[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84476-en":3,"doc-seo-84476-105":29,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},84476,687197100911,"Himbo","https://ap-avatar.wpscdn.com/avatar/a000239b6f1da00475?x-image-process=image/resize,m_fixed,w_180,h_180&k=1782698725881665579",8,"Research & Report","Critical Damping as a Momentum Schedule: Multi-Seed Validation, a Hybrid Recipe, and an Exhaustive Negative Result on Surgical Layer Selection","Critical damping applied to the damped harmonic oscillator model of SGD with momentum yields a tuning-free momentum schedule, fixing µ(t)=1−2√α(t) (with a fixed, never-tuned stability clamp). The schedule is validated on ResNet-18/CIFAR-10 with a 200-epoch cosine schedule across five seeds, reaching 90% test accuracy 2.34× faster than µ=0.9, but with a measurable final-accuracy deficit of 0.46 pp. A Hybrid-90 strategy removes the deficit while keeping the speedup. An exhaustive study finds gradient-attribution-based error signals do not select optimal surgical layers. The full layer-combination landscape is released.","arXiv :2603 .2892 1v 3 [ cs .LG] 12 Jul 2026  \nCritical Damping as a Momentum Schedule: Multi-Seed Validation, a Hybrid Recipe, and an Exhaustive Negative Result on Surgical Layer Selection  \nIvan Pasichnyk  \nWe Label Data Inc.  \n[ivan@welabeldata. com](ivan@welabeldata. com)  \n[https://orcid. org/0009-0004-8154-3345](https://orcid. org/0009-0004-8154-3345)  \nAbstract  \nThe critical damping condition of the damped harmonic oscillator model of SGD with momentum (Qian, 1999) yields a momentum schedule with no tuned hyperparameters: µ(t ) = 1 −2 pα (t )(a fixed, never-tuned stability clamp aside) . We validate this schedule across five random seeds on ResNet-18/CIFAR-10 (200-epoch cosine learning-rate schedule): it reaches 90% test accuracy  \n2.34 × faster than constant µ = 0.9 (range 1 .71–2.86 ×, 5/5 seeds, one-sided paired t-test p = 4 × 10−4), at the cost of a real final-accuracy deficit of 0 .46 pp (5/5 seeds, p = 0.009) . A shortschedule control shows the speedup is not a schedule-length artifact: baselines with the cosine schedule compressed to 60/100 epochs either pay 0 .5–0.9 pp of final accuracy or remain slower to 90% than the physics schedule at equal accuracy—physics Pareto-dominates the accuracy-matched compressed baseline. A hybrid recipe—critical-damping momentum until 90% accuracy, then constant µ = 0.9—removes the deficit while keeping the speedup: 95 .45 ± 0.05% final accuracy (indistinguishable from baseline) at 2 .4 × faster progress to 90%(n = 5) . The speedup generalizes across architectures—VGG-16 without skip connections reaches 90% test accuracy 1 .72 × faster (n = 3)—while on CIFAR-100 the effect is smaller: early-training gains persist (2–4 × to mid-training thresholds) but the pure schedule pays a larger final-accuracy cost (−1 .7 pp), compressing the accuracymatched gain to 1 .14 × and delimiting the schedule’s scope.  \nWe also report an exhaustive negative result on surgical layer selection. Version 2 of this paper claimed that gradient attribution computed on misclassified images selects which layers to retrain. Running the identical surgical-correction protocol on all 35 combinations of 3-of-7 layer groups shows the diagnostically selected triple ranks 11th of 35 (exact p = 0.31)—no better than random selection. What survives is weaker but real: combinations containing the top-ranked layer outperform those without it (+6 .2 vs −1 .4 mean net error reduction), and the bottom of the gradient-norm ranking reliably predicts the most harmful interventions (down to −20 net errors) . Gradient attribution on errors is a harm-avoidance signal, not an optimizer of layer selection. We release the full 35-combination landscape as a baseline we believe should accompany any layer-selection claim.  \nKeywords: momentum scheduling, critical damping, tuning-free momentum schedule, surgical finetuning, negative results, layer attribution  \n1 Introduction  \nMomentum and learning rate are usually tuned as independent hyperparameters. The damped harmonic oscillator model of SGD with momentum [15] says they should not be: for any learning-rate schedule α (t ), the critical damping condition fixes the momentum trajectory completely, µ (t ) = 1−2 pα (t ), with no tuned hyperparameters (the implementation applies a fixed, never-tuned stability clamp; Section 4) . This paper asks two questions about that formula, and answers a third one we did not plan to ask.  \nDoes it work? Yes, with a quantified trade-off (Section 4) . Across five random seeds on ResNet- 18/CIFAR-10 under a fixed 200-epoch cosine schedule, critical-damping momentum reaches 90% test accuracy 2 .34 × faster than the standard constant µ = 0.9 (range 1 .71–2.86 ×, 5/5 seeds, one-sided paired t-test p = 4 × 10−4) . The price is real, not seed noise: a final-accuracy deficit of 0 .46 pp (5/5 seeds, p = 0.009) . A short-schedule control (Section 4.2) rules out the schedule-length explanation: physics is faster to 90% than any compressed baseline of equal final ","cbCaihbYH9OxjmQ7","https://ap.wps.com/l/cbCaihbYH9OxjmQ7","pdf",311488,1,23,"English","en",105,"# Abstract\n# Introduction\n## Momentum–learning-rate coupling and critical damping\n## Speed–accuracy validation and trade-off\n## Hybrid-90 schedule to remove deficit\n## Negative result on surgical layer selection\n# Contributions","[{\"question\":\"What does the critical damping condition imply for SGD with momentum?\",\"answer\":\"It fixes the momentum trajectory from the learning-rate schedule as µ(t)=1−2√α(t) without tuned hyperparameters, aside from a fixed stability clamp.\"},{\"question\":\"How does the tuning-free critical-damping momentum schedule perform versus constant momentum?\",\"answer\":\"On ResNet-18/CIFAR-10 it reaches 90% test accuracy 2.34× faster than µ=0.9, but incurs a final-accuracy deficit of 0.46 pp.\"},{\"question\":\"Does gradient attribution on misclassified images correctly identify layers to retrain?\",\"answer\":\"No. Across all 35 combinations of 3-of-7 layer groups, the diagnostically selected triple ranks 11th of 35 (p=0.31), indicating no better-than-random selection.\"}]",1784195892,58,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":27},"critical-damping-as-a-momentum-schedule-multi-seed-validation-a-hybrid-recipe-and-an-exhaustive-negative-result-on-surgical-layer-selection","",{"@graph":35,"@context":85},[36,53,68],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/critical-damping-as-a-momentum-schedule-multi-seed-validation-a-hybrid-recipe-and-an-exhaustive-negative-result-on-surgical-layer-selection/84476/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-17","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What does the critical damping condition imply for SGD with momentum?","Question",{"text":75,"@type":76},"It fixes the momentum trajectory from the learning-rate schedule as µ(t)=1−2√α(t) without tuned hyperparameters, aside from a fixed stability clamp.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the tuning-free critical-damping momentum schedule perform versus constant momentum?",{"text":80,"@type":76},"On ResNet-18/CIFAR-10 it reaches 90% test accuracy 2.34× faster than µ=0.9, but incurs a final-accuracy deficit of 0.46 pp.",{"name":82,"@type":73,"acceptedAnswer":83},"Does gradient attribution on misclassified images correctly identify layers to retrain?",{"text":84,"@type":76},"No. Across all 35 combinations of 3-of-7 layer groups, the diagnostically selected triple ranks 11th of 35 (p=0.31), indicating no better-than-random 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