Sløring for nybegynnere Introduksjon Dette er en kort veiledning om uskarphetsteknikker for nybegynnere. Da jeg lærte dette, var det svært lite tilgjengelig materiale som var nyttig. Det var ikke sant selvfølgelig - det var masser av materiale, men halvparten av det var altfor enkelt og den andre halvdelen begynte La T være en vektorfunksjon evaluert over det halvåpne intervallet. og var full av veldig skummel multi-line likninger med de store sigma symboler og ting. Denne artikkelen er ment å rette opp det. Jeg snakker om ulike typer uskarphet og effektene du kan bruke dem til, med kildekoden i Java. En ansvarsfraskrivelse Når det blir sagt uskarphet, er det alltid noen som sier Hei, Det er ikke en ekte bevegelsesskarphet, eller skriver sint bokstaver i grønn blekk som klager på at matematikken er tvilsom, eller at det er en mye raskere måte å gjøre dette ved hjelp av sponglerizer-registret på HAL -9000. Ignorer disse menneskene. Dette er et stort emne, og denne artikkelen er bare for nybegynnere (som jeg stolt kan si jeg er en). Det som betyr noe, er at du får de resultatene du setter etter, og hvis resultatene du satser på krever tvilsom matematikk, så vær så snill. Hvis resultatene du prøver å se forferdelige mot meg, så er det bra, så lenge de ser bra ut til deg. En annen ansvarsfraskrivelse Det er kildekode i Java for ganske bra alt jeg snakker om her. Jeg gjør ingen påstander om at disse er optimalisert på noen måte - Jeg har valgt enkelhet over fart overalt, og du vil sannsynligvis kunne gjøre det meste av disse tingene gå raskere med litt innsats. Du kan bruke kildekoden for alt du vil, inkludert kommersielle formål, men det er ingen ansvar. Hvis ditt atomkraftverk eller missilsystem mislykkes på grunn av feil blur, er det ikke min feil. Hva er uskarpt Vi vet alle hva som er uskarpt, ikke vi Det er det som skjer når kameraet er ute av fokus eller hunden stjeler brillene dine. Hva som skjer er at det som skal sees som et skarpt punkt, blir smurt ut, vanligvis til en plateform. I bildevilkår betyr dette at hver piksel i kildebildet blir spredt over og blandet inn i omkringliggende piksler. En annen måte å se på dette er at hver piksel i destinasjonsbildet består av en blanding av omkringliggende piksler fra kildebildet. Operasjonen vi trenger for dette kalles convolution. Dette høres komplisert, men det er bare fordi matematikere liker å få ting til å lyde komplisert for å opprettholde den magiske luften og holde finansieringen rullende inn. Vel, jeg er på dem og jeg kan avsløre at konvolusjonen ikke er så komplisert (på mitt nivå uansett) . Måten det virker på, er dette: Vi tenker på å skyve et rektangulært antall tall over vårt bilde. Denne gruppen kalles konvolusjonskjernen. For hver piksel i bildet tar vi de tilsvarende tallene fra kjernen og pikslene de er over, multipliserer dem sammen og legger alle resultatene sammen for å lage den nye piksel. For eksempel, tenk vi ønsker å gjøre en veldig enkel uskarphet hvor vi bare gjennomsnittlig sammen hver piksel og dens åtte umiddelbare naboer. Kjernen vi trenger er: Legg merke til at disse alle legger opp til 1, noe som betyr at vårt resulterende bilde blir like lyst som originalen. Uten videre ado, kan vi sløre et bilde i Java. Alt det convolution-innholdet virker vanskelig å implementere, men heldigvis kommer Java med en innebygd og klar til bruk-operatør for å gjøre akkurat det. Jeg snakker ConvolveOp her. Heres koden: Fantastisk Et uklart bilde Det er ikke veldig uklart skjønt. La oss gjøre en veldig stor uskarphet som dette: Hmmmmmm. Vel, det er ikke så bra. Ikke bare tok det veldig lang tid, men resultatet er litt rart - alt ser bra ut, en slags firkant, og hva i verden har skjedd rundt kantene. Først kan kantene: ConvolveOp er en skummel namby-pamby-ting som er redd å falle utenfor kanten av bildet. Hvis kjerne overlapper kanten av bildet, gir den bare opp og bare etterlater pikselet uendret. Du kan endre dette ved å sende EDGEZEROFILL i stedet for EDGENOOP, men det er enda verre - pikslene rundt kanten blir bare satt til null og forsvinner effektivt. Hva skal vi gjøre Vel, vi kan kaste bildet ut rundt kantene før det blir uskarpt og beskjære resultatet, men det er bare å gi inn, og i tillegg ville vi ikke lære noe. I stedet vel skrive en skikkelig, fryktløs, no-nonsense operatør som ikke er redd for kanter. Velkall det ConvolveFilter å skille det fra ConvolveOp. Jeg skal ikke gjøre med informasjon om kilden i denne artikkelen - det er ikke nok tid eller plass, og vi har mye flere filtre å skrive enda, men du kan laste ned eller se kilden, og det skal være ganske selvforklarende. Nå er problemet med sperrighet: Årsaken til at alt ser firkantet ut er at det som ble gjort her, kalles en eskefarging - kjernen er formet som en firkant, som om det var et kamera med en firkantet blenderåpning. Forresten, ikke la noen fortelle deg at boksen uklarhet er ubrukelig - faktisk hvis du simulerer skyggen kastet av et kvadrat lys, er det akkurat det du vil. Uansett, vil de komme til nytte videre. En annen ting: Ikke bli forvirret - Jeg bruker begrepet eske uskarphet for å referere til formen på kjernen, ikke dens profil, som jeg skal ringe et eskefilter. Mer om dette senere. For å få en mer realistisk uskarphet, det vi burde ha gjort, er brukt en sirkelformet kjerne. Dette simulerer mye bedre hva et ekte kamera gjør. Det er mye bedre. Vel kom tilbake til dette senere, men først en avledning tilbake til boksen uskarphet. Weve løst kanten piksel problem, men vår uskarp går fortsatt veldig sakte, og ting kommer bare til å bli verre. Problemet er at antallet multiplikasjoner i konvolusjonen går opp som kvadratet i kjernen radius. Med en 100x100-kjernen skulle det bli 10000 multiplikasjoner og legger til per piksel (ca.). Hvordan kan vi komme rundt dette? Det viser seg at det er flere måter å gå om dette enn jeg har muligens tid til å skrive om, eller til og med plage å se på. En måte jeg vil nevne raskt før du feier det under teppet, er dette: Du kan gjøre en eske uskarphet ved å krympe bildet ditt, sløre det og skalere det opp igjen. Dette kan være bra for dine formål, og du bør ha det i tankene. Ett problem er at det ikke animerer veldig bra, men kan ikke være et problem for deg. La oss se på boksen uskarphet igjen: Det viser seg at det er et par veldig enkle måter å øke hastigheten på. For det første viser det seg at boksen uskarphet er separerbar. Dette betyr at vi kan gjøre 2D-uskarphet ved å gjøre to 1D-blurs, en gang i horisontal retning og en gang i vertikal retning. Dette er mye raskere enn å gjøre 2D-uskarphet fordi tiden som går, går opp i forhold til kjernestørrelsen, ikke som sin firkant. For det andre, Tenk på vinduet som glir over bildet. Når vi beveger den fra venstre mot høyre, kommer piksler inn i høyre kant og legges til totalt og samtidig gir piksler venstre kant og trekkes fra total. Alt vi trenger å gjøre er bare å legge til og trekke fra for å legge inn og utelate piksler ved hvert trinn i stedet for å legge sammen alle piksler i vinduet. Vi trenger bare å lagre et sett med kjørende totals som er bredden eller høyden til kjernen. Dette gir en enorm hastighetsforbedring på bekostning av å måtte skrive noen kode. Heldigvis har jeg skrevet koden for deg, så du vinner hele tiden. Vi trenger to pass, en gang for å uskarpe horisontalt og en gang vertikalt. Koden for disse er selvfølgelig ganske annerledes. Men vent Theres et triks vi kan gjøre som tillater oss bare å skrive koden en gang. Hvis vi skriver en uskarphet-funksjon som gjør det horisontale uskarphetet, men skriver utgangsbildet transponert, så kan vi bare kalle det to ganger. Første pass blurser horisontalt og transponerer, det andre passet gjør det samme, men som bildet er nå gjennomført, gjør det virkelig en vertikal uskarphet. Den andre transposisjonen gjør bildet riktig vei opp igjen og voila - en veldig rask boks uskarphet. Prøv det i denne appleten: Og heres kildekoden. Du har kanskje lagt merke til at vi bare har brukt en heltallradius så langt som gjør det enkelt å trene matriseindeksene for uskarpheten. Vi kan utvide teknikken til å gjøre uskarphet uskarphet (dvs. en ikke-integrert radius) bare ved lineær interpolering mellom arrayverdiene. Min kildekode gjør ikke dette, men det er enkelt å legge til. Gaussian Blur Nå er det på tide å ta opp hastigheten og firkantet utrykksspørsmål samtidig. For å kvitte seg med torget, se på uskarpheten, trenger vi en sirkulær formet kjerne. Dessverre fungerer det trikset vi brukte for bokseklær, men det er et smutthull: Hvis kjernen har riktig profil - Gauss-profilen - da kan vi gjøre 2D-uskarphet ved å utføre to 1D-blurs, akkurat som vi gjorde med boks uskarphet. Det er ikke så fort fordi skyvevinduet trick ikke fungerer, men det er fortsatt mye raskere enn å gjøre 2D-konvolusjonen. Profilen vi trenger er den kjente klokkeformede eller gaussiske kurven som du har hørt om: Her er noen kode for å lage en 1D Gaussisk kjernen for en gitt radius. Alt vi trenger å gjøre er å bruke dette to ganger, en gang horisontalt og en gang vertikalt. Som en bonus, har jeg pakket det opp i en GaussianFilter for å gjøre det enkelt å bruke. Det er derfor Gaussian-uskarpheten finnes i hver grafikkpakke - det er mye raskere enn andre typer uskarphet. Det eneste problemet er at det ikke er veldig realistisk når det gjelder å simulere kameralinser, men mer om det senere. Hvis du vil gjøre ting som å simulere skygger, så kan den gaussiske uskarpheten, eller kassens uskarphet, være bra. Det er et sted for alle disse effektene - bare fordi de ikke er realistiske, betyr det ikke at de ikke er nyttige. Gaussisk uskarphet er mye raskere, men det er ikke så nært som vår eskefarge vi gjorde tidligere. Hvis det bare var noen måte å kombinere de to. Jeg antar at du har giset nå at det kan være en, så jeg holder ikke spenningen lenger: Hvis du gjør mye av bokser, blir resultatet mer og mer som en Gauss-uskarphet. Faktisk kan du bevise det matematisk hvis du har et ekstra øyeblikk (men ikke fortell meg hvordan - jeg er ikke interessert). I praksis ser 3 til 5 boksene ut ganske bra. Ikke bare ta mitt ord for det: Skjermbildeappleten ovenfor har en glidebryter, slik at du kan prøve det selv. Alfa-kanaler En rask omledning her for å diskutere et problem som ofte kaster opp: Tenk deg at du vil uklare en form som er på gjennomsiktig bakgrunn. Du har et tomt bilde, og du tegner en form på den, og deretter slør bildet. Henger på - hvorfor ser uskarpheten ut for mørk Årsaken er at vi har uskarpt hver kanal, men hvor alfakanalen er null (de gjennomsiktige bitene), er de røde, grønne og blå kanalene null eller svarte. Når du gjør uskarpheten, blir sorten blandet med de ugjennomsiktige biter, og du får en mørk skygge. Løsningen er først og fremst at bildet alfa før det blir uskarpt og unpremultiply det etterpå. Selvfølgelig, hvis bildene dine allerede er forutbestemt, så er du klar. Bevegelsesskarphetstid for retningsendring. Så langt har vi bare snakket om uniform blurs, men det finnes andre typer. Bevegelsesskarphet er uskarphet du får når et objekt (eller kameraet) beveger seg under eksponeringen. Bildet blir uskarpt langs objektets tilsynelatende bane. Her skulle bare snakke om å simulere bevegelsesskarphet på et eksisterende stillbilde - å gjøre bevegelsesskarmen i animasjoner er et helt annet område. Var også bare å sløre hele bildet - ville ikke prøve å sløre et objekt i bildet. Den gode nyheten er at vi allerede har gjort enkle bevegelsesskarphet. Gå tilbake til boksen uskarpt applet over og sett den horisontale radiusen til, si 10, og den vertikale radiusen til null. Dette gir deg en fin horisontal bevegelsesskarphet. For noen formål kan dette være alt du trenger. En måte å lage en børstet metalltekstur på er for eksempel å ta et bilde som består av tilfeldig støy og bruke en bevegelsesskarphet. Hvis vi vil uskarpe i en annen retning enn horisontal eller vertikal, blir tingene mer kompliserte. En teknikk kan være å rotere bildet, uskarpe og deretter rotere tilbake. Hva gjør du her, er å gjøre det på den harde og sakte måten. Det vi trenger å gjøre er å løse over bildet, og for hver piksel legger du opp alle pikslene langs bevegelsesbanen. For en glatt bevegelsesskarphet betyr dette bare at du følger en rett linje fra pikselet, men du kan følge en wiggly-sti hvis du vil simulere kameraskakning med lang eksponering, sier. Spin og Zoom Blur Når vi har fått koden for bevegelsesskarphet på plass, er det et enkelt spørsmål å endre det for å gjøre zoom og spin blurs, eller til og med en kombinasjon av alle tre. Det er bare et spørsmål om å følge den rette veien for hver piksel. For radial blurs, følg bare en sti som går fra uskarphet. For en sperngryt, følg en tangentiell bane. Prøv det i denne appleten: Heres kildekoden for å gjøre disse tre typer bevegelsesskarphet: Faster Motion Blur Du har kanskje lagt merke til at det å gjøre bevegelsesskarmen er en ganske treg virksomhet - alle de sines og cosines gjør det veldig sakte. Hvis ikke var så bekymret for kvalitet skjønt, kan vi få fart på dette. Alt vi trenger å gjøre er å legge sammen mange forvandlede versjoner av bildet på en smart måte. Den klare delen er at vi kan gjøre en 1-piksers bevegelsesskarphet ved å gjennomsnittsføre bildet og det samme bildet oversatt av en piksel. Vi kan gjøre en 2-piks-uskarphet ved å gjenta dette med 1-pikslede uskarpe bilder. Ved å gjenta dette kan vi gjøre en N-pixel uskarphet i log2 (N) operasjoner, noe som er mye bedre enn å gjøre det på hard og langsom måte. Zoom og spin blurs kan gjøres ved å skalere og rotere i stedet for å oversette. Et filter vil gjøre alle tre ved hjelp av en AffineTransform. Prøv det i denne appleten: Domain Shifting Theres enda en annen måte å gjøre disse bevegelsene uskarphet på. Husk at du kunne gjøre den lineære bevegelsesskarmen ved å rotere bildet, gjøre en horisontal eske uskarpe og rotere tilbake. Det samme gjelder for zoom og spin blurs, bortsett fra at du trenger noe mer komplisert enn rotasjon. Det du trenger er polar transformasjonen. Når du har forvandlet bildet ditt, er en horisontal eske uskarphet et snurre når du forvandler tilbake, og en vertikal eskefarging gir deg en zoomfarging. En detalj er at du trenger en spesiell horisontal eske uskarphet som bryter på kantene ellers får du en skarp vertikal linje i ditt uskarpe bilde hvor spinnvinkelen skal vikle rundt. Uklarhet ved Fourier Transform Gaussisk uskarphet er veldig bra når du vil ha den gaussiske uskarphetseffekten, men hva hvis du vil ha en skikkelig linsskarphet som simulerer en ekte kameraåpning. Se på film eller tv-program for en stund, spesielt noe skutt om natten med lys i bakgrunnen, og du vil se at ting som er ute av fokus danner diskformer, eller kanskje pentagoner. Det er også et fenomen som kalles blomstrende hvor lyse deler av bildet vasker ut bildet, blir enda lysere i forhold til resten. Disse figurene kalles Bokeh. Noen mennesker elsker det og noen hater det. Vi bryr oss ikke om folk elsker det eller hater det, vi vil bare gjenskape det. Du vil ikke få disse diskformene med Gaussisk sløring - det er bare for fuzzy rundt kantene. Hva du trenger å gjøre det, bruk en fin, skarpkantet konvolusjonskjerne i form av kameraåpningen. Problemet du kommer over her er at alle de triksene som skal gjøres med separable kjerner, iterated box blurs og lignende, vil ikke fungere her - det er ingen separable kjerner som vil gi deg en pentagon (vel, sannsynligvis - jeg er ikke matematiker) - var tilbake til Det gamle problemet med uskarphetens tid går opp som torget i uskarphetradiusen. Frykt ikke, vi kan slå de tunge matematiske våpenene på problemet. Jeg vet ikke hvordan de tunge våpenene fungerer, men jeg kan sikte mot dem. De tunge våpen er Fourier Transforms. Jeg vet ikke hvordan de jobber fordi jeg ikke lyttet på universitetsforelesningene mine, men det er et stort antall om emnet du finner på Internett, selv om det er praktisk talt ingenting praktisk (dvs. med kildekoden) på temaet for uskarphet. Med Fourier Transforms kan du gjøre en uskarphet som tar en stund upåvirket av uskarphetradiusen (i praksis er det ikke helt riktig å håndtere bildekanter). Dessverre betyr det at det for en liten radius er langsom, men du vinner virkelig med en stor radius. En måte å håndtere dette på er å bruke den enkle convolutionen for små radier, og bytte til Fourier Transforms når du når til crossover-punkt i tid, forutsatt at du har gjort forsøkene for å bestemme hvor det er. Men vær forsiktig, hvis du animerer en uskarphet, må du sørge for at du ikke får noen synlige gjenstander ved det punktet du bytter algoritme - øyet er veldig godt å spotte dem. Av den grunn kan du foretrekke å holde fast med en algoritme for hele animasjonen. For stillbilder, kommer ingen til å legge merke til. Egentlig. Ser det virkelig annerledes ut. Sikkert, vi kan komme seg unna med en Gauss-uskarpe. Nå, heres et eksempel som vil hjelpe deg å gjøre tankene dine. Prinsippet bak å gjøre uskarphet er ikke for vanskelig, selv om det virker som magi. Hva vi gjør er å ta bildet og kjernen, og utfør Fourier-transformasjonen på dem begge. Vi multipliserer de to sammen og omvendte transformasjonen tilbake. Dette er akkurat det samme som å utføre den lange konvolusjonen ovenfor (bortsett fra avrundingsfeil). Du trenger egentlig ikke å vite hva en Fourier-transformasjon gjør for å implementere dette, men uansett, hva det gjør er å konvertere bildet til frekvensområdet - det resulterende bildet er en merkelig utseende av romlige frekvenser i bildet. Den omvendte, selvfølgelig, forvandler seg tilbake til verdensrommet. er, plass. Tenk på det som en grafisk equalizer for bilder. Du kan tenke på å sløre et bilde som å fjerne høyfrekvenser fra det, så det er hvordan Fourier-transformasjoner kommer inn i bildet. Implementering av dette er faktisk ganske grei, men det er mange ubehagelige detaljer å bekymre seg for. Først av alt trenger vi noen funksjoner for å gjøre transformasjonen og dens inverse. Disse finner du i klassen FFT. Dette er ikke på noen måte en superoptimalisert implementering - du kan finne mange av dem andre steder på Internett. Deretter må vi konvertere kjernen til et bilde i samme størrelse som bildet var uskarpt (jeg er sikker på at det finnes måter å unngå dette på, men jeg vet ikke nok matematikk - hvis bare Id hadde lyttet i disse forelesningene). Vi må også kaste ut vårt kildebilde av raden av uskarpheten, duplisere kanten piksler som det er vanskelig å få FFT til å håndtere kanter som dette. Nå fungerer FFT på komplekse tall, så vi må kopiere bildet og kjernen til float arrays. Vi kan gjøre et triks her - bildene våre har fire kanaler (alfa, rød, grønn og blå), så vi må gjøre fire transformer pluss en for kjernen, og gjør fem, men siden vi brukte komplekse tall kan vi gjøre to transformasjoner samtidig ved å sette en kanal i den reelle delen av gruppen og en kanal i den imaginære delen. Nå blir ting enkle, bare forvandle bildet og kjernen, komplekse multiplisere dem sammen og omvendt transformere, og vi har vårt bilde tilbake, men samvittet med kjernen. En siste liten detalj er at transformasjonsprosessen bytter over kvadranter av bildet, slik at vi må unwap. Bare en liten detalj gjenstår: FFT fungerer bare på bilder som er en kraft på 2 i hver retning. Det vi må gjøre er å legge to ganger uklarheten til bredden og høyden, finn den nest høyeste effekten på 2 og lag våre størrelser slik. For store bilder har dette et par problemer: En er som brukte mye minne. Husk at vi har våre bilder i float arrays, og vi trenger 6 av disse arrays, hver av dem er 4 ganger størrelsen på bildet når det er utvidet til en kraft på to. Din virtuelle Java-maskin kan godt klage på deg hvis du prøver dette på et stort bilde (jeg vet, jeg har prøvd). Det andre problemet er relatert: Ting går bare tregere med de store bildene på grunn av minneproblemer. Svaret er å dele bildet opp i fliser og uklare hver flis separat. Å velge en god flisestørrelse er et alternativforskningsproblem (dvs. jeg har ikke vært plaget til å eksperimentere mye), men er vanskelig - vi må overlappe flisene ved uklarheten, så hvis vi valgte en flisestørrelse på 256 med en uklarhet på 127 , vi må bare uklare 4 piksler med hver flis. Prøv det i denne appleten: Threshold Blurs Noe som ofte er ønsket, er en uskarphet som forvirrer deler av bildet som er svært liknende, men bevarer skarpe kanter. Dette er digital rynkekrem, og du kan se dette i en hvilken som helst filmplakat som alltid er skrevet ut - stjernens ansikter har alle de ekle flekkene utstrykt uten at bildet ser ut som uklart. Ofte er dette så overdone at skuespillerne ser ut som voksverk eller datagenererte figurer. Måten vi gjør dette på er å gjøre en vanlig konvolusjon, men bare telle i omkringliggende piksler som ligner målpiksel. Nærmere bestemt har vi en terskel og inkluderer bare en piksel i konvolusjonen hvis den avviger fra sentrumspiksel med mindre enn terskelen. Dessverre, de korte kuttene vi tok over, vil ikke fungere her da vi trenger å inkludere et annet sett med omkringliggende piksler for hver målpiksel, så var tilbake til full convolution igjen. Nå, selv om dette er ekstremt tvilsomt, fungerer det egentlig ganske bra for å fortsatt gjøre de to 1D-omviklingene for en Gauss-uskarphet som er raskere enn å gjøre hele 2D-konvolusjonen, så det er det jeg har gjort her. Du er velkommen til å endre kilden for å gjøre det hele. Prøv det ut i denne appleten: Variable Blurs Så langt har vi bare snakket om uniform blurs - hvor uklarheten er den samme på hvert punkt. For noen formål er det fint å ha blurs som har en annen radius på hvert punkt i bildet. Et eksempel er å simulere dybdeskarphet: Du kan ta et bilde som er i fokus overalt, og bruk en variabel uskarphet for å gjøre delene ute av fokus. Virkelig dybdeskarphet er mer komplisert enn dette fordi et objekt som ligger bak et annet objekt, ikke bør motta noe uklarhet fra objektet foran, men godt ignorere det og legge det til fagfolkene. Nå vil våre fancy triks over arent hjelpe oss mye her da alt involverer forutregning av kjerner eller stoler på at uklarheten er den samme over bildet og ved første øyekast ser det ut til at vi ikke har noe annet valg enn å falle tilbake på full convolution på hver piksel, bare denne gangen, er det mye verre da kjernen kan ha endret seg fra forrige piksel. Men alt er ikke tapt. Husk at triks med boks uklare hvor vi nettopp lagt til i piksler når de kom inn i kjernen og trukket dem tilbake etter hvert som de forlot. Det virker som om dette ikke vil fungere i variabel radius sak fordi wed må beholde totals for hver mulig radius, men det er en modifikasjon vi kan gjøre til trikset som gjør at vi magisk trekker ut totalene for en hvilken som helst radius med bare en subtraksjon. Hva vi gjør er preprocess bildet og erstatt hver piksel med summen av alle pikslene til venstre. På den måten når vi vil finne summen av alle pikslene mellom to punkter i en scanline, trenger vi bare å trekke den første fra den andre. Dette gjør at vi kan gjøre en rask variabel uskarphet ved å bruke en modifisert versjon av boksen uskarphetskode ovenfor. Å håndtere kantene er litt mer komplisert, fordi det bare er å trekke totalen, fungerer ikke for piksler av kanten, men dette er en mindre detalj. Vi trenger også litt mer lagringsplass fordi totalene vil gå over den maksimale verdien av en piksel - vel må du bruke en int per kanal i stedet for å lagre fire kanaler i ett int. Vel, OK, men dette er en Gaussisk (ish) uskarphet, det er ikke det. Hva med å gjøre det objektivet uklart ting med variabel radius Dessverre, du er uheldig her. Jeg sier ikke at det ikke er en super rask måte å gjøre dette på, men så langt jeg vet at du kommer til å gjøre den fulle konvolusjonen. Prøv det i denne appleten, noe som blurs mer når du beveger deg til høyre: Skarphet ved uklarhet Du kan bruke uskarphet til å skarpere et bilde, og sløre det med en teknikk som kalles uskarpe maskering. Det du gjør er å ta bildet og trekke en uskarpt versjon, slik at du kompenserer for tap av lysstyrke. Dette høres ut som magi, men det fungerer egentlig: sammenlign dette bildet med originalen. Prøv det i denne appleten: Hvis du trekker en uskarpt versjon av et bilde fra seg selv, skjer det, hva legger det til. Som noensinne trenger du ikke å gjette - jeg er her for å informere deg. Det du får er en slags glødende effekt som kan se ganske fin ut, eller ganske ost, avhengig av synspunktet ditt. Varierer mengden av uskarphet tilsatt varierer den glødende effekten. Du kan se denne effekten brukt mye på tv for drømmende utseende overganger. Prøv det i denne appleten: Å lage skygger Skygge er bare et spørsmål om å skape et bilde som ser ut som silhuetten til skyggeobjektet, slør det, muligens forvrenger eller flytter det, og limer opp det opprinnelige bildet over toppen. Siden dette er en veldig vanlig ting å ønske å gjøre, bør det være et filter for å gjøre det, og her er det. Dette er faktisk en veldig enkel implementering - det blur bare skyggen og trekker det opprinnelige bildet over toppen. I praksis er det bedre å ikke bry seg om å pusse pikslene som er helt skjult av objektet. Casting Stråler Vi kan gjøre det samme trikset for at lysstråler ser ut til å komme ut av et objekt, bare denne gangen gjør skyggen farge hvit og bruker en zoomskarmen i stedet for den vanlige uskarpheten, og legger til resultatet på toppen av originalen. Strålene ser ofte bedre ut hvis du bare støtter dem fra lyse deler av bildet, så filteret har en terskel som du kan angi for å begrense stråler til lyse områder. Dette er en god effekt å animere: gjør senteret av strålene bevege seg over bildet, og du får effekten av en bevegelig lyskilde bak bildet. Konklusjon Vel, det er det, og jeg har ikke engang nevnt andre uskarphetiske metoder som IIR-filtre, rekursive filtre og alle de andre ubehagelige tingene. Jeg håper du kommer unna med noe nyttig med dette, selv om det bare er et brennende ønske om å kjøpe litt grønn blekk og skrive meg et brev. Til slutt har du kanskje lagt merke til at kilden ovenfor er avhengig av noen andre klasser. Ikke bekymre deg, her er de: Bell Curve BREAKING DOWN Bell Curve Bell kurve er et generelt begrep som brukes til å beskrive en grafisk avbildning av en normal sannsynlighet distribusjon. De normale sannsynlighetsfordelingene som ligger til grunn for standardavvik fra medianen, eller fra det høyeste punktet på kurven, er det som gir den formen av en buet klokke. Et standardavvik er en måling som brukes til å kvantifisere variasjonen i datadispersjonen i et sett med verdier. Gjennomsnittet er gjennomsnittet av alle datapunkter i datasettet eller sekvensen. Standardavvik beregnes etter at gjennomsnittet er beregnet og representerer en prosentandel av samlet data. For eksempel, hvis en serie på 100 testresultater samles inn og brukes i en normal sannsynlighetsfordeling, bør 68 av 100 testpoengene falle innenfor en standardavvik over eller under gjennomsnittet. Ved å flytte to standardavvik bort fra gjennomsnittet, bør det inkluderes 95 av de 100 testpoengene som er samlet, og å flytte tre standardavvik bort fra gjennomsnittet skal utgjøre 99,7 av 100 testpoeng. Eventuelle testresultater som er ekstreme avvikere, for eksempel en poengsum på 100 eller 0, vil bli vurdert som langt hale datapunkter og ligger utenfor de tre standardavviksintervallene. Bruke datafordeling i finans Finansanalytikere og investorer bruker ofte en normal sannsynlighetsfordeling når de analyserer avkastningen på en sikkerhet eller av den generelle markedsfølsomheten. Standardavvik som skildrer avkastningen på et sikkerhet er kjent i finansverdenen som volatilitet. For eksempel, aksjer som viser en bellkurve er normalt blåchipsaksjer og har lavere og forutsigbar volatilitet. Investorer bruker normal sannsynlighetsfordeling av en aksjeavkastning før avkastning for å gjøre antagelser om forventet fremtidig avkastning. Men aksjer og andre verdipapirer viser noen ganger ikke-normale distribusjoner, noe som betyr at de ikke ser ut som en bellkurve. Ikke-normale distribusjoner har større svikt enn en normal sannsynlighetsfordeling. Hvis den svakere halen er skjev negativ, er det et signal til investorer at det er større sannsynlighet for negativ avkastning og omvendt. Positivt skjeve fette haler kan være et tegn på unormal fremtidig avkastning. Kryptering er et veldig populært middel for å skaffe seg store mengder merket data som moderne maskinlæringsmetoder krever. Selv om det er billig og rask å få tak i folkemerkede etiketter, lider det av betydelige mengder feil, og derved forringer ytelsen til maskinopplæringsoppgaver i nedstrøms. Med målet om å forbedre kvaliteten på de merkede dataene, søker vi å redusere de mange feilene som oppstår på grunn av dumme feil eller utilsiktede feil ved crowdsourcing-arbeidere. Vi foreslår en to-trinns innstilling for crowdsourcing hvor arbeideren først svarer på spørsmålene, og får da endre svarene sine etter å ha sett et (støyende) referansesvar. Vi formulerer matematisk denne prosessen og utvikler mekanismer for å stimulere arbeidstakere til å handle hensiktsmessig. Våre matematiske garantier viser at vår mekanisme stimulerer arbeiderne til å svare ærlig i begge trinn, og avstå fra å svare tilfeldig i første fase eller bare kopiere i det andre. Numeriske eksperimenter avslører en betydelig forbedring i ytelse som en slik 8220self-correction8221 kan gi når man bruker crowdsourcing til å trene maskinlæringsalgoritmer. Det finnes ulike parametriske modeller for å analysere parvise sammenligningsdata, inkludert Bradley-Terry-Luce (BTL) og Thurstone-modellene, men deres avhengighet av sterke parametriske antagelser er begrensende. I dette arbeidet studerer vi en fleksibel modell for parvise sammenligninger, der sannsynlighetene for utfall bare kreves for å tilfredsstille en naturlig form for stokastisk transitivitet. Denne klassen inkluderer parametriske modeller, inkludert BTL og Thurstone modeller som spesielle tilfeller, men er betydelig mer generelt. Vi gir ulike eksempler på modeller i denne bredere stokastisk transitive klassen, for hvilke klassiske parametriske modeller gir dårlig passform. Til tross for denne større fleksibiliteten viser vi at matrisen av sannsynligheter kan estimeres i samme takt som i standard parametriske modeller. På den annen side, i motsetning til i BTL - og Thurstone-modellene, er beregning av den minimalaksimale optimal estimatoren i den stokastisk transitive modellen ikke-trivial, og vi undersøker ulike beregningsmessige traktable alternativer. Vi viser at en enkel enverdigærskelgrenseralgoritme er statistisk konsistent, men ikke oppnår minimaksatsen. Vi foreslår og studerer algoritmer som oppnår minimax-hastigheten over interessante underklasser av den fulle stokastisk transitive klassen. Vi utfyller våre teoretiske resultater med grundige numeriske simuleringer. Vi viser hvordan en hvilken som helst binær parvismodell kan bli opprotet til en helt symmetrisk modell, der originale singleton-potensialer blir transformert til potensialer på kanter til en ekstra variabel, og deretter reetorert til en ny modell på det opprinnelige antall variabler. The new model is essentially equivalent to the original model, with the same partition function and allowing recovery of the original marginals or a MAP conguration, yet may have very different computational properties that allow much more efficient inference. This meta-approach deepens our understanding, may be applied to any existing algorithm to yield improved methods in practice, generalizes earlier theoretical results, and reveals a remarkable interpretation of the triplet-consistent polytope. We show how deep learning methods can be applied in the context of crowdsourcing and unsupervised ensemble learning. First, we prove that the popular model of Dawid and Skene, which assumes that all classifiers are conditionally independent, is to a Restricted Boltzmann Machine (RBM) with a single hidden node. Hence, under this model, the posterior probabilities of the true labels can be instead estimated via a trained RBM. Next, to address the more general case, where classifiers may strongly violate the conditional independence assumption, we propose to apply RBM-based Deep Neural Net (DNN). Experimental results on various simulated and real-world datasets demonstrate that our proposed DNN approach outperforms other state-of-the-art methods, in particular when the data violates the conditional independence assumption. Revisiting Semi-Supervised Learning with Graph Embeddings Zhilin Yang Carnegie Mellon University . William Cohen CMU . Ruslan Salakhudinov U. of Toronto Paper AbstractWe present a semi-supervised learning framework based on graph embeddings. Given a graph between instances, we train an embedding for each instance to jointly predict the class label and the neighborhood context in the graph. We develop both transductive and inductive variants of our method. In the transductive variant of our method, the class labels are determined by both the learned embeddings and input feature vectors, while in the inductive variant, the embeddings are defined as a parametric function of the feature vectors, so predictions can be made on instances not seen during training. On a large and diverse set of benchmark tasks, including text classification, distantly supervised entity extraction, and entity classification, we show improved performance over many of the existing models. Reinforcement learning can acquire complex behaviors from high-level specifications. However, defining a cost function that can be optimized effectively and encodes the correct task is challenging in practice. We explore how inverse optimal control (IOC) can be used to learn behaviors from demonstrations, with applications to torque control of high-dimensional robotic systems. Our method addresses two key challenges in inverse optimal control: first, the need for informative features and effective regularization to impose structure on the cost, and second, the difficulty of learning the cost function under unknown dynamics for high-dimensional continuous systems. To address the former challenge, we present an algorithm capable of learning arbitrary nonlinear cost functions, such as neural networks, without meticulous feature engineering. To address the latter challenge, we formulate an efficient sample-based approximation for MaxEnt IOC. We evaluate our method on a series of simulated tasks and real-world robotic manipulation problems, demonstrating substantial improvement over prior methods both in terms of task complexity and sample efficiency. In learning latent variable models (LVMs), it is important to effectively capture infrequent patterns and shrink model size without sacrificing modeling power. Various studies have been done to 8220diversify8221 a LVM, which aim to learn a diverse set of latent components in LVMs. Most existing studies fall into a frequentist-style regularization framework, where the components are learned via point estimation. In this paper, we investigate how to 8220diversify8221 LVMs in the paradigm of Bayesian learning, which has advantages complementary to point estimation, such as alleviating overfitting via model averaging and quantifying uncertainty. We propose two approaches that have complementary advantages. One is to define diversity-promoting mutual angular priors which assign larger density to components with larger mutual angles based on Bayesian network and von Mises-Fisher distribution and use these priors to affect the posterior via Bayes rule. We develop two efficient approximate posterior inference algorithms based on variational inference and Markov chain Monte Carlo sampling. The other approach is to impose diversity-promoting regularization directly over the post-data distribution of components. These two methods are applied to the Bayesian mixture of experts model to encourage the 8220experts8221 to be diverse and experimental results demonstrate the effectiveness and efficiency of our methods. High dimensional nonparametric regression is an inherently difficult problem with known lower bounds depending exponentially in dimension. A popular strategy to alleviate this curse of dimensionality has been to use additive models of emph , which model the regression function as a sum of independent functions on each dimension. Though useful in controlling the variance of the estimate, such models are often too restrictive in practical settings. Between non-additive models which often have large variance and first order additive models which have large bias, there has been little work to exploit the trade-off in the middle via additive models of intermediate order. In this work, we propose salsa, which bridges this gap by allowing interactions between variables, but controls model capacity by limiting the order of interactions. salsas minimises the residual sum of squares with squared RKHS norm penalties. Algorithmically, it can be viewed as Kernel Ridge Regression with an additive kernel. When the regression function is additive, the excess risk is only polynomial in dimension. Using the Girard-Newton formulae, we efficiently sum over a combinatorial number of terms in the additive expansion. Via a comparison on 15 real datasets, we show that our method is competitive against 21 other alternatives. We propose an extension to Hawkes processes by treating the levels of self-excitation as a stochastic differential equation. Our new point process allows better approximation in application domains where events and intensities accelerate each other with correlated levels of contagion. We generalize a recent algorithm for simulating draws from Hawkes processes whose levels of excitation are stochastic processes, and propose a hybrid Markov chain Monte Carlo approach for model fitting. Our sampling procedure scales linearly with the number of required events and does not require stationarity of the point process. A modular inference procedure consisting of a combination between Gibbs and Metropolis Hastings steps is put forward. We recover expectation maximization as a special case. Our general approach is illustrated for contagion following geometric Brownian motion and exponential Langevin dynamics. Rank aggregation systems collect ordinal preferences from individuals to produce a global ranking that represents the social preference. To reduce the computational complexity of learning the global ranking, a common practice is to use rank-breaking. Individuals preferences are broken into pairwise comparisons and then applied to efficient algorithms tailored for independent pairwise comparisons. However, due to the ignored dependencies, naive rank-breaking approaches can result in inconsistent estimates. The key idea to produce unbiased and accurate estimates is to treat the paired comparisons outcomes unequally, depending on the topology of the collected data. In this paper, we provide the optimal rank-breaking estimator, which not only achieves consistency but also achieves the best error bound. This allows us to characterize the fundamental tradeoff between accuracy and complexity in some canonical scenarios. Further, we identify how the accuracy depends on the spectral gap of a corresponding comparison graph. Dropout distillation Samuel Rota Bul FBK . Lorenzo Porzi FBK . Peter Kontschieder Microsoft Research Cambridge Paper AbstractDropout is a popular stochastic regularization technique for deep neural networks that works by randomly dropping (i. e. zeroing) units from the network during training. This randomization process allows to implicitly train an ensemble of exponentially many networks sharing the same parametrization, which should be averaged at test time to deliver the final prediction. A typical workaround for this intractable averaging operation consists in scaling the layers undergoing dropout randomization. This simple rule called 8216standard dropout8217 is efficient, but might degrade the accuracy of the prediction. In this work we introduce a novel approach, coined 8216dropout distillation8217, that allows us to train a predictor in a way to better approximate the intractable, but preferable, averaging process, while keeping under control its computational efficiency. We are thus able to construct models that are as efficient as standard dropout, or even more efficient, while being more accurate. Experiments on standard benchmark datasets demonstrate the validity of our method, yielding consistent improvements over conventional dropout. Metadata-conscious anonymous messaging Giulia Fanti UIUC . Peter Kairouz UIUC . Sewoong Oh UIUC . Kannan Ramchandran UC Berkeley . Pramod Viswanath UIUC Paper AbstractAnonymous messaging platforms like Whisper and Yik Yak allow users to spread messages over a network (e. g. a social network) without revealing message authorship to other users. The spread of messages on these platforms can be modeled by a diffusion process over a graph. Recent advances in network analysis have revealed that such diffusion processes are vulnerable to author deanonymization by adversaries with access to metadata, such as timing information. In this work, we ask the fundamental question of how to propagate anonymous messages over a graph to make it difficult for adversaries to infer the source. In particular, we study the performance of a message propagation protocol called adaptive diffusion introduced in (Fanti et al. 2015). We prove that when the adversary has access to metadata at a fraction of corrupted graph nodes, adaptive diffusion achieves asymptotically optimal source-hiding and significantly outperforms standard diffusion. We further demonstrate empirically that adaptive diffusion hides the source effectively on real social networks. The Teaching Dimension of Linear Learners Ji Liu University of Rochester . Xiaojin Zhu University of Wisconsin . Hrag Ohannessian University of Wisconsin-Madison Paper AbstractTeaching dimension is a learning theoretic quantity that specifies the minimum training set size to teach a target model to a learner. Previous studies on teaching dimension focused on version-space learners which maintain all hypotheses consistent with the training data, and cannot be applied to modern machine learners which select a specific hypothesis via optimization. This paper presents the first known teaching dimension for ridge regression, support vector machines, and logistic regression. We also exhibit optimal training sets that match these teaching dimensions. Our approach generalizes to other linear learners. Truthful Univariate Estimators Ioannis Caragiannis University of Patras . Ariel Procaccia Carnegie Mellon University . Nisarg Shah Carnegie Mellon University Paper AbstractWe revisit the classic problem of estimating the population mean of an unknown single-dimensional distribution from samples, taking a game-theoretic viewpoint. In our setting, samples are supplied by strategic agents, who wish to pull the estimate as close as possible to their own value. In this setting, the sample mean gives rise to manipulation opportunities, whereas the sample median does not. Our key question is whether the sample median is the best (in terms of mean squared error) truthful estimator of the population mean. We show that when the underlying distribution is symmetric, there are truthful estimators that dominate the median. Our main result is a characterization of worst-case optimal truthful estimators, which provably outperform the median, for possibly asymmetric distributions with bounded support. Why Regularized Auto-Encoders learn Sparse Representation Devansh Arpit SUNY Buffalo . Yingbo Zhou SUNY Buffalo . Hung Ngo SUNY Buffalo . Venu Govindaraju SUNY Buffalo Paper AbstractSparse distributed representation is the key to learning useful features in deep learning algorithms, because not only it is an efficient mode of data representation, but also 8212 more importantly 8212 it captures the generation process of most real world data. While a number of regularized auto-encoders (AE) enforce sparsity explicitly in their learned representation and others don8217t, there has been little formal analysis on what encourages sparsity in these models in general. Our objective is to formally study this general problem for regularized auto-encoders. We provide sufficient conditions on both regularization and activation functions that encourage sparsity. We show that multiple popular models (de-noising and contractive auto encoders, e. g.) and activations (rectified linear and sigmoid, e. g.) satisfy these conditions thus, our conditions help explain sparsity in their learned representation. Thus our theoretical and empirical analysis together shed light on the properties of regularizationactivation that are conductive to sparsity and unify a number of existing auto-encoder models and activation functions under the same analytical framework. k-variates: more pluses in the k-means Richard Nock Nicta 038 ANU . Raphael Canyasse Ecole Polytechnique and The Technion . Roksana Boreli Data61 . Frank Nielsen Ecole Polytechnique and Sony CS Labs Inc. Paper Abstractk-means seeding has become a de facto standard for hard clustering algorithms. In this paper, our first contribution is a two-way generalisation of this seeding, k-variates, that includes the sampling of general densities rather than just a discrete set of Dirac densities anchored at the point locations, textit a generalisation of the well known Arthur-Vassilvitskii (AV) approximation guarantee, in the form of a textit approximation bound of the textit optimum. This approximation exhibits a reduced dependency on the 8220noise8221 component with respect to the optimal potential 8212 actually approaching the statistical lower bound. We show that k-variates textit to efficient (biased seeding) clustering algorithms tailored to specific frameworks these include distributed, streaming and on-line clustering, with textit approximation results for these algorithms. Finally, we present a novel application of k-variates to differential privacy. For either the specific frameworks considered here, or for the differential privacy setting, there is little to no prior results on the direct application of k-means and its approximation bounds 8212 state of the art contenders appear to be significantly more complex and or display less favorable (approximation) properties. We stress that our algorithms can still be run in cases where there is textit closed form solution for the population minimizer. We demonstrate the applicability of our analysis via experimental evaluation on several domains and settings, displaying competitive performances vs state of the art. Multi-Player Bandits 8212 a Musical Chairs Approach Jonathan Rosenski Weizmann Institute of Science . Ohad Shamir Weizmann Institute of Science . Liran Szlak Weizmann Institute of Science Paper AbstractWe consider a variant of the stochastic multi-armed bandit problem, where multiple players simultaneously choose from the same set of arms and may collide, receiving no reward. This setting has been motivated by problems arising in cognitive radio networks, and is especially challenging under the realistic assumption that communication between players is limited. We provide a communication-free algorithm (Musical Chairs) which attains constant regret with high probability, as well as a sublinear-regret, communication-free algorithm (Dynamic Musical Chairs) for the more difficult setting of players dynamically entering and leaving throughout the game. Moreover, both algorithms do not require prior knowledge of the number of players. To the best of our knowledge, these are the first communication-free algorithms with these types of formal guarantees. The Information Sieve Greg Ver Steeg Information Sciences Institute . Aram Galstyan Information Sciences Institute Paper AbstractWe introduce a new framework for unsupervised learning of representations based on a novel hierarchical decomposition of information. Intuitively, data is passed through a series of progressively fine-grained sieves. Each layer of the sieve recovers a single latent factor that is maximally informative about multivariate dependence in the data. The data is transformed after each pass so that the remaining unexplained information trickles down to the next layer. Ultimately, we are left with a set of latent factors explaining all the dependence in the original data and remainder information consisting of independent noise. We present a practical implementation of this framework for discrete variables and apply it to a variety of fundamental tasks in unsupervised learning including independent component analysis, lossy and lossless compression, and predicting missing values in data. Deep Speech 2. End-to-End Speech Recognition in English and Mandarin Dario Amodei . Rishita Anubhai . Eric Battenberg . Carl Case . Jared Casper . Bryan Catanzaro . JingDong Chen . Mike Chrzanowski Baidu USA, Inc. . Adam Coates . Greg Diamos Baidu USA, Inc. . Erich Elsen Baidu USA, Inc. . Jesse Engel . Linxi Fan . Christopher Fougner . Awni Hannun Baidu USA, Inc. . Billy Jun . Tony Han . Patrick LeGresley . Xiangang Li Baidu . Libby Lin . Sharan Narang . Andrew Ng . Sherjil Ozair . Ryan Prenger . Sheng Qian Baidu . Jonathan Raiman . Sanjeev Satheesh Baidu SVAIL . David Seetapun . Shubho Sengupta . Chong Wang . Yi Wang . Zhiqian Wang . Bo Xiao . Yan Xie Baidu . Dani Yogatama . Jun Zhan . zhenyao Zhu Paper AbstractWe show that an end-to-end deep learning approach can be used to recognize either English or Mandarin Chinese speechtwo vastly different languages. Because it replaces entire pipelines of hand-engineered components with neural networks, end-to-end learning allows us to handle a diverse variety of speech including noisy environments, accents and different languages. Key to our approach is our application of HPC techniques, enabling experiments that previously took weeks to now run in days. This allows us to iterate more quickly to identify superior architectures and algorithms. As a result, in several cases, our system is competitive with the transcription of human workers when benchmarked on standard datasets. Finally, using a technique called Batch Dispatch with GPUs in the data center, we show that our system can be inexpensively deployed in an online setting, delivering low latency when serving users at scale. An important question in feature selection is whether a selection strategy recovers the 8220true8221 set of features, given enough data. We study this question in the context of the popular Least Absolute Shrinkage and Selection Operator (Lasso) feature selection strategy. In particular, we consider the scenario when the model is misspecified so that the learned model is linear while the underlying real target is nonlinear. Surprisingly, we prove that under certain conditions, Lasso is still able to recover the correct features in this case. We also carry out numerical studies to empirically verify the theoretical results and explore the necessity of the conditions under which the proof holds. We propose minimum regret search (MRS), a novel acquisition function for Bayesian optimization. MRS bears similarities with information-theoretic approaches such as entropy search (ES). However, while ES aims in each query at maximizing the information gain with respect to the global maximum, MRS aims at minimizing the expected simple regret of its ultimate recommendation for the optimum. While empirically ES and MRS perform similar in most of the cases, MRS produces fewer outliers with high simple regret than ES. We provide empirical results both for a synthetic single-task optimization problem as well as for a simulated multi-task robotic control problem. CryptoNets: Applying Neural Networks to Encrypted Data with High Throughput and Accuracy Ran Gilad-Bachrach Microsoft Research . Nathan Dowlin Princeton . Kim Laine Microsoft Research . Kristin Lauter Microsoft Research . Michael Naehrig Microsoft Research . John Wernsing Microsoft Research Paper AbstractApplying machine learning to a problem which involves medical, financial, or other types of sensitive data, not only requires accurate predictions but also careful attention to maintaining data privacy and security. Legal and ethical requirements may prevent the use of cloud-based machine learning solutions for such tasks. In this work, we will present a method to convert learned neural networks to CryptoNets, neural networks that can be applied to encrypted data. This allows a data owner to send their data in an encrypted form to a cloud service that hosts the network. The encryption ensures that the data remains confidential since the cloud does not have access to the keys needed to decrypt it. Nevertheless, we will show that the cloud service is capable of applying the neural network to the encrypted data to make encrypted predictions, and also return them in encrypted form. These encrypted predictions can be sent back to the owner of the secret key who can decrypt them. Therefore, the cloud service does not gain any information about the raw data nor about the prediction it made. We demonstrate CryptoNets on the MNIST optical character recognition tasks. CryptoNets achieve 99 accuracy and can make around 59000 predictions per hour on a single PC. Therefore, they allow high throughput, accurate, and private predictions. Spectral methods for dimensionality reduction and clustering require solving an eigenproblem defined by a sparse affinity matrix. When this matrix is large, one seeks an approximate solution. The standard way to do this is the Nystrom method, which first solves a small eigenproblem considering only a subset of landmark points, and then applies an out-of-sample formula to extrapolate the solution to the entire dataset. We show that by constraining the original problem to satisfy the Nystrom formula, we obtain an approximation that is computationally simple and efficient, but achieves a lower approximation error using fewer landmarks and less runtime. We also study the role of normalization in the computational cost and quality of the resulting solution. As a widely used non-linear activation, Rectified Linear Unit (ReLU) separates noise and signal in a feature map by learning a threshold or bias. However, we argue that the classification of noise and signal not only depends on the magnitude of responses, but also the context of how the feature responses would be used to detect more abstract patterns in higher layers. In order to output multiple response maps with magnitude in different ranges for a particular visual pattern, existing networks employing ReLU and its variants have to learn a large number of redundant filters. In this paper, we propose a multi-bias non-linear activation (MBA) layer to explore the information hidden in the magnitudes of responses. It is placed after the convolution layer to decouple the responses to a convolution kernel into multiple maps by multi-thresholding magnitudes, thus generating more patterns in the feature space at a low computational cost. It provides great flexibility of selecting responses to different visual patterns in different magnitude ranges to form rich representations in higher layers. Such a simple and yet effective scheme achieves the state-of-the-art performance on several benchmarks. We propose a novel multi-task learning method that can minimize the effect of negative transfer by allowing asymmetric transfer between the tasks based on task relatedness as well as the amount of individual task losses, which we refer to as Asymmetric Multi-task Learning (AMTL). To tackle this problem, we couple multiple tasks via a sparse, directed regularization graph, that enforces each task parameter to be reconstructed as a sparse combination of other tasks, which are selected based on the task-wise loss. We present two different algorithms to solve this joint learning of the task predictors and the regularization graph. The first algorithm solves for the original learning objective using alternative optimization, and the second algorithm solves an approximation of it using curriculum learning strategy, that learns one task at a time. We perform experiments on multiple datasets for classification and regression, on which we obtain significant improvements in performance over the single task learning and symmetric multitask learning baselines. This paper illustrates a novel approach to the estimation of generalization error of decision tree classifiers. We set out the study of decision tree errors in the context of consistency analysis theory, which proved that the Bayes error can be achieved only if when the number of data samples thrown into each leaf node goes to infinity. For the more challenging and practical case where the sample size is finite or small, a novel sampling error term is introduced in this paper to cope with the small sample problem effectively and efficiently. Extensive experimental results show that the proposed error estimate is superior to the well known K-fold cross validation methods in terms of robustness and accuracy. Moreover it is orders of magnitudes more efficient than cross validation methods. We study the convergence properties of the VR-PCA algorithm introduced by cite for fast computation of leading singular vectors. We prove several new results, including a formal analysis of a block version of the algorithm, and convergence from random initialization. We also make a few observations of independent interest, such as how pre-initializing with just a single exact power iteration can significantly improve the analysis, and what are the convexity and non-convexity properties of the underlying optimization problem. We consider the problem of principal component analysis (PCA) in a streaming stochastic setting, where our goal is to find a direction of approximate maximal variance, based on a stream of i. i.d. data points in realsd. A simple and computationally cheap algorithm for this is stochastic gradient descent (SGD), which incrementally updates its estimate based on each new data point. However, due to the non-convex nature of the problem, analyzing its performance has been a challenge. In particular, existing guarantees rely on a non-trivial eigengap assumption on the covariance matrix, which is intuitively unnecessary. In this paper, we provide (to the best of our knowledge) the first eigengap-free convergence guarantees for SGD in the context of PCA. This also partially resolves an open problem posed in cite . Moreover, under an eigengap assumption, we show that the same techniques lead to new SGD convergence guarantees with better dependence on the eigengap. Dealbreaker: A Nonlinear Latent Variable Model for Educational Data Andrew Lan Rice University . Tom Goldstein University of Maryland . Richard Baraniuk Rice University . Christoph Studer Cornell University Paper AbstractStatistical models of student responses on assessment questions, such as those in homeworks and exams, enable educators and computer-based personalized learning systems to gain insights into students knowledge using machine learning. Popular student-response models, including the Rasch model and item response theory models, represent the probability of a student answering a question correctly using an affine function of latent factors. While such models can accurately predict student responses, their ability to interpret the underlying knowledge structure (which is certainly nonlinear) is limited. In response, we develop a new, nonlinear latent variable model that we call the dealbreaker model, in which a students success probability is determined by their weakest concept mastery. We develop efficient parameter inference algorithms for this model using novel methods for nonconvex optimization. We show that the dealbreaker model achieves comparable or better prediction performance as compared to affine models with real-world educational datasets. We further demonstrate that the parameters learned by the dealbreaker model are interpretablethey provide key insights into which concepts are critical (i. e. the dealbreaker) to answering a question correctly. We conclude by reporting preliminary results for a movie-rating dataset, which illustrate the broader applicability of the dealbreaker model. We derive a new discrepancy statistic for measuring differences between two probability distributions based on combining Stein8217s identity and the reproducing kernel Hilbert space theory. We apply our result to test how well a probabilistic model fits a set of observations, and derive a new class of powerful goodness-of-fit tests that are widely applicable for complex and high dimensional distributions, even for those with computationally intractable normalization constants. Both theoretical and empirical properties of our methods are studied thoroughly. Variable Elimination in the Fourier Domain Yexiang Xue Cornell University . Stefano Ermon . Ronan Le Bras Cornell University . Carla . Bart Paper AbstractThe ability to represent complex high dimensional probability distributions in a compact form is one of the key insights in the field of graphical models. Factored representations are ubiquitous in machine learning and lead to major computational advantages. We explore a different type of compact representation based on discrete Fourier representations, complementing the classical approach based on conditional independencies. We show that a large class of probabilistic graphical models have a compact Fourier representation. This theoretical result opens up an entirely new way of approximating a probability distribution. We demonstrate the significance of this approach by applying it to the variable elimination algorithm. Compared with the traditional bucket representation and other approximate inference algorithms, we obtain significant improvements. Low-rank matrix approximation has been widely adopted in machine learning applications with sparse data, such as recommender systems. However, the sparsity of the data, incomplete and noisy, introduces challenges to the algorithm stability 8212 small changes in the training data may significantly change the models. As a result, existing low-rank matrix approximation solutions yield low generalization performance, exhibiting high error variance on the training dataset, and minimizing the training error may not guarantee error reduction on the testing dataset. In this paper, we investigate the algorithm stability problem of low-rank matrix approximations. We present a new algorithm design framework, which (1) introduces new optimization objectives to guide stable matrix approximation algorithm design, and (2) solves the optimization problem to obtain stable low-rank approximation solutions with good generalization performance. Experimental results on real-world datasets demonstrate that the proposed work can achieve better prediction accuracy compared with both state-of-the-art low-rank matrix approximation methods and ensemble methods in recommendation task. Given samples from two densities p and q, density ratio estimation (DRE) is the problem of estimating the ratio pq. Two popular discriminative approaches to DRE are KL importance estimation (KLIEP), and least squares importance fitting (LSIF). In this paper, we show that KLIEP and LSIF both employ class-probability estimation (CPE) losses. Motivated by this, we formally relate DRE and CPE, and demonstrate the viability of using existing losses from one problem for the other. For the DRE problem, we show that essentially any CPE loss (eg logistic, exponential) can be used, as this equivalently minimises a Bregman divergence to the true density ratio. We show how different losses focus on accurately modelling different ranges of the density ratio, and use this to design new CPE losses for DRE. For the CPE problem, we argue that the LSIF loss is useful in the regime where one wishes to rank instances with maximal accuracy at the head of the ranking. In the course of our analysis, we establish a Bregman divergence identity that may be of independent interest. We study nonconvex finite-sum problems and analyze stochastic variance reduced gradient (SVRG) methods for them. SVRG and related methods have recently surged into prominence for convex optimization given their edge over stochastic gradient descent (SGD) but their theoretical analysis almost exclusively assumes convexity. In contrast, we prove non-asymptotic rates of convergence (to stationary points) of SVRG for nonconvex optimization, and show that it is provably faster than SGD and gradient descent. We also analyze a subclass of nonconvex problems on which SVRG attains linear convergence to the global optimum. We extend our analysis to mini-batch variants of SVRG, showing (theoretical) linear speedup due to minibatching in parallel settings. Hierarchical Variational Models Rajesh Ranganath . Dustin Tran Columbia University . Blei David Columbia Paper AbstractBlack box variational inference allows researchers to easily prototype and evaluate an array of models. Recent advances allow such algorithms to scale to high dimensions. However, a central question remains: How to specify an expressive variational distribution that maintains efficient computation To address this, we develop hierarchical variational models (HVMs). HVMs augment a variational approximation with a prior on its parameters, which allows it to capture complex structure for both discrete and continuous latent variables. The algorithm we develop is black box, can be used for any HVM, and has the same computational efficiency as the original approximation. We study HVMs on a variety of deep discrete latent variable models. HVMs generalize other expressive variational distributions and maintains higher fidelity to the posterior. The field of mobile health (mHealth) has the potential to yield new insights into health and behavior through the analysis of continuously recorded data from wearable health and activity sensors. In this paper, we present a hierarchical span-based conditional random field model for the key problem of jointly detecting discrete events in such sensor data streams and segmenting these events into high-level activity sessions. Our model includes higher-order cardinality factors and inter-event duration factors to capture domain-specific structure in the label space. We show that our model supports exact MAP inference in quadratic time via dynamic programming, which we leverage to perform learning in the structured support vector machine framework. We apply the model to the problems of smoking and eating detection using four real data sets. Our results show statistically significant improvements in segmentation performance relative to a hierarchical pairwise CRF. Binary embeddings with structured hashed projections Anna Choromanska Courant Institute, NYU . Krzysztof Choromanski Google Research NYC . Mariusz Bojarski NVIDIA . Tony Jebara Columbia . Sanjiv Kumar . Yann Paper AbstractWe consider the hashing mechanism for constructing binary embeddings, that involves pseudo-random projections followed by nonlinear (sign function) mappings. The pseudorandom projection is described by a matrix, where not all entries are independent random variables but instead a fixed budget of randomness is distributed across the matrix. Such matrices can be efficiently stored in sub-quadratic or even linear space, provide reduction in randomness usage (i. e. number of required random values), and very often lead to computational speed ups. We prove several theoretical results showing that projections via various structured matrices followed by nonlinear mappings accurately preserve the angular distance between input high-dimensional vectors. To the best of our knowledge, these results are the first that give theoretical ground for the use of general structured matrices in the nonlinear setting. In particular, they generalize previous extensions of the Johnson - Lindenstrauss lemma and prove the plausibility of the approach that was so far only heuristically confirmed for some special structured matrices. Consequently, we show that many structured matrices can be used as an efficient information compression mechanism. Our findings build a better understanding of certain deep architectures, which contain randomly weighted and untrained layers, and yet achieve high performance on different learning tasks. We empirically verify our theoretical findings and show the dependence of learning via structured hashed projections on the performance of neural network as well as nearest neighbor classifier. A Variational Analysis of Stochastic Gradient Algorithms Stephan Mandt Columbia University . Matthew Hoffman Adobe Research . Blei David Columbia Paper AbstractStochastic Gradient Descent (SGD) is an important algorithm in machine learning. With constant learning rates, it is a stochastic process that, after an initial phase of convergence, generates samples from a stationary distribution. We show that SGD with constant rates can be effectively used as an approximate posterior inference algorithm for probabilistic modeling. Specifically, we show how to adjust the tuning parameters of SGD such as to match the resulting stationary distribution to the posterior. This analysis rests on interpreting SGD as a continuous-time stochastic process and then minimizing the Kullback-Leibler divergence between its stationary distribution and the target posterior. (This is in the spirit of variational inference.) In more detail, we model SGD as a multivariate Ornstein-Uhlenbeck process and then use properties of this process to derive the optimal parameters. This theoretical framework also connects SGD to modern scalable inference algorithms we analyze the recently proposed stochastic gradient Fisher scoring under this perspective. We demonstrate that SGD with properly chosen constant rates gives a new way to optimize hyperparameters in probabilistic models. This paper proposes a new mechanism for sampling training instances for stochastic gradient descent (SGD) methods by exploiting any side-information associated with the instances (for e. g. class-labels) to improve convergence. Previous methods have either relied on sampling from a distribution defined over training instances or from a static distribution that fixed before training. This results in two problems a) any distribution that is set apriori is independent of how the optimization progresses and b) maintaining a distribution over individual instances could be infeasible in large-scale scenarios. In this paper, we exploit the side information associated with the instances to tackle both problems. More specifically, we maintain a distribution over classes (instead of individual instances) that is adaptively estimated during the course of optimization to give the maximum reduction in the variance of the gradient. Intuitively, we sample more from those regions in space that have a textit gradient contribution. Our experiments on highly multiclass datasets show that our proposal converge significantly faster than existing techniques. Tensor regression has shown to be advantageous in learning tasks with multi-directional relatedness. Given massive multiway data, traditional methods are often too slow to operate on or suffer from memory bottleneck. In this paper, we introduce subsampled tensor projected gradient to solve the problem. Our algorithm is impressively simple and efficient. It is built upon projected gradient method with fast tensor power iterations, leveraging randomized sketching for further acceleration. Theoretical analysis shows that our algorithm converges to the correct solution in fixed number of iterations. The memory requirement grows linearly with the size of the problem. We demonstrate superior empirical performance on both multi-linear multi-task learning and spatio-temporal applications. This paper presents a novel distributed variational inference framework that unifies many parallel sparse Gaussian process regression (SGPR) models for scalable hyperparameter learning with big data. To achieve this, our framework exploits a structure of correlated noise process model that represents the observation noises as a finite realization of a high-order Gaussian Markov random process. By varying the Markov order and covariance function for the noise process model, different variational SGPR models result. This consequently allows the correlation structure of the noise process model to be characterized for which a particular variational SGPR model is optimal. We empirically evaluate the predictive performance and scalability of the distributed variational SGPR models unified by our framework on two real-world datasets. Online Stochastic Linear Optimization under One-bit Feedback Lijun Zhang Nanjing University . Tianbao Yang University of Iowa . Rong Jin Alibaba Group . Yichi Xiao Nanjing University . Zhi-hua Zhou Paper AbstractIn this paper, we study a special bandit setting of online stochastic linear optimization, where only one-bit of information is revealed to the learner at each round. This problem has found many applications including online advertisement and online recommendation. We assume the binary feedback is a random variable generated from the logit model, and aim to minimize the regret defined by the unknown linear function. Although the existing method for generalized linear bandit can be applied to our problem, the high computational cost makes it impractical for real-world applications. To address this challenge, we develop an efficient online learning algorithm by exploiting particular structures of the observation model. Specifically, we adopt online Newton step to estimate the unknown parameter and derive a tight confidence region based on the exponential concavity of the logistic loss. Our analysis shows that the proposed algorithm achieves a regret bound of O(dsqrt ), which matches the optimal result of stochastic linear bandits. We present an adaptive online gradient descent algorithm to solve online convex optimization problems with long-term constraints, which are constraints that need to be satisfied when accumulated over a finite number of rounds T, but can be violated in intermediate rounds. For some user-defined trade-off parameter beta in (0, 1), the proposed algorithm achieves cumulative regret bounds of O(Tmax ) and O(T ), respectively for the loss and the constraint violations. Our results hold for convex losses, can handle arbitrary convex constraints and rely on a single computationally efficient algorithm. Our contributions improve over the best known cumulative regret bounds of Mahdavi et al. (2012), which are respectively O(T12) and O(T34) for general convex domains, and respectively O(T23) and O(T23) when the domain is further restricted to be a polyhedral set. We supplement the analysis with experiments validating the performance of our algorithm in practice. Motivated by an application of eliciting users8217 preferences, we investigate the problem of learning hemimetrics, i. e. pairwise distances among a set of n items that satisfy triangle inequalities and non-negativity constraints. In our application, the (asymmetric) distances quantify private costs a user incurs when substituting one item by another. We aim to learn these distances (costs) by asking the users whether they are willing to switch from one item to another for a given incentive offer. Without exploiting structural constraints of the hemimetric polytope, learning the distances between each pair of items requires Theta(n2) queries. We propose an active learning algorithm that substantially reduces this sample complexity by exploiting the structural constraints on the version space of hemimetrics. Our proposed algorithm achieves provably-optimal sample complexity for various instances of the task. For example, when the items are embedded into K tight clusters, the sample complexity of our algorithm reduces to O(n K). Extensive experiments on a restaurant recommendation data set support the conclusions of our theoretical analysis. We present an approach for learning simple algorithms such as copying, multi-digit addition and single digit multiplication directly from examples. Our framework consists of a set of interfaces, accessed by a controller. Typical interfaces are 1-D tapes or 2-D grids that hold the input and output data. For the controller, we explore a range of neural network-based models which vary in their ability to abstract the underlying algorithm from training instances and generalize to test examples with many thousands of digits. The controller is trained using Q-learning with several enhancements and we show that the bottleneck is in the capabilities of the controller rather than in the search incurred by Q-learning. Learning Physical Intuition of Block Towers by Example Adam Lerer Facebook AI Research . Sam Gross Facebook AI Research . Rob Fergus Facebook AI Research Paper AbstractWooden blocks are a common toy for infants, allowing them to develop motor skills and gain intuition about the physical behavior of the world. In this paper, we explore the ability of deep feed-forward models to learn such intuitive physics. Using a 3D game engine, we create small towers of wooden blocks whose stability is randomized and render them collapsing (or remaining upright). This data allows us to train large convolutional network models which can accurately predict the outcome, as well as estimating the trajectories of the blocks. The models are also able to generalize in two important ways: (i) to new physical scenarios, e. g. towers with an additional block and (ii) to images of real wooden blocks, where it obtains a performance comparable to human subjects. Structure Learning of Partitioned Markov Networks Song Liu The Inst. of Stats. Math. . Taiji Suzuki . Masashi Sugiyama University of Tokyo . Kenji Fukumizu The Institute of Statistical Mathematics Paper AbstractWe learn the structure of a Markov Network between two groups of random variables from joint observations. Since modelling and learning the full MN structure may be hard, learning the links between two groups directly may be a preferable option. We introduce a novel concept called the emph whose factorization directly associates with the Markovian properties of random variables across two groups. A simple one-shot convex optimization procedure is proposed for learning the emph factorizations of the partitioned ratio and it is theoretically guaranteed to recover the correct inter-group structure under mild conditions. The performance of the proposed method is experimentally compared with the state of the art MN structure learning methods using ROC curves. Real applications on analyzing bipartisanship in US congress and pairwise DNAtime-series alignments are also reported. This work focuses on dynamic regret of online convex optimization that compares the performance of online learning to a clairvoyant who knows the sequence of loss functions in advance and hence selects the minimizer of the loss function at each step. By assuming that the clairvoyant moves slowly (i. e. the minimizers change slowly), we present several improved variation-based upper bounds of the dynamic regret under the true and noisy gradient feedback, which are in light of the presented lower bounds. The key to our analysis is to explore a regularity metric that measures the temporal changes in the clairvoyant8217s minimizers, to which we refer as path variation. Firstly, we present a general lower bound in terms of the path variation, and then show that under full information or gradient feedback we are able to achieve an optimal dynamic regret. Secondly, we present a lower bound with noisy gradient feedback and then show that we can achieve optimal dynamic regrets under a stochastic gradient feedback and two-point bandit feedback. Moreover, for a sequence of smooth loss functions that admit a small variation in the gradients, our dynamic regret under the two-point bandit feedback matches that is achieved with full information. Beyond CCA: Moment Matching for Multi-View Models Anastasia Podosinnikova INRIA 8211 ENS . Francis Bach Inria . Simon Lacoste-Julien INRIA Paper AbstractWe introduce three novel semi-parametric extensions of probabilistic canonical correlation analysis with identifiability guarantees. We consider moment matching techniques for estimation in these models. For that, by drawing explicit links between the new models and a discrete version of independent component analysis (DICA), we first extend the DICA cumulant tensors to the new discrete version of CCA. By further using a close connection with independent component analysis, we introduce generalized covariance matrices, which can replace the cumulant tensors in the moment matching framework, and, therefore, improve sample complexity and simplify derivations and algorithms significantly. As the tensor power method or orthogonal joint diagonalization are not applicable in the new setting, we use non-orthogonal joint diagonalization techniques for matching the cumulants. We demonstrate performance of the proposed models and estimation techniques on experiments with both synthetic and real datasets. We present two computationally inexpensive techniques for estimating the numerical rank of a matrix, combining powerful tools from computational linear algebra. These techniques exploit three key ingredients. The first is to approximate the projector on the non-null invariant subspace of the matrix by using a polynomial filter. Two types of filters are discussed, one based on Hermite interpolation and the other based on Chebyshev expansions. The second ingredient employs stochastic trace estimators to compute the rank of this wanted eigen-projector, which yields the desired rank of the matrix. In order to obtain a good filter, it is necessary to detect a gap between the eigenvalues that correspond to noise and the relevant eigenvalues that correspond to the non-null invariant subspace. The third ingredient of the proposed approaches exploits the idea of spectral density, popular in physics, and the Lanczos spectroscopic method to locate this gap. Unsupervised Deep Embedding for Clustering Analysis Junyuan Xie University of Washington . Ross Girshick Facebook . Ali Farhadi University of Washington Paper AbstractClustering is central to many data-driven application domains and has been studied extensively in terms of distance functions and grouping algorithms. Relatively little work has focused on learning representations for clustering. In this paper, we propose Deep Embedded Clustering (DEC), a method that simultaneously learns feature representations and cluster assignments using deep neural networks. DEC learns a mapping from the data space to a lower-dimensional feature space in which it iteratively optimizes a clustering objective. Our experimental evaluations on image and text corpora show significant improvement over state-of-the-art methods. Dimensionality reduction is a popular approach for dealing with high dimensional data that leads to substantial computational savings. Random projections are a simple and effective method for universal dimensionality reduction with rigorous theoretical guarantees. In this paper, we theoretically study the problem of differentially private empirical risk minimization in the projected subspace (compressed domain). Empirical risk minimization (ERM) is a fundamental technique in statistical machine learning that forms the basis for various learning algorithms. Starting from the results of Chaudhuri et al. (NIPS 2009, JMLR 2011), there is a long line of work in designing differentially private algorithms for empirical risk minimization problems that operate in the original data space. We ask: is it possible to design differentially private algorithms with small excess risk given access to only projected data In this paper, we answer this question in affirmative, by showing that for the class of generalized linear functions, we can obtain excess risk bounds of O(w(Theta) n ) under eps-differential privacy, and O((w(Theta)n) ) under (eps, delta)-differential privacy, given only the projected data and the projection matrix. Here n is the sample size and w(Theta) is the Gaussian width of the parameter space that we optimize over. Our strategy is based on adding noise for privacy in the projected subspace and then lifting the solution to original space by using high-dimensional estimation techniques. A simple consequence of these results is that, for a large class of ERM problems, in the traditional setting (i. e. with access to the original data), under eps-differential privacy, we improve the worst-case risk bounds of Bassily et al. (FOCS 2014). We consider the maximum likelihood parameter estimation problem for a generalized Thurstone choice model, where choices are from comparison sets of two or more items. We provide tight characterizations of the mean square error, as well as necessary and sufficient conditions for correct classification when each item belongs to one of two classes. These results provide insights into how the estimation accuracy depends on the choice of a generalized Thurstone choice model and the structure of comparison sets. We find that for a priori unbiased structures of comparisons, e. g. when comparison sets are drawn independently and uniformly at random, the number of observations needed to achieve a prescribed estimation accuracy depends on the choice of a generalized Thurstone choice model. For a broad set of generalized Thurstone choice models, which includes all popular instances used in practice, the estimation error is shown to be largely insensitive to the cardinality of comparison sets. On the other hand, we found that there exist generalized Thurstone choice models for which the estimation error decreases much faster with the cardinality of comparison sets. Large-Margin Softmax Loss for Convolutional Neural Networks Weiyang Liu Peking University . Yandong Wen South China University of Technology . Zhiding Yu Carnegie Mellon University . Meng Yang Shenzhen University Paper AbstractCross-entropy loss together with softmax is arguably one of the most common used supervision components in convolutional neural networks (CNNs). Despite its simplicity, popularity and excellent performance, the component does not explicitly encourage discriminative learning of features. In this paper, we propose a generalized large-margin softmax (L-Softmax) loss which explicitly encourages intra-class compactness and inter-class separability between learned features. Moreover, L-Softmax not only can adjust the desired margin but also can avoid overfitting. We also show that the L-Softmax loss can be optimized by typical stochastic gradient descent. Extensive experiments on four benchmark datasets demonstrate that the deeply-learned features with L-softmax loss become more discriminative, hence significantly boosting the performance on a variety of visual classification and verification tasks. A Random Matrix Approach to Echo-State Neural Networks Romain Couillet CentraleSupelec . Gilles Wainrib ENS Ulm, Paris, France . Hafiz Tiomoko Ali CentraleSupelec, Gif-sur-Yvette, France . Harry Sevi ENS Lyon, Lyon, Paris Paper AbstractRecurrent neural networks, especially in their linear version, have provided many qualitative insights on their performance under different configurations. This article provides, through a novel random matrix framework, the quantitative counterpart of these performance results, specifically in the case of echo-state networks. Beyond mere insights, our approach conveys a deeper understanding on the core mechanism under play for both training and testing. One-hot CNN (convolutional neural network) has been shown to be effective for text categorization (Johnson 038 Zhang, 2015). We view it as a special case of a general framework which jointly trains a linear model with a non-linear feature generator consisting of text region embedding pooling8217. Under this framework, we explore a more sophisticated region embedding method using Long Short-Term Memory (LSTM). LSTM can embed text regions of variable (and possibly large) sizes, whereas the region size needs to be fixed in a CNN. We seek effective and efficient use of LSTM for this purpose in the supervised and semi-supervised settings. The best results were obtained by combining region embeddings in the form of LSTM and convolution layers trained on unlabeled data. The results indicate that on this task, embeddings of text regions, which can convey complex concepts, are more useful than embeddings of single words in isolation. We report performances exceeding the previous best results on four benchmark datasets. Crowdsourcing systems are popular for solving large-scale labelling tasks with low-paid (or even non-paid) workers. We study the problem of recovering the true labels from noisy crowdsourced labels under the popular Dawid-Skene model. To address this inference problem, several algorithms have recently been proposed, but the best known guarantee is still significantly larger than the fundamental limit. We close this gap under a simple but canonical scenario where each worker is assigned at most two tasks. In particular, we introduce a tighter lower bound on the fundamental limit and prove that Belief Propagation (BP) exactly matches this lower bound. The guaranteed optimality of BP is the strongest in the sense that it is information-theoretically impossible for any other algorithm to correctly la - bel a larger fraction of the tasks. In the general setting, when more than two tasks are assigned to each worker, we establish the dominance result on BP that it outperforms other existing algorithms with known provable guarantees. Experimental results suggest that BP is close to optimal for all regimes considered, while existing state-of-the-art algorithms exhibit suboptimal performances. Learning control has become an appealing alternative to the derivation of control laws based on classic control theory. However, a major shortcoming of learning control is the lack of performance guarantees which prevents its application in many real-world scenarios. As a step in this direction, we provide a stability analysis tool for controllers acting on dynamics represented by Gaussian processes (GPs). We consider arbitrary Markovian control policies and system dynamics given as (i) the mean of a GP, and (ii) the full GP distribution. For the first case, our tool finds a state space region, where the closed-loop system is provably stable. In the second case, it is well known that infinite horizon stability guarantees cannot exist. Instead, our tool analyzes finite time stability. Empirical evaluations on simulated benchmark problems support our theoretical results. Learning a classifier from private data distributed across multiple parties is an important problem that has many potential applications. How can we build an accurate and differentially private global classifier by combining locally-trained classifiers from different parties, without access to any partys private data We propose to transfer the knowledge of the local classifier ensemble by first creating labeled data from auxiliary unlabeled data, and then train a global differentially private classifier. We show that majority voting is too sensitive and therefore propose a new risk weighted by class probabilities estimated from the ensemble. Relative to a non-private solution, our private solution has a generalization error bounded by O(epsilon M ). This allows strong privacy without performance loss when the number of participating parties M is large, such as in crowdsensing applications. We demonstrate the performance of our framework with realistic tasks of activity recognition, network intrusion detection, and malicious URL detection. Network Morphism Tao Wei University at Buffalo . Changhu Wang Microsoft Research . Yong Rui Microsoft Research . Chang Wen Chen Paper AbstractWe present a systematic study on how to morph a well-trained neural network to a new one so that its network function can be completely preserved. We define this as network morphism in this research. After morphing a parent network, the child network is expected to inherit the knowledge from its parent network and also has the potential to continue growing into a more powerful one with much shortened training time. The first requirement for this network morphism is its ability to handle diverse morphing types of networks, including changes of depth, width, kernel size, and even subnet. To meet this requirement, we first introduce the network morphism equations, and then develop novel morphing algorithms for all these morphing types for both classic and convolutional neural networks. The second requirement is its ability to deal with non-linearity in a network. We propose a family of parametric-activation functions to facilitate the morphing of any continuous non-linear activation neurons. Experimental results on benchmark datasets and typical neural networks demonstrate the effectiveness of the proposed network morphism scheme. Second-order optimization methods such as natural gradient descent have the potential to speed up training of neural networks by correcting for the curvature of the loss function. Unfortunately, the exact natural gradient is impractical to compute for large models, and most approximations either require an expensive iterative procedure or make crude approximations to the curvature. We present Kronecker Factors for Convolution (KFC), a tractable approximation to the Fisher matrix for convolutional networks based on a structured probabilistic model for the distribution over backpropagated derivatives. Similarly to the recently proposed Kronecker-Factored Approximate Curvature (K-FAC), each block of the approximate Fisher matrix decomposes as the Kronecker product of small matrices, allowing for efficient inversion. KFC captures important curvature information while still yielding comparably efficient updates to stochastic gradient descent (SGD). We show that the updates are invariant to commonly used reparameterizations, such as centering of the activations. In our experiments, approximate natural gradient descent with KFC was able to train convolutional networks several times faster than carefully tuned SGD. Furthermore, it was able to train the networks in 10-20 times fewer iterations than SGD, suggesting its potential applicability in a distributed setting. Budget constrained optimal design of experiments is a classical problem in statistics. Although the optimal design literature is very mature, few efficient strategies are available when these design problems appear in the context of sparse linear models commonly encountered in high dimensional machine learning and statistics. In this work, we study experimental design for the setting where the underlying regression model is characterized by a ell1-regularized linear function. We propose two novel strategies: the first is motivated geometrically whereas the second is algebraic in nature. We obtain tractable algorithms for this problem and also hold for a more general class of sparse linear models. We perform an extensive set of experiments, on benchmarks and a large multi-site neuroscience study, showing that the proposed models are effective in practice. The latter experiment suggests that these ideas may play a small role in informing enrollment strategies for similar scientific studies in the short-to-medium term future. Minding the Gaps for Block Frank-Wolfe Optimization of Structured SVMs Anton Osokin . Jean-Baptiste Alayrac ENS . Isabella Lukasewitz INRIA . Puneet Dokania INRIA and Ecole Centrale Paris . Simon Lacoste-Julien INRIA Paper AbstractIn this paper, we propose several improvements on the block-coordinate Frank-Wolfe (BCFW) algorithm from Lacoste-Julien et al. (2013) recently used to optimize the structured support vector machine (SSVM) objective in the context of structured prediction, though it has wider applications. The key intuition behind our improvements is that the estimates of block gaps maintained by BCFW reveal the block suboptimality that can be used as an adaptive criterion. First, we sample objects at each iteration of BCFW in an adaptive non-uniform way via gap-based sampling. Second, we incorporate pairwise and away-step variants of Frank-Wolfe into the block-coordinate setting. Third, we cache oracle calls with a cache-hit criterion based on the block gaps. Fourth, we provide the first method to compute an approximate regularization path for SSVM. Finally, we provide an exhaustive empirical evaluation of all our methods on four structured prediction datasets. Exact Exponent in Optimal Rates for Crowdsourcing Chao Gao Yale University . Yu Lu Yale University . Dengyong Zhou Microsoft Research Paper AbstractCrowdsourcing has become a popular tool for labeling large datasets. This paper studies the optimal error rate for aggregating crowdsourced labels provided by a collection of amateur workers. Under the Dawid-Skene probabilistic model, we establish matching upper and lower bounds with an exact exponent mI(pi), where m is the number of workers and I(pi) is the average Chernoff information that characterizes the workers8217 collective ability. Such an exact characterization of the error exponent allows us to state a precise sample size requirement m ge frac logfrac in order to achieve an epsilon misclassification error. In addition, our results imply optimality of various forms of EM algorithms given accurate initializers of the model parameters. Unsupervised learning and supervised learning are key research topics in deep learning. However, as high-capacity supervised neural networks trained with a large amount of labels have achieved remarkable success in many computer vision tasks, the availability of large-scale labeled images reduced the significance of unsupervised learning. Inspired by the recent trend toward revisiting the importance of unsupervised learning, we investigate joint supervised and unsupervised learning in a large-scale setting by augmenting existing neural networks with decoding pathways for reconstruction. First, we demonstrate that the intermediate activations of pretrained large-scale classification networks preserve almost all the information of input images except a portion of local spatial details. Then, by end-to-end training of the entire augmented architecture with the reconstructive objective, we show improvement of the network performance for supervised tasks. We evaluate several variants of autoencoders, including the recently proposed 8220what-where8221 autoencoder that uses the encoder pooling switches, to study the importance of the architecture design. Taking the 16-layer VGGNet trained under the ImageNet ILSVRC 2012 protocol as a strong baseline for image classification, our methods improve the validation-set accuracy by a noticeable margin. (LRR) has been a significant method for segmenting data that are generated from a union of subspaces. It is also known that solving LRR is challenging in terms of time complexity and memory footprint, in that the size of the nuclear norm regularized matrix is n-by-n (where n is the number of samples). In this paper, we thereby develop a novel online implementation of LRR that reduces the memory cost from O(n2) to O(pd), with p being the ambient dimension and d being some estimated rank (d 20 reduction in the model size without any loss in accuracy on CIFAR-10 benchmark. We also demonstrate that fine-tuning can further enhance the accuracy of fixed point DCNs beyond that of the original floating point model. In doing so, we report a new state-of-the-art fixed point performance of 6.78 error-rate on CIFAR-10 benchmark. Provable Algorithms for Inference in Topic Models Sanjeev Arora Princeton University . Rong Ge . Frederic Koehler Princeton University . Tengyu Ma Princeton University . Ankur Moitra Paper AbstractRecently, there has been considerable progress on designing algorithms with provable guarantees 8212typically using linear algebraic methods8212for parameter learning in latent variable models. Designing provable algorithms for inference has proved more difficult. Here we tak e a first step towards provable inference in topic models. We leverage a property of topic models that enables us to construct simple linear estimators for the unknown topic proportions that have small variance, and consequently can work with short documents. Our estimators also correspond to finding an estimate around which the posterior is well-concentrated. We show lower bounds that for shorter documents it can be information theoretically impossible to find the hidden topics. Finally, we give empirical results that demonstrate that our algorithm works on realistic topic models. It yields good solutions on synthetic data and runs in time comparable to a single iteration of Gibbs sampling. This paper develops an approach for efficiently solving general convex optimization problems specified as disciplined convex programs (DCP), a common general-purpose modeling framework. Specifically we develop an algorithm based upon fast epigraph projections, projections onto the epigraph of a convex function, an approach closely linked to proximal operator methods. We show that by using these operators, we can solve any disciplined convex program without transforming the problem to a standard cone form, as is done by current DCP libraries. We then develop a large library of efficient epigraph projection operators, mirroring and extending work on fast proximal algorithms, for many common convex functions. Finally, we evaluate the performance of the algorithm, and show it often achieves order of magnitude speedups over existing general-purpose optimization solvers. We study the fixed design segmented regression problem: Given noisy samples from a piecewise linear function f, we want to recover f up to a desired accuracy in mean-squared error. Previous rigorous approaches for this problem rely on dynamic programming (DP) and, while sample efficient, have running time quadratic in the sample size. As our main contribution, we provide new sample near-linear time algorithms for the problem that 8211 while not being minimax optimal 8211 achieve a significantly better sample-time tradeoff on large datasets compared to the DP approach. Our experimental evaluation shows that, compared with the DP approach, our algorithms provide a convergence rate that is only off by a factor of 2 to 4, while achieving speedups of three orders of magnitude. Energetic Natural Gradient Descent Philip Thomas CMU . Bruno Castro da Silva . Christoph Dann Carnegie Mellon University . Emma Paper AbstractWe propose a new class of algorithms for minimizing or maximizing functions of parametric probabilistic models. These new algorithms are natural gradient algorithms that leverage more information than prior methods by using a new metric tensor in place of the commonly used Fisher information matrix. This new metric tensor is derived by computing directions of steepest ascent where the distance between distributions is measured using an approximation of energy distance (as opposed to Kullback-Leibler divergence, which produces the Fisher information matrix), and so we refer to our new ascent direction as the energetic natural gradient. Partition Functions from Rao-Blackwellized Tempered Sampling David Carlson Columbia University . Patrick Stinson Columbia University . Ari Pakman Columbia University . Liam Paper AbstractPartition functions of probability distributions are important quantities for model evaluation and comparisons. We present a new method to compute partition functions of complex and multimodal distributions. Such distributions are often sampled using simulated tempering, which augments the target space with an auxiliary inverse temperature variable. Our method exploits the multinomial probability law of the inverse temperatures, and provides estimates of the partition function in terms of a simple quotient of Rao-Blackwellized marginal inverse temperature probability estimates, which are updated while sampling. We show that the method has interesting connections with several alternative popular methods, and offers some significant advantages. In particular, we empirically find that the new method provides more accurate estimates than Annealed Importance Sampling when calculating partition functions of large Restricted Boltzmann Machines (RBM) moreover, the method is sufficiently accurate to track training and validation log-likelihoods during learning of RBMs, at minimal computational cost. In this paper we address the identifiability and efficient learning problems of finite mixtures of Plackett-Luce models for rank data. We prove that for any kgeq 2, the mixture of k Plackett-Luce models for no more than 2k-1 alternatives is non-identifiable and this bound is tight for k2. For generic identifiability, we prove that the mixture of k Plackett-Luce models over m alternatives is if kleqlfloorfrac 2rfloor. We also propose an efficient generalized method of moments (GMM) algorithm to learn the mixture of two Plackett-Luce models and show that the algorithm is consistent. Our experiments show that our GMM algorithm is significantly faster than the EMM algorithm by Gormley 038 Murphy (2008), while achieving competitive statistical efficiency. The combinatorial explosion that plagues planning and reinforcement learning (RL) algorithms can be moderated using state abstraction. Prohibitively large task representations can be condensed such that essential information is preserved, and consequently, solutions are tractably computable. However, exact abstractions, which treat only fully-identical situations as equivalent, fail to present opportunities for abstraction in environments where no two situations are exactly alike. In this work, we investigate approximate state abstractions, which treat nearly-identical situations as equivalent. We present theoretical guarantees of the quality of behaviors derived from four types of approximate abstractions. Additionally, we empirically demonstrate that approximate abstractions lead to reduction in task complexity and bounded loss of optimality of behavior in a variety of environments. Power of Ordered Hypothesis Testing Lihua Lei Lihua . William Fithian UC Berkeley, Department of Statistics Paper AbstractOrdered testing procedures are multiple testing procedures that exploit a pre-specified ordering of the null hypotheses, from most to least promising. We analyze and compare the power of several recent proposals using the asymptotic framework of Li 038 Barber (2015). While accumulation tests including ForwardStop can be quite powerful when the ordering is very informative, they are asymptotically powerless when the ordering is weaker. By contrast, Selective SeqStep, proposed by Barber 038 Candes (2015), is much less sensitive to the quality of the ordering. We compare the power of these procedures in different regimes, concluding that Selective SeqStep dominates accumulation tests if either the ordering is weak or non-null hypotheses are sparse or weak. Motivated by our asymptotic analysis, we derive an improved version of Selective SeqStep which we call Adaptive SeqStep, analogous to Storeys improvement on the Benjamini-Hochberg proce - dure. We compare these methods using the GEO-Query data set analyzed by (Li 038 Barber, 2015) and find Adaptive SeqStep has favorable performance for both good and bad prior orderings. PHOG: Probabilistic Model for Code Pavol Bielik ETH Zurich . Veselin Raychev ETH Zurich . Martin Vechev ETH Zurich Paper AbstractWe introduce a new generative model for code called probabilistic higher order grammar (PHOG). PHOG generalizes probabilistic context free grammars (PCFGs) by allowing conditioning of a production rule beyond the parent non-terminal, thus capturing rich contexts relevant to programs. Even though PHOG is more powerful than a PCFG, it can be learned from data just as efficiently. We trained a PHOG model on a large JavaScript code corpus and show that it is more precise than existing models, while similarly fast. As a result, PHOG can immediately benefit existing programming tools based on probabilistic models of code. We consider the problem of online prediction in changing environments. In this framework the performance of a predictor is evaluated as the loss relative to an arbitrarily changing predictor, whose individual components come from a base class of predictors. Typical results in the literature consider different base classes (experts, linear predictors on the simplex, etc.) separately. Introducing an arbitrary mapping inside the mirror decent algorithm, we provide a framework that unifies and extends existing results. As an example, we prove new shifting regret bounds for matrix prediction problems. Hyperparameter selection generally relies on running multiple full training trials, with selection based on validation set performance. We propose a gradient-based approach for locally adjusting hyperparameters during training of the model. Hyperparameters are adjusted so as to make the model parameter gradients, and hence updates, more advantageous for the validation cost. We explore the approach for tuning regularization hyperparameters and find that in experiments on MNIST, SVHN and CIFAR-10, the resulting regularization levels are within the optimal regions. The additional computational cost depends on how frequently the hyperparameters are trained, but the tested scheme adds only 30 computational overhead regardless of the model size. Since the method is significantly less computationally demanding compared to similar gradient-based approaches to hyperparameter optimization, and consistently finds good hyperparameter values, it can be a useful tool for training neural network models. Many of the recent Trajectory Optimization algorithms alternate between local approximation of the dynamics and conservative policy update. However, linearly approximating the dynamics in order to derive the new policy can bias the update and prevent convergence to the optimal policy. In this article, we propose a new model-free algorithm that backpropagates a local quadratic time-dependent Q-Function, allowing the derivation of the policy update in closed form. Our policy update ensures exact KL-constraint satisfaction without simplifying assumptions on the system dynamics demonstrating improved performance in comparison to related Trajectory Optimization algorithms linearizing the dynamics. Due to its numerous applications, rank aggregation has become a problem of major interest across many fields of the computer science literature. In the vast majority of situations, Kemeny consensus(es) are considered as the ideal solutions. It is however well known that their computation is NP-hard. Many contributions have thus established various results to apprehend this complexity. In this paper we introduce a practical method to predict, for a ranking and a dataset, how close the Kemeny consensus(es) are to this ranking. A major strength of this method is its generality: it does not require any assumption on the dataset nor the ranking. Furthermore, it relies on a new geometric interpretation of Kemeny aggregation that, we believe, could lead to many other results. Horizontally Scalable Submodular Maximization Mario Lucic ETH Zurich . Olivier Bachem ETH Zurich . Morteza Zadimoghaddam Google Research . Andreas Krause Paper AbstractA variety of large-scale machine learning problems can be cast as instances of constrained submodular maximization. Existing approaches for distributed submodular maximization have a critical drawback: The capacity 8211 number of instances that can fit in memory 8211 must grow with the data set size. In practice, while one can provision many machines, the capacity of each machine is limited by physical constraints. We propose a truly scalable approach for distributed submodular maximization under fixed capacity. The proposed framework applies to a broad class of algorithms and constraints and provides theoretical guarantees on the approximation factor for any available capacity. We empirically evaluate the proposed algorithm on a variety of data sets and demonstrate that it achieves performance competitive with the centralized greedy solution. Group Equivariant Convolutional Networks Taco Cohen University of Amsterdam . Max Welling University of Amsterdam CIFAR Paper AbstractWe introduce Group equivariant Convolutional Neural Networks (G-CNNs), a natural generalization of convolutional neural networks that reduces sample complexity by exploiting symmetries. G-CNNs use G-convolutions, a new type of layer that enjoys a substantially higher degree of weight sharing than regular convolution layers. G-convolutions increase the expressive capacity of the network without increasing the number of parameters. Group convolution layers are easy to use and can be implemented with negligible computational overhead for discrete groups generated by translations, reflections and rotations. G-CNNs achieve state of the art results on CIFAR10 and rotated MNIST. The partition function is fundamental for probabilistic graphical models8212it is required for inference, parameter estimation, and model selection. Evaluating this function corresponds to discrete integration, namely a weighted sum over an exponentially large set. This task quickly becomes intractable as the dimensionality of the problem increases. We propose an approximation scheme that, for any discrete graphical model whose parameter vector has bounded norm, estimates the partition function with arbitrarily small error. Our algorithm relies on a near minimax optimal polynomial approximation to the potential function and a Clenshaw-Curtis style quadrature. Furthermore, we show that this algorithm can be randomized to split the computation into a high-complexity part and a low-complexity part, where the latter may be carried out on small computational devices. Experiments confirm that the new randomized algorithm is highly accurate if the parameter norm is small, and is otherwise comparable to methods with unbounded error. Correcting Forecasts with Multifactor Neural Attention Matthew Riemer IBM . Aditya Vempaty IBM . Flavio Calmon IBM . Fenno Heath IBM . Richard Hull IBM . Elham Khabiri IBM Paper AbstractAutomatic forecasting of time series data is a challenging problem in many industries. Current forecast models adopted by businesses do not provide adequate means for including data representing external factors that may have a significant impact on the time series, such as weather, national events, local events, social media trends, promotions, etc. This paper introduces a novel neural network attention mechanism that naturally incorporates data from multiple external sources without the feature engineering needed to get other techniques to work. We demonstrate empirically that the proposed model achieves superior performance for predicting the demand of 20 commodities across 107 stores of one of America8217s largest retailers when compared to other baseline models, including neural networks, linear models, certain kernel methods, Bayesian regression, and decision trees. Our method ultimately accounts for a 23.9 relative improvement as a result of the incorporation of external data sources, and provides an unprecedented level of descriptive ability for a neural network forecasting model. Observational studies are rising in importance due to the widespread accumulation of data in fields such as healthcare, education, employment and ecology. We consider the task of answering counterfactual questions such as, 8220Would this patient have lower blood sugar had she received a different medication8221. We propose a new algorithmic framework for counterfactual inference which brings together ideas from domain adaptation and representation learning. In addition to a theoretical justification, we perform an empirical comparison with previous approaches to causal inference from observational data. Our deep learning algorithm significantly outperforms the previous state-of-the-art. Gaussian Processes (GPs) provide a general and analytically tractable way of modeling complex time-varying, nonparametric functions. The Automatic Bayesian Covariance Discovery (ABCD) system constructs natural-language description of time-series data by treating unknown time-series data nonparametrically using GP with a composite covariance kernel function. Unfortunately, learning a composite covariance kernel with a single time-series data set often results in less informative kernel that may not give qualitative, distinctive descriptions of data. We address this challenge by proposing two relational kernel learning methods which can model multiple time-series data sets by finding common, shared causes of changes. We show that the relational kernel learning methods find more accurate models for regression problems on several real-world data sets US stock data, US house price index data and currency exchange rate data. We introduce a new approach for amortizing inference in directed graphical models by learning heuristic approximations to stochastic inverses, designed specifically for use as proposal distributions in sequential Monte Carlo methods. We describe a procedure for constructing and learning a structured neural network which represents an inverse factorization of the graphical model, resulting in a conditional density estimator that takes as input particular values of the observed random variables, and returns an approximation to the distribution of the latent variables. This recognition model can be learned offline, independent from any particular dataset, prior to performing inference. The output of these networks can be used as automatically-learned high-quality proposal distributions to accelerate sequential Monte Carlo across a diverse range of problem settings. Slice Sampling on Hamiltonian Trajectories Benjamin Bloem-Reddy Columbia University . John Cunningham Columbia University Paper AbstractHamiltonian Monte Carlo and slice sampling are amongst the most widely used and studied classes of Markov Chain Monte Carlo samplers. We connect these two methods and present Hamiltonian slice sampling, which allows slice sampling to be carried out along Hamiltonian trajectories, or transformations thereof. Hamiltonian slice sampling clarifies a class of model priors that induce closed-form slice samplers. More pragmatically, inheriting properties of slice samplers, it offers advantages over Hamiltonian Monte Carlo, in that it has fewer tunable hyperparameters and does not require gradient information. We demonstrate the utility of Hamiltonian slice sampling out of the box on problems ranging from Gaussian process regression to Pitman-Yor based mixture models. Noisy Activation Functions Caglar Glehre . Marcin Moczulski . Misha Denil . Yoshua Bengio U. of Montreal Paper AbstractCommon nonlinear activation functions used in neural networks can cause training difficulties due to the saturation behavior of the activation function, which may hide dependencies that are not visible to vanilla-SGD (using first order gradients only). Gating mechanisms that use softly saturating activation functions to emulate the discrete switching of digital logic circuits are good examples of this. We propose to exploit the injection of appropriate noise so that the gradients may flow easily, even if the noiseless application of the activation function would yield zero gradients. Large noise will dominate the noise-free gradient and allow stochastic gradient descent to explore more. By adding noise only to the problematic parts of the activation function, we allow the optimization procedure to explore the boundary between the degenerate saturating) and the well-behaved parts of the activation function. We also establish connections to simulated annealing, when the amount of noise is annealed down, making it easier to optimize hard objective functions. We find experimentally that replacing such saturating activation functions by noisy variants helps optimization in many contexts, yielding state-of-the-art or competitive results on different datasets and task, especially when training seems to be the most difficult, e. g. when curriculum learning is necessary to obtain good results. PD-Sparse. A Primal and Dual Sparse Approach to Extreme Multiclass and Multilabel Classification Ian En-Hsu Yen University of Texas at Austin . Xiangru Huang UTaustin . Pradeep Ravikumar UT Austin . Kai Zhong ICES department, University of Texas at Austin . Inderjit Paper AbstractWe consider Multiclass and Multilabel classification with extremely large number of classes, of which only few are labeled to each instance. In such setting, standard methods that have training, prediction cost linear to the number of classes become intractable. State-of-the-art methods thus aim to reduce the complexity by exploiting correlation between labels under assumption that the similarity between labels can be captured by structures such as low-rank matrix or balanced tree. However, as the diversity of labels increases in the feature space, structural assumption can be easily violated, which leads to degrade in the testing performance. In this work, we show that a margin-maximizing loss with l1 penalty, in case of Extreme Classification, yields extremely sparse solution both in primal and in dual without sacrificing the expressive power of predictor. We thus propose a Fully-Corrective Block-Coordinate Frank-Wolfe (FC-BCFW) algorithm that exploits both primal and dual sparsity to achieve a complexity sublinear to the number of primal and dual variables. A bi-stochastic search method is proposed to further improve the efficiency. In our experiments on both Multiclass and Multilabel problems, the proposed method achieves significant higher accuracy than existing approaches of Extreme Classification with very competitive training and prediction time.
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