{"id":"562a5184-559b-416f-a694-16af819a2b76","arxiv_id":"2506.13234","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A single-weight perturbation at the very start of training makes otherwise identical neural networks diverge to different loss basins, and this sensitivity drops sharply within the first fraction of training.","lead":"Training a neural network twice from nearly the same starting point can send it to completely different solutions, even if the only difference is a tiny change to a single weight early in training. The paper measures exactly how small a perturbation can be and how training time, architecture, and pre-training change that sensitivity, which matters for anyone who wants to merge or ensemble models.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Section 3.1's unproved inequality connecting deterministic same-noise perturbations to stochastic training instability is the weakest load-bearing step; a simple multiplicative-noise counterexample shows it is not a general theorem.","rationale":"The paper's central empirical finding—that a controlled deterministic perturbation applied near initialization sends otherwise identical training runs to different loss basins—is supported by consistent trends across four metrics and many settings, and the authors provide code. I do not see an internal contradiction or a circular step. The weakest point is the unproved inequality in §3.1 connecting these deterministic experiments to ordinary stochastic training. The reader already identified this; I agree it is the most load-bearing concern. Without the inequality, the paper's practical implications for fine-tuning, merging, and ensembling rest on an extrapolation from same-noise runs to independent-noise runs that is not established. A simple multiplicative-noise counterexample shows the statement is not true for arbitrary stochastic maps, and Figure 10 checks only one setting, so the concern is concrete rather than stylistic. Error-bar and seed-count concerns are secondary: the absence of error bars on main figures weakens 'reliably' but does not threaten the core observation, and the authors report three-seed averaging. Non-convergence of the endpoints is a limitation, but barriers are defined at the measured time and are consistent with prior spawning work. Therefore the reader's CONDITIONAL verdict stands; the authors should either prove or qualify the inequality and add the missing variance information.","tokens_in":30479,"tokens_out":11565,"duration_ms":140855,"concrete_test":"Recompute the Figure 10 comparison for all main regimes—ResNet-20 with each hyperparameter set, ResNet-50 CIFAR transfer, ViT fine-tuning, MultiBERT on QNLI, and OLMo on GSM8K—using both same-noise and independent-noise couplings at the same t, σ, and dissimilarity metric (train barrier and L2), with at least five seeds per condition. If any setting yields an average independent-noise barrier significantly below the average same-noise barrier, the §3.1 inequality is empirically violated there, and the paper must either prove the inequality under explicit conditions or soften the 'lower bound' claim to a per-setting empirical observation.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing step is the unproved inequality in §3.1: E[d(T(θ,ξ),T(θ+ε,ξ'))] ≥ E[d(T(θ,ξ),T(θ+ε,ξ))] for independently sampled noise ξ and ξ'. The paper uses this to claim that its deterministic same-noise experiments lower-bound the barrier induced by independent SGD noise in parent-child spawning. The inequality is not a general theorem: for a stochastic map T(θ,ξ)=ξ·θ with scalar θ and ξ,ξ' iid ±1, take θ=−ε/2; same-noise runs always differ by |ε|, while independent-noise runs differ by 0 or |ε| with equal probability, giving expected distance |ε|/2 < |ε|. Nothing in §3 rules out such behavior for SGD maps, and Figure 10 verifies the inequality for a single ResNet-20 setting only. Because the abstract's practical framing (perturbations drive fine-tuning and merging outcomes) relies on transferring deterministic divergence to stochastic training, this missing proof is the most load-bearing assumption; if the inequality fails in any studied regime, the deterministic results remain valid as measurements but do not automatically speak to stochastic training instability.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a deterministic 'spawn-and-perturb' protocol to measure the sensitivity of neural network training trajectories to small perturbations of the weights. From an initial state, a network is trained to time t, one copy is perturbed by noise of controlled magnitude and direction, and both copies are then trained with identical noise; divergence is measured by L2 parameter distance, training-loss barriers, permutation-aligned barriers, and Angular CKA. Across ResNet, ViT, BERT, and OLMo experiments, the authors report that perturbations as small as a single weight applied at initialization reliably cause large barriers, that stability increases rapidly during early training, that certain hyperparameters (warm-up, wider architectures) improve stability, that pre-trained networks are generally more stable but longer language-model pre-training can reduce fine-tuning stability, and that barrier and L2 growth are not exponential. The paper interprets the results as evidence for a 'butterfly effect' in neural network training with implications for fine-tuning, model merging, and ensembles.","tokens_in":30597,"tokens_out":4715,"duration_ms":57987,"significance":"If the claims hold, this is a clean empirical characterization of trajectory sensitivity that complements prior noise-based spawning experiments and extends them to controlled perturbations of arbitrary size and timing. Strengths of the paper include the carefully designed protocol, the use of multiple complementary divergence measures, the breadth of architectures and tasks, reproducibility-oriented practices (deterministic training, released code), and the absence of fitted free parameters in the reported measurements. The central 'single-weight perturbation causes divergence' finding is striking and falsifiable. However, the paper's stated bridge from deterministic to stochastic training rests on an unproved and in general false inequality, and some headline claims (non-exponential growth, several comparative fine-tuning findings) lack statistical support. These issues are addressable but currently weaken the interpretation of otherwise valuable measurements.","major_comments":[{"comment":"The inequality E[d(T(θ,ξ),T(θ+ε,ξ'))] ≥ E[d(T(θ,ξ),T(θ+ε,ξ))] is stated without proof as a general lower bound connecting deterministic same-noise experiments to stochastic training. It is not true for arbitrary stochastic maps and distance metrics: for scalar θ, T(θ,ξ)=ξθ with ξ,ξ' iid uniform ±1, and θ=−ε/2, the same-noise distance is always |ε| whereas the independent-noise expected distance is |ε|/2. Since this inequality is the only stated formal justification for claiming that the deterministic experiments 'lower bound' the instability of regular stochastic training, and since that claim is invoked in the abstract and in the practical fine-tuning/merging discussion, it must either be proved under explicit assumptions on T and d, or reformulated as an empirical observation restricted to the settings verified in Figure 10.","section":"§3.1, displayed inequality after Eq. (1)"},{"comment":"The claim that 'neither barrier nor L2 increase exponentially over training' is based on visual inspection of curves that are averaged over perturbation magnitudes, with no statistical test or fitted growth model. This is contribution #5 of the paper and directly contradicts the linearized-dynamics expectation derived in Section B.4, so it needs quantitative support: for example, slopes of log(L2) and log(barrier) versus step with confidence intervals, or a comparison of exponential versus sub-exponential models, and a statement of whether the non-exponential pattern holds per perturbation magnitude or only after averaging.","section":"§5, Figure 6"},{"comment":"The main figures do not show error bars or other variance information even though Appendix B reports that evaluations are averaged over only three seeds (two for fine-tuning) per configuration. Several comparative claims in Section 4.4, such as the final MultiBERT checkpoint (2000k) having the largest QNLI barriers and the ImageNet-1k-fine-tuned ViT-Base having larger barriers than its in21k counterpart, depend on differences that may be within seed variance. Adding error bars or shaded confidence regions, or reporting effect sizes, is necessary for the reader to assess the reliability of these quantitative comparisons.","section":"Main figures, especially Figures 2, 4, 5, 6, 7"}],"minor_comments":[{"comment":"The phrase 'even extremely small perturbations reliably causes otherwise identical training trajectories to diverge' has a subject-verb agreement error ('perturbations ... causes'); please change 'causes' to 'cause'.","section":"Abstract"},{"comment":"There is an extra closing parenthesis in 'dCKA(θT, θ′T)) = CKA [...]'; also, the notation dCKA is used for a distance while CKA is defined as arccos(...), so please make the composition explicit.","section":"Equation (5)"},{"comment":"The symbol T is overloaded: T denotes both the single-step stochastic map and the total number of training steps in T^T, and the displayed inequality uses a single T while referring to full training. Please use distinct notation for the number of steps and the map.","section":"§3.1, Eq. (1)"},{"comment":"In the right panel, the y-axis label 'Noise L2' is not defined in the caption; please specify whether this is the L2 norm of the perturbation, the expected SGD noise magnitude, or something else.","section":"Figure 10"},{"comment":"There are typos: 'corraborate' should be 'corroborate' in Section C.3, and 'tarnsformer' should be 'transformer' in Section D.2.","section":"Section C.3 and Section D.2"},{"comment":"Barrier values are reported as '0.00 ± 0.00'; please state whether these are exactly zero or simply below the rounding threshold, and if so, give the threshold or report more significant digits.","section":"Tables 4 and 5"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Read this one for the method and for one counterexample that matters. The protocol is a real advance over the old noise-based spawning: a deterministic, controlled perturbation at a chosen time, with batch and Gaussian directions and a scale normalized to initialization. The single-weight result is notable—one weight at initialization is enough to send otherwise identical ResNet runs into different basins—and the trend across CKA, barriers, and permutation alignment is consistent. The finding that more pre-training can reduce fine-tuning stability (MultiBERT, OLMo, and the ViT-1k case) is genuinely surprising and worth taking seriously. The negative results are also useful: permutation matching does not reduce barriers for identically initialized runs, and divergence is not exponential.\n\nThe soft spot is exactly where the stress-test note points. Section 3.1 states without proof that deterministic same-noise experiments lower-bound stochastic training instability. That inequality is not a general theorem; the multiplicative-noise counterexample is correct. The paper only checks one ResNet setting in Figure 10, and the abstract's practical framing leans on the transfer. This is load-bearing for the practical story, though not for the raw empirical observations. Fixing it is straightforward: either prove a restricted version for SGD under reasonable assumptions, or soften the wording to say that stochastic instability is at least as large in the settings tested. I would not block acceptance on that alone, but it should be addressed.\n\nMinor issues: main figures show no error bars despite three seeds (appendix tables do include them, which helps), and the non-exponential divergence claim rests on visual inspection rather than a rate test. Both are small; the trends look consistent, and the code and hyperparameters are public.\n\nThis paper deserves a serious referee. I would send it out, flagging the inequality and the error bars as revision items; the central empirical results are careful, reproducible, and worth building on.","headline":"A genuinely useful measurement framework for training sensitivity, with a real empirical counterexample about pre-training stability, but the deterministic-to-stochastic lower-bound claim is too strong as stated.","tokens_in":31192,"tokens_out":2186,"would_cite":true,"duration_ms":24397,"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":"A single perturbed weight early in training can send identical neural networks to different loss basins.","keywords":["butterfly effect","neural network training stability","loss barriers","linear mode connectivity","permutation alignment","fine-tuning stability","model ensembling","chaotic training phase"],"falsifier":"A direct test is to run the spawn-and-perturb protocol at initialization with a single perturbed weight, but after the perturbation train the two copies with independently sampled minibatches rather than identical ones; if the resulting barrier at convergence is not at least as large as the barrier from the identical-noise run, the claimed lower bound fails. A second test is to search even one architecture, task, or optimizer for which the same single-weight perturbation at $t=0$ produces a zero training-loss barrier, which would contradict the claim that early training is universally chaotic.","tokens_in":30167,"feed_emoji":"🦋","tokens_out":10125,"duration_ms":109272,"temperature":0.7,"pith_summary":"Neural network training is often blamed on stochastic gradient descent noise when two runs diverge. This paper argues that the training dynamics themselves are chaotic near the start: during an initial \"chaotic\" phase, perturbing as little as a single weight in an otherwise identically initialized network reliably sends the two copies to different loss basins, separated by a measurable loss barrier along the linear path between their final weights. The effect decays rapidly with training time, so the same perturbation applied after roughly the first half percent of training leaves the trajectories essentially together. These barriers survive permutation alignment and are visible in representational similarity, which means the two networks are genuinely different in function, not just in how neurons are ordered. If the claim holds, training instability is an inherent property of early optimization rather than a byproduct of noise, with direct consequences for fine-tuning, model merging, and ensemble diversity.","feed_headline":"One weight is enough to derail identical neural network runs","feed_subtitle":"A perturbation below typical hardware noise sends identical runs into different loss basins early in training.","key_machinery":"The central instrument is the spawn-and-perturb experiment. A parent network is trained deterministically from $\\theta_0$ to time $t$, split into two copies, one copy is given a perturbation $\\varepsilon$ of controlled magnitude $\\sigma$, and both copies are trained identically to time $T$; instability is the size of $d(\\theta_T, \\theta'_T)$ under four dissimilarity measures: $L^2$ parameter distance, the training-loss barrier $\\sup_\\alpha[\\ell(\\alpha\\theta_T+(1-\\alpha)\\theta'_T)-\\alpha\\ell(\\theta_T)-(1-\\alpha)\\ell(\\theta'_T)]$, the same barrier after weight matching over neuron permutations, and Angular CKA between penultimate-layer activations. The load-bearing theoretical object is the deterministic training map $T^T(\\theta_0; \\xi)$ with fixed noise $\\xi$, together with the claim that divergence after a single controlled perturbation lower-bounds divergence after independent stochastic noise: $E[d(T(\\theta,\\xi), T(\\theta+\\varepsilon,\\xi'))]\\ge E[d(T(\\theta,\\xi), T(\\theta+\\varepsilon,\\xi))]$. The perturbations are chosen either in the direction of a batch gradient or as initialization-scaled Gaussian noise, so that stability is probed along both training-relevant and generic directions.","core_discovery":"Viewed as a dynamical system, training is an iterated map $T$ on the parameter space; with the random seed fixed, it is deterministic. The paper claims that this map exhibits a butterfly effect: a perturbation $\\varepsilon$ applied at $\\theta_0$, scaled to as little as one weight (roughly $10^{-4}$ relative to the initialization scale), grows under training so that the final network $\\theta'_T$ has a large training-loss barrier from $\\theta_T$, even though both copies saw identical batches, augmentations, and seeds. Applying the same perturbation after only $0.5\\%$ of training yields a much smaller or negligible barrier, and later perturbations require orders-of-magnitude larger $\\varepsilon$ to produce any barrier at all. Weight matching that searches over neuron permutations fails to reduce the barrier, and Angular CKA shows the diverged networks have different intermediate representations, so the divergence is functional rather than a symmetry artifact. The paper presents this as evidence that early training is chaotic in the dynamical-systems sense, not merely noisy.","pith_inferences":["If the lower-bound inequality holds in general, then every stochastic training run is at least as unpredictable as these deterministic experiments show, and reproduction efforts should focus on avoiding the chaotic phase (for example, by starting pre-trained or extending warm-up) rather than on controlling noise alone.","The rapid stabilization within 0.5% of training suggests a sharp dynamical transition; estimating the local Lyapunov exponent (the rate at which nearby trajectories separate) along the training path would locate this transition and could predict the minimum perturbation that survives to convergence.","The task dependence of stability (CIFAR-100 to CIFAR-10 versus the reverse, and vision versus language) hints that the controlling factor is the geometry of the loss landscape near the starting checkpoint, not the architecture class; a testable prediction is that stability correlates with local curvature at the checkpoint.","Since weight matching cannot remove the barriers, a natural follow-up the paper leaves open is whether richer alignment methods, such as layer-wise or activation-normalized permutations, could reconnect the divergent networks, or whether the functional differences are irreducible."],"forward_implications":["A model that is stable to training noise can still be unstable to perturbations larger than that noise, so stability measured by seed agreement is not a guarantee of convergence to a single basin.","Hyperparameter choices that increase early stability, such as longer learning-rate warm-up and wider or shallower architectures, shift but do not eliminate the chaotic phase.","Fine-tuning stability is not a monotone function of pre-training: for several language-model checkpoints, the most pre-trained model is the most sensitive to perturbations during fine-tuning.","Because permutation alignment does not remove the barriers, the diverged networks are not equivalent up to neuron reordering, and permutation-based merging methods will not reconnect them by themselves.","In the vision settings tested, ensembling the original and perturbed networks yields accuracy gains that grow with representational dissimilarity, so deliberate perturbation is a candidate mechanism for ensemble diversity."],"supporting_citations":[{"why":"Introduced the spawn experiment and the linear mode connectivity barrier; the paper replaces their independent training noise with a single controlled perturbation.","marker":"Frankle et al. (2020a)"},{"why":"Established the chaotic-early and stable-late phases of training that the paper quantifies at perturbation scale.","marker":"Fort et al. (2020)"},{"why":"Provided the weight-matching permutation algorithm used to test whether the measured barriers vanish under neuron permutations.","marker":"Ainsworth et al. (2023)"},{"why":"Defined the loss-barrier setting and gave the prior evidence that pre-trained models fine-tune stably, which the paper qualifies.","marker":"Neyshabur et al. (2020)"},{"why":"Supplied the permutation-invariance hypothesis that motivates the barrier-modulo-permutation measurement.","marker":"Entezari et al. (2022)"},{"why":"Provided Angular CKA, the representational dissimilarity metric used to show the divergence is functional.","marker":"Williams et al. (2021)"},{"why":"Gave the dynamical-stability view of SGD and the argument that deterministic exponential divergence would dominate stochastic effects.","marker":"Wu et al. (2018)"},{"why":"Showed language-model fine-tuning can land in distinct basins, motivating the controlled perturbation study of BERT and OLMo.","marker":"Juneja et al. (2023)"},{"why":"Studied stability to training noise along transfer trajectories; the paper compares its deterministic perturbation results to this baseline.","marker":"Vlaar & Frankle (2022)"},{"why":"Showed warm-up, larger batches, and lower learning rates reduce early variability, the hyperparameter expectations tested here.","marker":"Altıntaş et al. (2023)"}],"fun_headline_variants":["One-weight nudge forks identical neural net runs","Butterfly effect in training: tiny change, big barrier","Tiny weight tweak sends identical runs to different minima","Training chaos: a single weight flips the trajectory","Early perturbation derails networks, later ones don't"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that a single deterministic perturbation cannot make two identically trained copies diverge less than independent random batches would; if independent noise can sometimes push the two copies closer together, then the deterministic butterfly experiments would not bound the instability of ordinary stochastic training.","fun_headline_variants_meta":{"raw":{"variants":["One-weight nudge forks identical neural net runs","Butterfly effect in training: tiny change, big barrier","Tiny weight tweak sends identical runs to different minima","Training chaos: a single weight flips the trajectory","Early perturbation derails networks, later ones don't"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00015,"raw_usage":{"total_tokens":1181,"prompt_tokens":916,"completion_tokens":265,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":532,"completion_tokens_details":{"reasoning_tokens":187}},"tokens_in":532,"tokens_out":265,"duration_ms":3806,"temperature":1.0,"reasoning_tokens":187,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T00:36:58.942885+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A direct test is to run the spawn-and-perturb protocol at initialization with a single perturbed weight, but after the perturbation train the two copies with independently sampled minibatches rather than identical ones; if the resulting barrier at convergence is not at least as large as the barrier from the identical-noise run, the claimed lower bound fails. A second test is to search even one architecture, task, or optimizer for which the same single-weight perturbation at $t=0$ produces a zero training-loss barrier, which would contradict the claim that early training is universally chaotic.","supporting_citations":[{"cited_title":"Git re-basin: Merging models modulo permutation symmetries","cited_arxiv_id":null,"evidence_quote":"Provided the weight-matching permutation algorithm used to test whether the measured barriers vanish under neuron permutations."},{"cited_title":"What is being transferred in transfer learning? In Advances in Neural Information Processing Systems, volume 33, pp.\\ 512--523, 2020","cited_arxiv_id":null,"evidence_quote":"Defined the loss-barrier setting and gave the prior evidence that pre-trained models fine-tune stably, which the paper qualifies."},{"cited_title":"The role of permutation invariance in linear mode connectivity of neural networks","cited_arxiv_id":null,"evidence_quote":"Supplied the permutation-invariance hypothesis that motivates the barrier-modulo-permutation measurement."},{"cited_title":"H., Kunz, E., Kornblith, S., and Linderman, S","cited_arxiv_id":null,"evidence_quote":"Provided Angular CKA, the representational dissimilarity metric used to show the divergence is functional."},{"cited_title":"Linear connectivity reveals generalization strategies","cited_arxiv_id":null,"evidence_quote":"Showed language-model fine-tuning can land in distinct basins, motivating the controlled perturbation study of BERT and OLMo."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Studied stability to training noise along transfer trajectories; the paper compares its deterministic perturbation results to this baseline."}],"review_version":1}