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Linear regression ndi imodzi mwama algorithms oyambira madera ambiri okhudzana ndi kusanthula deta. Chifukwa chake ndi chodziwikiratu. Ichi ndi njira yosavuta komanso yomveka bwino, yomwe yathandizira kuti igwiritsidwe ntchito kwambiri kwa zaka makumi ambiri, kapena zaka mazana ambiri. Lingaliro ndilakuti timatengera kudalira kwamtundu umodzi pamitundu ina, ndiyeno kuyesa kubwezeretsa kudalira uku.
Koma nkhaniyi sikutanthauza kugwiritsa ntchito kubwereza kwa mzere kuthetsa mavuto othandiza. Apa tiwona zinthu zosangalatsa za kukhazikitsidwa kwa ma aligorivimu ogawidwa kuti achire, zomwe tidakumana nazo polemba gawo lophunzirira makina mu. . Masamu ang'onoang'ono, kuphunzira pamakina, ndi makompyuta omwe amagawidwa angakuthandizeni kudziwa momwe mungayendetsere mizere ngakhale deta yanu igawidwe pa masauzande ambiri.
Kodi tikukamba za chiyani?
Tikuyang'anizana ndi ntchito yobwezeretsa kudalira kwa mzere. Monga deta yolowera, gulu la ma vectors omwe amati ndi odziyimira pawokha amaperekedwa, chilichonse chomwe chimalumikizidwa ndi mtengo wake wamitundu yodalira. Deta iyi ikhoza kuyimiridwa mu mawonekedwe a matrices awiri:
Tsopano, popeza kudalira kumaganiziridwa, ndipo, kuonjezeranso, mzere, tidzalemba malingaliro athu mu mawonekedwe a matrices (kuti muchepetse kujambula, apa ndi pansipa akuganiziridwa kuti nthawi yaulere ya equation yabisika kumbuyo. , ndi gawo lomaliza la matrix lili ndi mayunitsi):
Zikumveka ngati dongosolo la ma equation amzere, sichoncho? Zikuwoneka, koma mwachidziwikire sipadzakhala njira zothetsera dongosolo lotere la equations. Chifukwa cha izi ndi phokoso, lomwe limapezeka pafupifupi deta yeniyeni. Chifukwa china chingakhale kusowa kwa kudalira kwa mzere monga momwemo, komwe kungathe kulimbana ndi kubweretsa zosintha zina zomwe sizidalira zoyambazo. Taganizirani chitsanzo ichi:
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Ichi ndi chitsanzo chosavuta cha kusinthika kwa mzere komwe kumawonetsa ubale wamtundu umodzi (pambali mwa axis ) kuchokera kumtundu wina (pambali mwa axis ). Kuti dongosolo la ma equation a mzere wolingana ndi chitsanzo ichi likhale ndi yankho, mfundo zonse ziyenera kukhala pamzere wowongoka womwewo. Koma si zoona. Koma samanama pamzere wowongoka womwewo ndendende chifukwa cha phokoso (kapena chifukwa kuganiza kwa ubale wa mzere kunali kolakwika). Chifukwa chake, kuti mubwezeretse ubale wofananira kuchokera ku data yeniyeni, nthawi zambiri ndikofunikira kuyambitsa lingaliro linanso: zolowetsamo zili ndi phokoso ndipo phokoso ili . Mutha kupanga malingaliro okhudza mitundu ina ya kugawa phokoso, koma nthawi zambiri ndikugawa koyenera komwe kumaganiziridwa, komwe kudzakambidwenso.
Njira yofikira kwambiri
Chifukwa chake, tidaganiza zokhala ndi phokoso lomwe limagawidwa mwachisawawa. Zoyenera kuchita zikatero? Pankhani imeneyi masamu alipo ndipo amagwiritsidwa ntchito kwambiri . Mwachidule, tanthauzo lake lagona pa kusankha ndi kukulitsa kwake kotsatira.
Timabwereranso kubwezeretsa ubale wa mzere kuchokera ku deta ndi phokoso labwinobwino. Zindikirani kuti mgwirizano womwe umaganiziridwa kuti ndiwofanana ndi chiyembekezo cha masamu kugawa komwe kulipo. Pa nthawi yomweyo, mwayi kuti zimatenga mtengo umodzi kapena wina, kutengera kukhalapo kwa zowoneka , motere:
Tiyeni tsopano tilowe mmalo ΠΈ Zosintha zomwe timafunikira ndi:
Chotsalira ndikupeza vekitala , pomwe kuthekera uku ndikokwanira. Kuti muwonjezere ntchitoyi, ndibwino kuti mutenge logarithm yake (logarithm ya ntchitoyi idzafika pamtunda womwewo monga momwe ntchitoyo):
Zomwe, nazonso, zimatsikira kuchepetsa ntchito zotsatirazi:
Mwa njira, izi zimatchedwa njira . Nthawi zambiri zonse zomwe zili pamwambazi zimasiyidwa ndipo njirayi imangogwiritsidwa ntchito.
Kusintha kwa QR
Zochepa zomwe zili pamwambazi zitha kupezeka popeza pomwe gradient ya ntchitoyi ndi ziro. Ndipo gradient idzalembedwa motere:
ndi njira ya matrix yothetsera vuto lochepetsera lomwe limagwiritsidwa ntchito munjira yocheperako. Pachifukwa ichi, timalembanso equation mu mawonekedwe a matrix:
Kenako timachotsa matrix ku matrices ΠΈ ndikuchita masinthidwe angapo (algorithm yowola ya QR palokha sidzaganiziridwa pano, kungogwiritsa ntchito kwake pokhudzana ndi ntchito yomwe ilipo):
Matrix ndi orthogonal. Izi zimathandiza kuti tichotse ntchito :
Ndipo ngati mutasintha pa , ndiye zidzatheka . Poganizira zimenezo ndi matrix apamwamba katatu, amawoneka motere:
Izi zitha kuthetsedwa pogwiritsa ntchito njira yolowa m'malo. Chinthu ili ngati , chinthu cham'mbuyo ili ngati ndi zina zotero.
Ndikoyenera kudziwa apa kuti zovuta za algorithm yobwera chifukwa chogwiritsa ntchito kuwonongeka kwa QR ndizofanana . Komanso, ngakhale kuti ntchito yochulutsa matrix ikufanana bwino, sizingatheke kulemba mtundu wogawa bwino wa algorithm iyi.
Kutsika kwa Gradient
Polankhula za kuchepetsa ntchito, nthawi zonse ndi bwino kukumbukira njira yotsika (stochastic) gradient. Iyi ndi njira yosavuta komanso yothandiza yochepetsera kutengera kuwerengetsa mobwerezabwereza kupendekera kwa chinthu pamalo amodzi kenako ndikuchisunthira kunjira moyang'anizana ndi gradient. Chilichonse choterechi chimabweretsa yankho pafupi ndi zochepa. The gradient ikuwoneka chimodzimodzi:
Njirayi imafananizidwanso bwino ndikugawidwa chifukwa cha mawonekedwe amtundu wa woyendetsa gradient. Zindikirani kuti munjira yomwe ili pamwambapa, pansi pa chizindikiro chowerengera pali mawu odziyimira pawokha. Mwa kuyankhula kwina, tikhoza kuwerengera gradient mopanda ma indices onse kuyambira woyamba mpaka , kufananiza ndi izi, werengerani ma indices ndi mpaka . Kenaka yikani ma gradients omwe amachokera. Chotsatira cha kuwonjezera chidzakhala chimodzimodzi ngati tidawerengera nthawi yomweyo gradient ya indices kuyambira woyamba mpaka . Choncho, ngati deta igawidwa pakati pa zidutswa zingapo za deta, gradient ikhoza kuwerengedwa payokha pa chidutswa chilichonse, ndiyeno zotsatira za kuwerengera izi zikhoza kufotokozedwa mwachidule kuti mupeze zotsatira zomaliza:
Kuchokera pamawonedwe okhazikitsa, izi zikugwirizana ndi paradigm . Pa sitepe iliyonse ya kutsika kwa gradient, ntchito imatumizidwa ku node iliyonse ya data kuti iwerengere gradient, kenako ma gradients owerengeka amasonkhanitsidwa palimodzi, ndipo zotsatira za kuchuluka kwake zimagwiritsidwa ntchito kukonza zotsatira.
Ngakhale ndizosavuta kukhazikitsa komanso kuthekera kochita mu MapReduce paradigm, kutsika kwa gradient kumakhalanso ndi zovuta zake. Makamaka, kuchuluka kwa masitepe ofunikira kuti mukwaniritse kulumikizana ndikokwera kwambiri poyerekeza ndi njira zina zapadera.
Mtengo wa LSQR
ndi njira ina yothetsera vutoli, yomwe ili yoyenera kubwezeretsanso kubwereza kwa mzere komanso kuthetsa machitidwe a ma equation a mzere. Chinthu chake chachikulu ndi chakuti chimagwirizanitsa ubwino wa njira za matrix ndi njira yobwerezabwereza. Kukhazikitsa kwa njirayi kungapezeke m'malaibulale onse awiri , ndi . Kufotokozera kwa njirayi sikudzaperekedwa apa (ikhoza kupezeka m'nkhaniyi ). M'malo mwake, njira idzawonetsedwa yosinthira LSQR kuti ichitidwe m'malo ogawidwa.
Njira ya LSQR idakhazikitsidwa . Iyi ndi njira yobwerezabwereza, kubwereza kulikonse kumakhala ndi izi:
Koma ngati tikuganiza kuti matrix imagawika mozungulira, ndiye kubwereza kulikonse kutha kuyimiridwa ngati masitepe awiri a MapReduce. Mwanjira iyi, ndizotheka kuchepetsa kusamutsidwa kwa data nthawi iliyonse yobwereza (ma vector okha okhala ndi kutalika kofanana ndi kuchuluka kwa zosadziwika):
Ndi njira iyi yomwe imagwiritsidwa ntchito pokhazikitsa linear regression in .
Pomaliza
Pali ma aligorivimu ambiri obwerera kumbuyo, koma si onse omwe angagwiritsidwe ntchito muzochitika zonse. Chifukwa chake kuwonongeka kwa QR ndikwabwino pakuyankha molondola pamaseti ang'onoang'ono a data. Kutsika kwa gradient ndikosavuta kugwiritsa ntchito ndipo kumakupatsani mwayi wopeza yankho loyerekeza. Ndipo LSQR imaphatikiza zinthu zabwino kwambiri za ma aligorivimu awiri apitawo, popeza imatha kugawidwa, imasinthasintha mwachangu poyerekeza ndi kutsika kwa gradient, komanso imalola kuyimitsidwa koyambirira kwa aligorivimu, mosiyana ndi kuwonongeka kwa QR, kuti mupeze yankho lachidule.
Source: www.habr.com