{"id":"https://openalex.org/W7165146692","doi":"https://doi.org/10.48550/arxiv.2606.18524","title":"On the Residual Scaling of Looped Transformers: Stability and Transferability","display_name":"On the Residual Scaling of Looped Transformers: Stability and Transferability","publication_year":2026,"publication_date":"2026-06-16","ids":{"openalex":"https://openalex.org/W7165146692","doi":"https://doi.org/10.48550/arxiv.2606.18524"},"language":null,"primary_location":{"id":"doi:10.48550/arxiv.2606.18524","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2606.18524","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"},"type":"preprint","indexed_in":["datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://doi.org/10.48550/arxiv.2606.18524","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5026042163","display_name":"Shaowen Wang","orcid":"https://orcid.org/0000-0001-5848-590X"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Wang, Shaowen","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5101653513","display_name":"Bingrui Li","orcid":"https://orcid.org/0000-0002-4807-1452"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Li, Bingrui","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5138875694","display_name":"Ge Zhang","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Zhang, Ge","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5102688813","display_name":"W E I-Xing Huang","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Huang, Wenhao","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5111473372","display_name":"Su Yan","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Yan, Shen","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5138923099","display_name":"Jian Li","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Li, Jian","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":false,"cited_by_count":0,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10036","display_name":"Advanced Neural Network Applications","score":0.2802000045776367,"subfield":{"id":"https://openalex.org/subfields/1707","display_name":"Computer Vision and Pattern Recognition"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10036","display_name":"Advanced Neural Network Applications","score":0.2802000045776367,"subfield":{"id":"https://openalex.org/subfields/1707","display_name":"Computer Vision and Pattern Recognition"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11612","display_name":"Stochastic Gradient Optimization Techniques","score":0.14319999516010284,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11307","display_name":"Domain Adaptation and Few-Shot Learning","score":0.10790000110864639,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/scaling","display_name":"Scaling","score":0.9024999737739563},{"id":"https://openalex.org/keywords/residual","display_name":"Residual","score":0.8061000108718872},{"id":"https://openalex.org/keywords/transferability","display_name":"Transferability","score":0.6531000137329102},{"id":"https://openalex.org/keywords/hyperparameter","display_name":"Hyperparameter","score":0.6280999779701233},{"id":"https://openalex.org/keywords/stability","display_name":"Stability (learning theory)","score":0.42570000886917114},{"id":"https://openalex.org/keywords/loop","display_name":"Loop (graph theory)","score":0.41909998655319214},{"id":"https://openalex.org/keywords/control-theory","display_name":"Control theory (sociology)","score":0.4097000062465668}],"concepts":[{"id":"https://openalex.org/C99844830","wikidata":"https://www.wikidata.org/wiki/Q102441924","display_name":"Scaling","level":2,"score":0.9024999737739563},{"id":"https://openalex.org/C155512373","wikidata":"https://www.wikidata.org/wiki/Q287450","display_name":"Residual","level":2,"score":0.8061000108718872},{"id":"https://openalex.org/C61272859","wikidata":"https://www.wikidata.org/wiki/Q7834031","display_name":"Transferability","level":3,"score":0.6531000137329102},{"id":"https://openalex.org/C8642999","wikidata":"https://www.wikidata.org/wiki/Q4171168","display_name":"Hyperparameter","level":2,"score":0.6280999779701233},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.48019999265670776},{"id":"https://openalex.org/C112972136","wikidata":"https://www.wikidata.org/wiki/Q7595718","display_name":"Stability (learning theory)","level":2,"score":0.42570000886917114},{"id":"https://openalex.org/C184670325","wikidata":"https://www.wikidata.org/wiki/Q512604","display_name":"Loop (graph theory)","level":2,"score":0.41909998655319214},{"id":"https://openalex.org/C47446073","wikidata":"https://www.wikidata.org/wiki/Q5165890","display_name":"Control theory (sociology)","level":3,"score":0.4097000062465668},{"id":"https://openalex.org/C121864883","wikidata":"https://www.wikidata.org/wiki/Q677916","display_name":"Statistical physics","level":1,"score":0.4083000123500824},{"id":"https://openalex.org/C2777210771","wikidata":"https://www.wikidata.org/wiki/Q4927124","display_name":"Block (permutation group theory)","level":2,"score":0.36230000853538513},{"id":"https://openalex.org/C66882249","wikidata":"https://www.wikidata.org/wiki/Q169336","display_name":"Homogeneous","level":2,"score":0.3244999945163727},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.3199000060558319},{"id":"https://openalex.org/C2778755073","wikidata":"https://www.wikidata.org/wiki/Q10858537","display_name":"Scale (ratio)","level":2,"score":0.3075000047683716},{"id":"https://openalex.org/C66322947","wikidata":"https://www.wikidata.org/wiki/Q11658","display_name":"Transformer","level":3,"score":0.2996000051498413},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.28790000081062317},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.28769999742507935},{"id":"https://openalex.org/C192562407","wikidata":"https://www.wikidata.org/wiki/Q228736","display_name":"Materials science","level":0,"score":0.2775999903678894},{"id":"https://openalex.org/C2988430800","wikidata":"https://www.wikidata.org/wiki/Q428971","display_name":"Scaling law","level":3,"score":0.2709999978542328},{"id":"https://openalex.org/C132010649","wikidata":"https://www.wikidata.org/wiki/Q189222","display_name":"Intuition","level":2,"score":0.25119999051094055}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.48550/arxiv.2606.18524","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2606.18524","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"doi:10.48550/arxiv.2606.18524","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2606.18524","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":[],"abstract_inverted_index":{"Looped":[0],"(weight-tied)":[1],"Transformers":[2,135],"apply":[3],"a":[4,72],"shared":[5],"residual":[6,35,49],"block":[7],"$N$":[8,68,129],"times":[9],"($h":[10],"\\leftarrow":[11],"h":[12],"+":[13],"\\varepsilon\\,f(h)$,":[14],"same":[15],"$f$":[16],"at":[17],"each":[18],"step),":[19],"increasing":[20],"effective":[21],"depth":[22],"without":[23,130],"adding":[24],"parameters.":[25],"Prior":[26],"depth-scaling":[27],"analyses":[28],"prescribe":[29],"$\\varepsilon":[30,58,75],"=":[31,59,76],"1/\\!\\sqrt{L}$":[32],"for":[33,43],"depth-$L$":[34],"networks.":[36],"We":[37],"show":[38],"that":[39,78,101,137],"this":[40],"is":[41,100],"insufficient":[42],"looped":[44,67,134],"architectures:":[45],"weight":[46],"sharing":[47],"makes":[48],"updates":[50],"correlated":[51],"across":[52,149],"iterations,":[53],"requiring":[54],"the":[55,80,87,94,102,109,117],"stronger":[56],"scaling":[57,139,148],"1/N$.":[60],"For":[61],"multi-layer":[62],"blocks":[63],"($L$":[64],"unique":[65,112],"layers":[66,113],"times),":[69],"we":[70],"derive":[71],"factored":[73],"parameterization":[74],"\u03bb/(N\\!\\sqrt{L})$":[77],"separates":[79],"two":[81],"sources":[82],"of":[83,111],"growth:":[84],"$1/N$":[85,138],"controls":[86,93],"within-layer":[88],"loop":[89,118,150],"correlation,":[90],"and":[91,142],"$1/\\!\\sqrt{L}$":[92],"across-layer":[95],"variance.":[96],"A":[97],"key":[98],"consequence":[99],"optimal":[103],"learning":[104],"rate":[105],"depends":[106],"only":[107],"on":[108,116,133],"number":[110],"$L$,":[114],"not":[115],"count":[119],"$N$,":[120],"enabling":[121],"direct":[122],"hyperparameter":[123],"transfer":[124],"from":[125],"small":[126],"to":[127],"large":[128],"retuning.":[131],"Experiments":[132],"confirm":[136],"improves":[140],"trainability":[141],"yields":[143],"better":[144],"loss":[145],"than":[146],"$1/\\!\\sqrt{N}$":[147],"counts.":[151]},"counts_by_year":[],"updated_date":"2026-07-28T07:46:37.118299","created_date":"2026-06-19T00:00:00"}