{"id":"https://openalex.org/W7163539769","doi":"https://doi.org/10.48550/arxiv.2606.04031","title":"Transient Amplification in Cascaded Gradient Methods: Exact Minimax Laws and Tracking without Resonance","display_name":"Transient Amplification in Cascaded Gradient Methods: Exact Minimax Laws and Tracking without Resonance","publication_year":2026,"publication_date":"2026-06-01","ids":{"openalex":"https://openalex.org/W7163539769","doi":"https://doi.org/10.48550/arxiv.2606.04031"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2606.04031","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2606.04031","pdf_url":"https://arxiv.org/pdf/2606.04031","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","indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2606.04031","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5137832700","display_name":"Ahanaf Hasan Ariq","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Ariq, Ahanaf Hasan","raw_affiliation_strings":["School and College"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School and College","institution_ids":[]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5126257691","display_name":"Davide Wiest","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Wiest, Davide","raw_affiliation_strings":["School and College"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School and College","institution_ids":[]}]}],"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":0.9775000214576721,"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":0.9775000214576721,"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/T10792","display_name":"Matrix Theory and Algorithms","score":0.017000000923871994,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"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/T10963","display_name":"Advanced Optimization Algorithms Research","score":0.0010999999940395355,"subfield":{"id":"https://openalex.org/subfields/2612","display_name":"Numerical Analysis"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/transient-amplification","display_name":"transient amplification","score":0.9660000205039978},{"id":"https://openalex.org/keywords/non-normal-operators","display_name":"non-normal operators","score":0.753000020980835},{"id":"https://openalex.org/keywords/spectral-instability","display_name":"spectral instability","score":0.6710000038146973},{"id":"https://openalex.org/keywords/minimax-lower-bounds","display_name":"minimax lower bounds","score":0.7919999957084656},{"id":"https://openalex.org/keywords/bi-level-optimization","display_name":"bi-level optimization","score":0.8460000157356262},{"id":"https://openalex.org/keywords/generative-adversarial-networks","display_name":"generative adversarial networks","score":0.9819999933242798},{"id":"https://openalex.org/keywords/iteration-complexity","display_name":"iteration complexity","score":0.7770000100135803},{"id":"https://openalex.org/keywords/linear-quadratic-problem","display_name":"linear-quadratic problems","score":0.6230000257492065},{"id":"https://openalex.org/keywords/integral-quadratic-constraints","display_name":"integral quadratic constraints","score":0.6299999952316284},{"id":"https://openalex.org/keywords/neural-network-training-dynamics","display_name":"neural network training dynamics","score":0.7799999713897705}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6313999891281128},{"id":"https://openalex.org/C130367717","wikidata":"https://www.wikidata.org/wiki/Q189791","display_name":"Diagonal","level":2,"score":0.5569999814033508},{"id":"https://openalex.org/C99844830","wikidata":"https://www.wikidata.org/wiki/Q102441924","display_name":"Scaling","level":2,"score":0.5160999894142151},{"id":"https://openalex.org/C200331156","wikidata":"https://www.wikidata.org/wiki/Q506041","display_name":"Jacobian matrix and determinant","level":2,"score":0.5108000040054321},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.450300008058548},{"id":"https://openalex.org/C177918212","wikidata":"https://www.wikidata.org/wiki/Q803623","display_name":"Perturbation (astronomy)","level":2,"score":0.4496999979019165},{"id":"https://openalex.org/C112972136","wikidata":"https://www.wikidata.org/wiki/Q7595718","display_name":"Stability (learning theory)","level":2,"score":0.4332999885082245},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.4309999942779541},{"id":"https://openalex.org/C153258448","wikidata":"https://www.wikidata.org/wiki/Q1199743","display_name":"Gradient descent","level":3,"score":0.4253999888896942},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.4203000068664551},{"id":"https://openalex.org/C149728462","wikidata":"https://www.wikidata.org/wiki/Q751319","display_name":"Minimax","level":2,"score":0.40470001101493835},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.39239999651908875},{"id":"https://openalex.org/C2780799671","wikidata":"https://www.wikidata.org/wiki/Q17087362","display_name":"Transient (computer programming)","level":2,"score":0.3718000054359436},{"id":"https://openalex.org/C131584629","wikidata":"https://www.wikidata.org/wiki/Q4308705","display_name":"Coupling (piping)","level":2,"score":0.3569999933242798},{"id":"https://openalex.org/C140532419","wikidata":"https://www.wikidata.org/wiki/Q249748","display_name":"Spectral radius","level":3,"score":0.32170000672340393},{"id":"https://openalex.org/C165064840","wikidata":"https://www.wikidata.org/wiki/Q1321061","display_name":"Matching (statistics)","level":2,"score":0.310699999332428},{"id":"https://openalex.org/C177937620","wikidata":"https://www.wikidata.org/wiki/Q7254447","display_name":"Pseudo-spectral method","level":4,"score":0.30300000309944153},{"id":"https://openalex.org/C206688291","wikidata":"https://www.wikidata.org/wiki/Q7617819","display_name":"Stochastic gradient descent","level":3,"score":0.3009999990463257},{"id":"https://openalex.org/C63479239","wikidata":"https://www.wikidata.org/wiki/Q7353546","display_name":"Robustness (evolution)","level":3,"score":0.29010000824928284},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.2883000075817108},{"id":"https://openalex.org/C207821765","wikidata":"https://www.wikidata.org/wiki/Q405372","display_name":"Instability","level":2,"score":0.2858000099658966},{"id":"https://openalex.org/C167964875","wikidata":"https://www.wikidata.org/wiki/Q17011487","display_name":"Exponential stability","level":3,"score":0.28220000863075256},{"id":"https://openalex.org/C13662910","wikidata":"https://www.wikidata.org/wiki/Q193139","display_name":"Trajectory","level":2,"score":0.27000001072883606},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.2689000070095062},{"id":"https://openalex.org/C121864883","wikidata":"https://www.wikidata.org/wiki/Q677916","display_name":"Statistical physics","level":1,"score":0.2628999948501587}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2606.04031","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2606.04031","pdf_url":"https://arxiv.org/pdf/2606.04031","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.2606.04031","doi":"https://doi.org/10.48550/arxiv.2606.04031","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2606.04031","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:2606.04031","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2606.04031","pdf_url":"https://arxiv.org/pdf/2606.04031","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/W7163539769.pdf","grobid_xml":"https://content.openalex.org/works/W7163539769.grobid-xml"},"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W7162316364","https://openalex.org/W4221165655","https://openalex.org/W7082651057","https://openalex.org/W4415102927","https://openalex.org/W3127962651","https://openalex.org/W4416895182","https://openalex.org/W7079467925","https://openalex.org/W2483349545","https://openalex.org/W7151935849","https://openalex.org/W1989957966"],"abstract_inverted_index":{"Leader-follower":[0],"gradient":[1],"iterations,":[2],"including":[3],"bilevel":[4],"and":[5,35,81,125,155,170,191],"two-time-scale":[6],"methods,":[7],"can":[8,178],"exhibit":[9],"large":[10],"finite-time":[11,86],"excursions":[12],"despite":[13],"asymptotic":[14],"stability.":[15],"For":[16,105],"cascaded":[17],"iterations":[18],"with":[19],"block":[20],"matrix":[21],"J":[22],"=":[23,118],"[[A,0],[C,D]],":[24],"symmetric":[25],"diagonal":[26],"blocks":[27],"of":[28,59,77,141,144],"norm":[29,37,58,140],"at":[30,38,48,134],"most":[31,39,135],"gamma":[32],"<":[33],"1,":[34],"coupling":[36,109],"c,":[40],"we":[41],"prove":[42],"an":[43,65],"exact,":[44],"dimension-free":[45],"minimax":[46],"formula:":[47],"every":[49],"horizon":[50],"t,":[51],"the":[52,56,70,75,91,108,121,126,138,145,163,168,180],"worst-case":[53],"amplification":[54,132],"equals":[55],"spectral":[57,146],"[[gamma^t,0],[c":[60],"t":[61],"gamma^(t-1),gamma^t]],":[62],"attained":[63],"by":[64,116],"aligned":[66,92],"two-dimensional":[67],"quadratic.":[68],"Consequently,":[69],"worst":[71],"transient":[72],"is":[73,110,123,129,133,156],"on":[74],"order":[76],"1":[78,136],"+":[79],"c/(1-gamma),":[80],"eigenvalues":[82],"alone":[83],"cannot":[84],"predict":[85],"gain.":[87],"Under":[88],"martingale":[89],"noise,":[90],"system":[93],"has":[94],"stationary":[95],"follower":[96,113,127,153,176],"variance":[97],"sigma^2":[98],"c^2":[99],"(1+gamma^2)/(1-gamma^2)^3,":[100],"establishing":[101],"cubic":[102],"inverse-gap":[103],"scaling.":[104],"quadratic":[106],"followers,":[107],"linked":[111],"to":[112],"sensitivity":[114,181],"R":[115],"C":[117],"(I-D)R.":[119],"When":[120],"leader":[122,164],"isotropic":[124],"map":[128],"positive":[130],"semidefinite,":[131],"plus":[137],"operator":[139],"R,":[142],"independently":[143],"gaps.":[147],"A":[148],"look-ahead":[149],"Newton":[150],"update":[151],"eliminates":[152],"memory":[154],"minimax-optimal":[157],"among":[158],"linear":[159],"updates":[160],"that":[161,172],"track":[162],"exactly.":[165],"Experiments":[166],"validate":[167],"bounds":[169],"show":[171],"accelerated":[173],"or":[174],"inexact":[175],"solvers":[177],"exceed":[179],"bound":[182],"when":[183],"their":[184],"effective":[185],"maps":[186],"have":[187],"negative":[188],"eigenvalues.":[189],"Code":[190],"experiment":[192],"scripts:":[193],"https://github.com/Axym-Labs/transient-amplification":[194]},"counts_by_year":[],"updated_date":"2026-10-28T10:33:24.094632","created_date":"2026-06-05T00:00:00"}
