{"id":"https://openalex.org/W7208794479","doi":"https://doi.org/10.48550/arxiv.2609.02373","title":"Percolation Dynamics in Optimization: Variance Cascades and Nested Symmetry","display_name":"Percolation Dynamics in Optimization: Variance Cascades and Nested Symmetry","publication_year":2026,"publication_date":"2026-09-02","ids":{"openalex":"https://openalex.org/W7208794479","doi":"https://doi.org/10.48550/arxiv.2609.02373"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.02373","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.02373","pdf_url":"https://arxiv.org/pdf/2609.02373","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,"listed_in":[],"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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2609.02373","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5005132456","display_name":"Sai Niranjan Ramachandran","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Ramachandran, Sai Niranjan","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5058767558","display_name":"Suvrit Sra","orcid":"https://orcid.org/0000-0001-8516-4925","observed_orcids":["https://orcid.org/0000-0001-8516-4925"]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Sra, Suvrit","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":true,"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/T11612","display_name":"Stochastic Gradient Optimization Techniques","score":0.9955999851226807,"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"}},"topics":[{"id":"https://openalex.org/T11612","display_name":"Stochastic Gradient Optimization Techniques","score":0.9955999851226807,"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/T10320","display_name":"Neural Networks and Applications","score":0.0020000000949949026,"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/T12535","display_name":"Machine Learning and Data Classification","score":9.999999747378752e-05,"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/percolation","display_name":"percolation","score":0.7730000019073486},{"id":"https://openalex.org/keywords/subnetworks","display_name":"subnetworks","score":0.675000011920929},{"id":"https://openalex.org/keywords/architectural-symmetry","display_name":"architectural symmetry","score":0.703000009059906},{"id":"https://openalex.org/keywords/trapping","display_name":"trapping","score":0.32100000977516174},{"id":"https://openalex.org/keywords/adam-optimizer","display_name":"Adam optimizer","score":0.6600000262260437},{"id":"https://openalex.org/keywords/adamw","display_name":"AdamW","score":0.9850000143051147},{"id":"https://openalex.org/keywords/heavy-tailed-noise","display_name":"heavy-tailed noise","score":0.8640000224113464},{"id":"https://openalex.org/keywords/phase-transitions","display_name":"phase transitions","score":0.7459999918937683},{"id":"https://openalex.org/keywords/order-parameter","display_name":"order parameter","score":0.8880000114440918},{"id":"https://openalex.org/keywords/transformer","display_name":"transformer models","score":0.6000000238418579}],"concepts":[{"id":"https://openalex.org/C121864883","wikidata":"https://www.wikidata.org/wiki/Q677916","display_name":"Statistical physics","level":1,"score":0.7027000188827515},{"id":"https://openalex.org/C99844830","wikidata":"https://www.wikidata.org/wiki/Q102441924","display_name":"Scaling","level":2,"score":0.6003999710083008},{"id":"https://openalex.org/C15867958","wikidata":"https://www.wikidata.org/wiki/Q5280413","display_name":"Directed percolation","level":4,"score":0.5447999835014343},{"id":"https://openalex.org/C34146451","wikidata":"https://www.wikidata.org/wiki/Q5048094","display_name":"Cascade","level":2,"score":0.5285999774932861},{"id":"https://openalex.org/C153258448","wikidata":"https://www.wikidata.org/wiki/Q1199743","display_name":"Gradient descent","level":3,"score":0.4900999963283539},{"id":"https://openalex.org/C167879884","wikidata":"https://www.wikidata.org/wiki/Q727568","display_name":"Balanced flow","level":2,"score":0.4876999855041504},{"id":"https://openalex.org/C135593079","wikidata":"https://www.wikidata.org/wiki/Q1750766","display_name":"Scale invariance","level":2,"score":0.4821999967098236},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.4706999957561493},{"id":"https://openalex.org/C197129107","wikidata":"https://www.wikidata.org/wiki/Q1921621","display_name":"Merge (version control)","level":2,"score":0.45350000262260437},{"id":"https://openalex.org/C190470478","wikidata":"https://www.wikidata.org/wiki/Q2370229","display_name":"Invariant (physics)","level":2,"score":0.4478999972343445},{"id":"https://openalex.org/C2780457167","wikidata":"https://www.wikidata.org/wiki/Q17156484","display_name":"Percolation (cognitive psychology)","level":2,"score":0.4456000030040741},{"id":"https://openalex.org/C123757187","wikidata":"https://www.wikidata.org/wiki/Q9195957","display_name":"Network dynamics","level":2,"score":0.41019999980926514},{"id":"https://openalex.org/C96469262","wikidata":"https://www.wikidata.org/wiki/Q1324364","display_name":"Homogeneous space","level":2,"score":0.40070000290870667},{"id":"https://openalex.org/C55689738","wikidata":"https://www.wikidata.org/wiki/Q15963867","display_name":"Discrete time and continuous time","level":2,"score":0.39800000190734863},{"id":"https://openalex.org/C112633086","wikidata":"https://www.wikidata.org/wiki/Q381287","display_name":"White noise","level":2,"score":0.3871999979019165},{"id":"https://openalex.org/C145912823","wikidata":"https://www.wikidata.org/wiki/Q113558","display_name":"Dynamics (music)","level":2,"score":0.3756999969482422},{"id":"https://openalex.org/C206688291","wikidata":"https://www.wikidata.org/wiki/Q7617819","display_name":"Stochastic gradient descent","level":3,"score":0.37459999322891235},{"id":"https://openalex.org/C38349280","wikidata":"https://www.wikidata.org/wiki/Q1434290","display_name":"Flow (mathematics)","level":2,"score":0.3741999864578247},{"id":"https://openalex.org/C34947359","wikidata":"https://www.wikidata.org/wiki/Q665189","display_name":"Complex network","level":2,"score":0.34119999408721924},{"id":"https://openalex.org/C73301696","wikidata":"https://www.wikidata.org/wiki/Q5469984","display_name":"Formalism (music)","level":3,"score":0.3391999900341034},{"id":"https://openalex.org/C2776359362","wikidata":"https://www.wikidata.org/wiki/Q2145286","display_name":"Representation (politics)","level":3,"score":0.3375000059604645},{"id":"https://openalex.org/C8272713","wikidata":"https://www.wikidata.org/wiki/Q176737","display_name":"Stochastic process","level":2,"score":0.3357999920845032},{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.33399999141693115},{"id":"https://openalex.org/C39943821","wikidata":"https://www.wikidata.org/wiki/Q506346","display_name":"Semimartingale","level":2,"score":0.33230000734329224},{"id":"https://openalex.org/C61445026","wikidata":"https://www.wikidata.org/wiki/Q217608","display_name":"Fixed point","level":2,"score":0.3312000036239624},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.3287000060081482},{"id":"https://openalex.org/C2778755073","wikidata":"https://www.wikidata.org/wiki/Q10858537","display_name":"Scale (ratio)","level":2,"score":0.3264999985694885},{"id":"https://openalex.org/C99498987","wikidata":"https://www.wikidata.org/wiki/Q2210247","display_name":"Noise (video)","level":3,"score":0.32339999079704285},{"id":"https://openalex.org/C149288129","wikidata":"https://www.wikidata.org/wiki/Q185357","display_name":"Phase transition","level":2,"score":0.30970001220703125},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.29190000891685486},{"id":"https://openalex.org/C196083921","wikidata":"https://www.wikidata.org/wiki/Q7915758","display_name":"Variance (accounting)","level":2,"score":0.2874000072479248},{"id":"https://openalex.org/C163716315","wikidata":"https://www.wikidata.org/wiki/Q901177","display_name":"Gaussian","level":2,"score":0.2786000072956085},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.26499998569488525},{"id":"https://openalex.org/C47822265","wikidata":"https://www.wikidata.org/wiki/Q854457","display_name":"Complex system","level":2,"score":0.2630000114440918},{"id":"https://openalex.org/C177918212","wikidata":"https://www.wikidata.org/wiki/Q803623","display_name":"Perturbation (astronomy)","level":2,"score":0.2563000023365021},{"id":"https://openalex.org/C63479239","wikidata":"https://www.wikidata.org/wiki/Q7353546","display_name":"Robustness (evolution)","level":3,"score":0.2538999915122986},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.2524999976158142}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.02373","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.02373","pdf_url":"https://arxiv.org/pdf/2609.02373","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,"listed_in":[],"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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.2609.02373","doi":"https://doi.org/10.48550/arxiv.2609.02373","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.02373","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,"listed_in":[],"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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2609.02373","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.02373","pdf_url":"https://arxiv.org/pdf/2609.02373","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,"listed_in":[],"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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7208794479.pdf","grobid_xml":"https://content.openalex.org/works/W7208794479.grobid-xml"},"referenced_works_count":8,"referenced_works":["https://openalex.org/W2113839990","https://openalex.org/W2125930537","https://openalex.org/W3001279689","https://openalex.org/W3087768334","https://openalex.org/W4221155841","https://openalex.org/W4320170046","https://openalex.org/W4399136690","https://openalex.org/W4417464808"],"related_works":["https://openalex.org/W7133225648","https://openalex.org/W4411087767","https://openalex.org/W2765393161","https://openalex.org/W4415796479","https://openalex.org/W3170469037","https://openalex.org/W7165117747","https://openalex.org/W7131433015","https://openalex.org/W4226485870","https://openalex.org/W4404571285","https://openalex.org/W3215746499"],"abstract_inverted_index":{"We":[0,34,81],"study":[1],"the":[2,39,88],"dynamics":[3],"of":[4],"Stochastic":[5],"Gradient":[6],"Descent":[7],"(SGD),":[8],"which":[9,49,87],"is":[10],"known":[11],"to":[12,22,55,93],"steer":[13],"deep":[14],"neural":[15],"networks":[16],"toward":[17],"invariant":[18],"sets":[19],"that":[20],"correspond":[21],"simpler":[23],"subnetworks.":[24],"How":[25],"this":[26,36],"steering":[27],"unfolds":[28],"over":[29,92],"time":[30],"remains":[31],"poorly":[32],"understood.":[33],"answer":[35],"by":[37,62],"modeling":[38],"stochastic":[40],"gradient":[41,99],"flow":[42],"(SGF)":[43],"as":[44,69],"a":[45,73,105],"percolation":[46],"process,":[47],"in":[48,57,72],"nested":[50],"architectural":[51],"symmetries":[52],"force":[53],"subnetworks":[54],"merge":[56],"discrete":[58],"blocks":[59],"rather":[60],"than":[61],"single-edge":[63],"attachment.":[64],"These":[65],"structural":[66],"transitions":[67],"register":[68],"variance":[70],"spikes":[71],"macroscopic":[74],"order":[75],"parameter":[76],"echoing":[77],"physical":[78],"phase":[79],"transitions.":[80],"further":[82],"state":[83],"sufficient":[84],"conditions":[85],"under":[86,97],"trapping":[89],"argument":[90],"carries":[91],"Adam":[94],"and":[95,101],"AdamW":[96],"heavy-tailed":[98],"noise":[100],"measure":[102],"them":[103],"on":[104],"trained":[106],"Transformer.":[107]},"counts_by_year":[],"updated_date":"2026-11-08T12:12:28.789561","created_date":"2026-09-05T00:00:00"}