturbation Theta is written without a layer superscript; clarifying that Theta is added to the pre-activation of the layer would remove ambiguity.","section":"Eq. (5)"}],"recommendation":"major_revision","confidential_remarks":"The main gap is the missing backpropagation baseline at the larger network sizes and the arbitrary 80% threshold used for the scaling claim. If the authors can provide time-to-accuracy at BP-comparable accuracy, state the hardware time model, and resolve the pseudocode/theory mismatch, the paper would be suitable for publication. The derivation in Appendix A is a solid contribution and should be preserved."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The takeaway is mixed. The paper makes a genuinely useful distinction between gradient-estimation time and time-to-target-accuracy for perturbative training, and shows that in their FashionMNIST simulations the latter does not grow with network size. That is a real observation worth building on, and the node-versus-weight perturbation comparison plus the MGD-with-Adam demonstration are new relative to the cited literature. The estimator math in Appendix A is standard simultaneous-perturbation material, and it is correct; the pseudocode and the GitHub link help replicability.\n\nThe soft spot is the gap between the headline and the evidence. The abstract and Section 3.2 claim that a network with over a million parameters trains to the same testing accuracy as backpropagation. What Fig. 4 actually shows is time to 80% test accuracy. Backprop in Fig. 4a reaches 91.6%, and no backprop curve is shown above 2.55e5 parameters. So the flat scaling could be specific to an undemanding threshold; time to a BP-comparable accuracy might still grow with network size. The paper should report time to a common accuracy level that BP actually attains, or at least plot the full accuracy-versus-iterations curves for the largest networks. Error bars in Fig. 4b/c would help too, since the quartile shading used in Fig. 3 disappears in the main scaling figure.\n\nThere is also an unstated hardware assumption in the wall-clock gloss: each MGD iteration is treated as O(1) time regardless of N, meaning perturbations, inference, and updates are fully parallel. That may be realistic for some analog designs, but it should be explicit, because the time-to-train claim depends on it. If per-iteration time grows with N, the scaling curves need to be rescaled before drawing hardware conclusions.\n\nNone of this is fatal. The gradient-accuracy versus training-accuracy distinction is well motivated, the simulations are consistent with the broader point that accurate gradients are not necessary for online training, and the Adam compatibility result is a practical step. The paper is aimed at the neuromorphic hardware community and people studying zeroth-order optimization scaling; they will get value from it even if the headline is overstated.\n\nMy recommendation: send it to peer review. A serious referee should ask for a BP-comparable accuracy threshold, error bars, hyperparameter details, and a clear statement of the parallel-hardware time model. That would put the scaling claim on solid ground.","headline":"Makes a useful distinction between gradient-estimation time and time-to-target-accuracy for perturbative training, but the claim of backprop-matching accuracy at million-parameter scale outruns the evidence.","tokens_in":18531,"tokens_out":3912,"would_cite":true,"duration_ms":31869,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["68T07","90C56"],"pacs":[],"model":"deepseek-v4-flash","headline":"A perturbative training method can match backpropagation accuracy on million-parameter networks, because time-to-accuracy does not scale with size.","keywords":["multiplexed gradient descent","perturbative training","zeroth-order optimization","weight perturbation","node perturbation","neuromorphic hardware","gradient estimation scaling","time-to-accuracy"],"falsifier":"Run the same FashionMNIST architecture sweep on a physical MGD chip and measure wall-clock time to 80% testing accuracy for networks of $10^4$ and $10^6$ parameters; if the larger network takes roughly $10^2$ times longer, the claimed time-to-accuracy scaling does not survive real hardware costs.","tokens_in":17545,"feed_emoji":"🧠","tokens_out":9139,"duration_ms":76202,"temperature":0.7,"pith_summary":"Multiplexed gradient descent (MGD) is a zeroth-order training method that treats a neural network as a black box: it applies small random perturbations to weights or nodes, reads the resulting change in loss, and correlates the two to estimate the gradient. The paper's central claim is that the time to estimate that gradient grows with network size, but the time to actually train the network to a target accuracy does not—in simulations, iterations to reach 80% accuracy vary by less than an order of magnitude while network size changes by more than three orders of magnitude. This matters because the presumed linear scaling of gradient estimation has been the main reason perturbative methods were dismissed as impractical for training large analog neuromorphic hardware. The paper further shows that the MGD gradient can replace the backpropagation gradient inside standard optimizers such as Adam, and that a network with over one million parameters trains to the same testing accuracy as backpropagation.","feed_headline":"Perturbative training matches backprop at one million parameters","feed_subtitle":"Time to target accuracy, not gradient-estimation time, stays flat as networks grow.","key_machinery":"The central object is the MGD estimator $G = (1/\\tau_\\theta)\\Lambda^{-1}\\sum_{t=1}^{\\tau_\\theta}\\Delta C(t)\\Theta(t)$, where $\\Theta(t)$ is a vector of mean-zero random perturbations, $\\Delta C(t)$ is the resulting change in loss, $\\Lambda$ is the perturbation covariance, and $\\tau_\\theta$ is a time constant controlling how many perturbation steps are averaged before a weight update. By Taylor expansion, $\\Delta C(t) \\approx \\nabla_\\Theta C \\cdot \\Theta(t)$, so correlating loss changes with perturbations recovers the gradient; the paper proves $G$ is unbiased and computes its covariance, which vanishes as $\\tau_\\theta\\to\\infty$. Two concrete perturbation schemes are studied: weight perturbation, which perturbs every parameter and costs $K=N$ perturbations per gradient estimate, and node perturbation, which perturbs the input to each activation and costs $K \\approx \\sqrt{N}$ for dense layers, at the price of a one-step local backpropagation and a multiplication per synapse. The three MGD time constants let the same framework interpolate between strategies such as coordinate descent and SPSA, and let an implementation trade more gradient averaging for fewer weight updates.","core_discovery":"On its own terms, the paper establishes a separation between two quantities that earlier analyses conflated: the number of iterations needed to estimate the gradient of the loss to a given accuracy, and the number of iterations needed to drive testing accuracy to a fixed target. For weight perturbation the former scales as $N$ and for node perturbation as $\\sqrt{N}$, where $N$ is the number of parameters; for the latter, across networks from roughly $10^4$ to $10^6$ parameters, the iteration count to 80% testing accuracy stays within one order of magnitude and in parts of the range decreases. The same flat pattern does not hold when task complexity is varied: training time rises strongly with the number of FashionMNIST classes. The paper also proves that the MGD estimator $G$ is an unbiased estimator of the gradient with covariance that shrinks as the integration time $\\tau_\\theta$ grows, and demonstrates numerically that using $G$ inside the Adam update rule reduces the number of weight updates to 80% accuracy by about 37%.","pith_inferences":["The flat iteration-count scaling becomes a flat wall-clock scaling only if one MGD iteration costs the same time for any network size; on hardware that perturbs parameters or reads losses sequentially, the time-to-accuracy curves in the paper would need to be rescaled by a size-dependent factor.","The near-independence of training time from network size is consistent with overparameterized networks being easier to optimize, but the paper does not isolate that mechanism; a direct test would hold the architecture and optimizer fixed and vary only width while tracking the full loss trajectory.","The results suggest a benchmark shift for analog neuromorphic hardware: comparisons should report time-to-target-accuracy on actual devices rather than per-iteration gradient fidelity, and task complexity should be controlled separately from network size.","Because MGD needs only forward propagation and a global cost-change broadcast, its success hints at a route to online learning in spiking and other non-differentiable hardware; the paper flags this as future work rather than claiming it."],"forward_implications":["Gradient-estimation time is the wrong benchmark for judging perturbative training; time-to-accuracy is the quantity that matters for whether the approach can solve a real problem.","A network with more than one million parameters can be trained with MGD to the same testing accuracy as backpropagation on the same architecture, at least on FashionMNIST.","Because $G$ is a drop-in replacement for the backpropagation gradient, standard optimizer machinery such as momentum and Adam can be combined with MGD; in the paper Adam cuts the number of updates to 80% accuracy by roughly 37%.","Increasing the gradient integration time $\\tau_\\theta$ can reduce the required number of weight updates by orders of magnitude, which matters for hardware with limited write endurance.","Node perturbation is not universally faster than weight perturbation: its advantage depends on the target accuracy, and convolutional layers can have more activations than weights, reversing the naive scaling comparison."],"supporting_citations":[{"why":"Establishes MGD, its three time constants, and earlier evidence that in-hardware training can be fast; this paper extends that framework.","marker":"[27]"},{"why":"Defines model-free distributed weight-perturbation learning, the template for MGD's weight perturbation.","marker":"[14]"},{"why":"Introduces summed-weight neuron perturbation, which this paper generalizes as node perturbation and compares against weight perturbation.","marker":"[17]"},{"why":"Introduces simultaneous perturbation stochastic approximation, the perturbation-and-correlation gradient estimator MGD builds on.","marker":"[32]"},{"why":"Exemplifies the field's claim that perturbative training scales poorly; the paper's target-accuracy results directly address it.","marker":"[2]"},{"why":"Cites the poor scaling of gradient-estimation time to question perturbative learning's relevance, the assumption the paper tests.","marker":"[3]"},{"why":"Shows an accurate gradient estimate is not required for online gradient descent, which motivates using time-to-accuracy rather than gradient time.","marker":"[33]"},{"why":"Supplies the Adam optimizer update rule used to demonstrate MGD works as a drop-in gradient replacement.","marker":"[35]"}],"fun_headline_variants":["Perturbative training time to accuracy stays flat up to 1M parameters","MGD training time to accuracy doesn't scale with network size","Flat time to accuracy: perturbative training scales to million-parameter nets","Perturbative training: time to accuracy flat from 10K to 1M parameters","Time to accuracy, not gradient estimation, dictates MGD scaling"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that one MGD iteration costs the same wall-clock time regardless of network size, because perturbations, loss readouts, and weight updates are assumed to happen in parallel on hardware.","fun_headline_variants_meta":{"raw":{"variants":["Perturbative training time to accuracy stays flat up to 1M parameters","MGD training time to accuracy doesn't scale with network size","Flat time to accuracy: perturbative training scales to million-parameter nets","Perturbative training: time to accuracy flat from 10K to 1M parameters","Time to accuracy, not gradient estimation, dictates MGD scaling"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000899,"raw_usage":{"total_tokens":3881,"prompt_tokens":965,"completion_tokens":2916,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":581,"completion_tokens_details":{"reasoning_tokens":2817}},"tokens_in":581,"tokens_out":2916,"duration_ms":17747,"temperature":1.0,"reasoning_tokens":2817,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T14:19:07.104194+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the same FashionMNIST architecture sweep on a physical MGD chip and measure wall-clock time to 80% testing accuracy for networks of $10^4$ and $10^6$ parameters; if the larger network takes roughly $10^2$ times longer, the claimed time-to-accuracy scaling does not survive real hardware costs.","supporting_citations":[{"cited_title":"Analog VLSI Autonomous Systems for Learning and Optimization PhD thesis (Caltech, 1994)","cited_arxiv_id":null,"evidence_quote":"Establishes MGD, its three time constants, and earlier evidence that in-hardware training can be fast; this paper extends that framework."},{"cited_title":"Frontiers in Neural Circuits 0, 53","cited_arxiv_id":null,"evidence_quote":"Defines model-free distributed weight-perturbation learning, the template for MGD's weight perturbation."},{"cited_title":"A Fast Stochastic Error-Descent Algorithm for Supervised Learning and Optimization in Advances in Neural Information Processing Systems 5 (NIPS 1992) (1992), 244–251","cited_arxiv_id":null,"evidence_quote":"Introduces summed-weight neuron perturbation, which this paper generalizes as node perturbation and compares against weight perturbation."},{"cited_title":"P., Santoro, A., Marris, L., Akerman, C","cited_arxiv_id":null,"evidence_quote":"Cites the poor scaling of gradient-estimation time to question perturbative learning's relevance, the assumption the paper tests."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Shows an accurate gradient estimate is not required for online gradient descent, which motivates using time-to-accuracy rather than gradient time."}],"review_version":1}