Comparison-based Choices

Transcription

1 Comprison-se Choies John Ugner Mngement Siene & Engineering Stnfor University Joint work with: Jon Kleinerg (Cornell) Senhil Mullinthn (Hrvr) EC 17 Boston June 28, 2017

2 Preiting isrete hoies Clssi prolem: onsumer preferenes [Thurstone 27, Lue 59], ommuting [MFen 78], shool hoie [Kohn-Mnski-Munel 76]

3 Preiting online isrete hoies

4 Preiting online isrete hoies How well n we lern/preit hoie set effets?.k.. violtions of the inepenene of irrelevnt lterntives (IIA) [Sheffet-Mishr-Ieong ICML 2012, Yin et l. WSDM 2014]

5 Choie set effets Bis towrs moertion, ompromise effet [Simonson 1989, Simonson-Tversky 1992, Kmeni 2008, Trueloo 2013]

6 Choie set effets Bis towrs moertion, ompromise effet megpixels weight [Simonson 1989, Simonson-Tversky 1992, Kmeni 2008, Trueloo 2013]

7 Choie set effets Bis towrs moertion, ompromise effet megpixels weight [Simonson 1989, Simonson-Tversky 1992, Kmeni 2008, Trueloo 2013]

8 Choie set effets Bis towrs moertion, ompromise effet megpixels Similrity version weight [Simonson 1989, Simonson-Tversky 1992, Kmeni 2008, Trueloo 2013] megpixels weight

9 Choie set effets Bis towrs moertion, ompromise effet megpixels Similrity version weight [Simonson 1989, Simonson-Tversky 1992, Kmeni 2008, Trueloo 2013] megpixels weight

10 Choie set effets Bis towrs moertion, ompromise effet megpixels Orinl omprisons Similrity version [Simonson 1989, Simonson-Tversky 1992, Kmeni 2008, Trueloo 2013] megpixels weight weight Similrity requires istne

11 The present work Fouse on omprison-se funtions. Investigte symptoti query omplexity: if n gent mkes omprison-se hoies, how hr to lern their hoie funtion? Assume popultion is not lerning, mening hoie set effets re not trnsient irrtionlity. Severl query frmeworks: Ative queries vs. pssive strem of queries Fixe hoie funtion vs. mixture of hoie funtions

12 The present work Fouse on omprison-se funtions. Investigte symptoti query omplexity: if n gent mkes omprison-se hoies, how hr to lern their hoie funtion? Assume popultion is not lerning, mening hoie set effets re not trnsient irrtionlity. Severl query frmeworks: Ative queries vs. pssive strem of queries Fixe hoie funtion vs. mixture of hoie funtions Bsi tkewy: omprison-se funtions in one imension (still rih!) re no hrer to lern thn inry omprisons (sorting).

13 Comprison-se hoie funtions Definition: Given set of lterntives U, hoie funtion f mps every non-empty S U to n element u S. Exmple: S U: f( ) = u

14 Comprison-se hoie funtions Definition: Given set of lterntives U, hoie funtion f mps every non-empty S U to n element u S. Exmple: S U: f( ) = u Emeing items: Consier U s emee in ttriute spe, h:u->x For X = R 1, h(ui) re utilities: e

15 Comprison-se hoie funtions Definition: Given set of lterntives U, hoie funtion f mps every non-empty S U to n element u S. Exmple: S U: f( ) = u Emeing items: Consier U s emee in ttriute spe, h:u->x For X = R 1, h(ui) re utilities: Comprison-se funtions: Definition: Choie funtions tht n e written s omprisons (<,>,=) over {h(ui): ui S}. e

16 Comprison-se hoie funtions In one imension, omprison-se funtions re ll position-seletion funtions: selet l-of-k. Exmple: k=4, l=2 f(s) =

17 Comprison-se hoie funtions In one imension, omprison-se funtions re ll position-seletion funtions: selet l-of-k. Exmple: k=4, l=2 f(s) = Seleting 1-of-2 is sorting. Fous on k-sets S with fixe k.

18 Comprison-se hoie funtions In one imension, omprison-se funtions re ll position-seletion funtions: selet l-of-k. Exmple: k=4, l=2 f(s) = f(s) = e e Seleting 1-of-2 is sorting. Fous on k-sets S with fixe k. Position-seletion funtions exhiit hoie set effets.

19 Query omplexity Oserve sequene of (hoie set, hoie) pirs (S, f(s)). How mny o we nee to oserve to report f(s) for (lmost) ll S?

20 Query omplexity Oserve sequene of (hoie set, hoie) pirs (S, f(s)). How mny o we nee to oserve to report f(s) for (lmost) ll S? Ative vs. pssive queries Ative: n hoose wht k-set S to query next, sequentilly. Pssive: Strem of rnom k-sets S. Fixe vs. mixe hoie funtions Fixe: ll queries of sme `-of-k funtion. Mixe: mixture ( 1,..., k ) of ifferent positions selete.

21 Query omplexity, inry hoies How oes sorting (1-of-2) fit in this query omplexity frmework? Mixe inry hoie funtions mp to (p,1-p) noisy sorting. Ative Fixe Sorting from omprisons O(n log n) Mixe Sorting with noisy omprisons (Feige et l. 1994) O(n log n) Pssive Sorting in one roun (Alon-Azr 1988) O(n log n loglog n)?

22 Query omplexity, k-set hoies Sorting results trnslte to position-seletion funtions: Ative Fixe Two-phse lgorithm O(n log n) Mixe Apttion of two-phse lgorithm O(n log n) Pssive Streming moel O(n k-1 log n loglog n)?

23 Query omplexity: tive, fixe Phse 1: fin ineligile lterntives vi isr lgorithm f(s) = = ineligile lterntives S = S 2 = { } { } k ` item(s) ` 1 item(s)

24 Query omplexity: tive, fixe Phse 1: fin ineligile lterntives vi isr lgorithm f(s) = = ineligile lterntives S = S 2 = { } { } k ` item(s) ` 1 item(s) Phse 2: P hoie set with ineligile lterntives, o inry sort.

25 Query omplexity: tive, fixe Phse 1: fin ineligile lterntives vi isr lgorithm f(s) = = ineligile lterntives S = S 2 = { } { } k ` item(s) ` 1 item(s) Phse 2: P hoie set with ineligile lterntives, o inry sort. O(n) queries in isr lgorithm, O(n log n) queries to sort. Only reovers orer, not orienttion: on t know if pe sort is mx or min, ut not neee to reover f(s) for ever S. Algorithm oesn t epen on wht position is eing selete for.

26 Query omplexity: tive, mixe Inste of `-of-k, mixture of positions with proilities ( 1,..., k ), onstnt seprtion. f(s) = 0: Estimte proilities of eh position y stuying k+1-set losely. 1: Run isr phse O(log n) times, fin mx-ineligile lterntives 2: Cn then p hoie set n run noisy mx with (mx, min, fil) outomes inste of (mx, min) outomes s in (Feige et l. 1994).

27 Query omplexity: tive, mixe Inste of `-of-k, mixture of positions with proilities ( 1,..., k ), onstnt seprtion. f(s) = 0: Estimte proilities of eh position y stuying k+1-set losely. 1: Run isr phse O(log n) times, fin mx-ineligile lterntives 2: Cn then p hoie set n run noisy mx with (mx, min, fil) outomes inste of (mx, min) outomes s in (Feige et l. 1994). O(1) queries estimte proilities, O(n log n) queries in isr lgorithm, O(n log n) queries to sort. Nee to ook-keep mny filure proilities, ut stright forwr.

28 Query omplexity: pssive, fixe Pssive query moel: Poisson proess where eh k-set enters the strem with equl rte α. See given k-set in intervl [0,T] with proility pt. How long n intervl [0,T] o we nee to oserve strem? Phse 1: use queries in [0,T1], with T1 lrge enough so tht ll items exept ineligile lterntives re hosen. Phse 2: Simulte pirwise omprisons using queries where k-2 of the elements re ineligile.

29 Query omplexity: pssive, fixe Pssive query moel: Poisson proess where eh k-set enters the strem with equl rte α. See given k-set in intervl [0,T] with proility pt. How long n intervl [0,T] o we nee to oserve strem? Phse 1: use queries in [0,T1], with T1 lrge enough so tht ll items exept ineligile lterntives re hosen. Phse 2: Simulte pirwise omprisons using queries where k-2 of the elements re ineligile. For Phse 2 to work, nee pt to e O(log n loglog n / n). En up seeing ~log(n)/n frtion of ll (n hoose k) hoie sets. For k 3, proof only works for positions 1<l<k, not l=1 or l=k, whih reks our nlysis (pt 0).

30 Query omplexity, k-set hoies Sorting results trnslte to position-seletion funtions: Ative Fixe Two-phse lgorithm O(n log n) Mixe No new iffiulties O(n log n) Pssive Streming moel O(n k-1 log n loglog n)? Immeite questions: Better lgo for pssive strem; sorting in one noisy roun ; higher-im omprison funtions; istne-omprison.

31 Distne-omprison-se hoie Distne-omprison-se funtions re omprison funtions on the set of pirwise istnes.

32 Distne-omprison-se hoie Distne-omprison-se funtions re omprison funtions on the set of pirwise istnes. Distne-omprison vs. omprison funtions re quite ifferent. istne omprison omprison

33 Distne-omprison-se hoie Distne-omprison-se funtions re omprison funtions on the set of pirwise istnes. Distne-omprison vs. omprison funtions re quite ifferent. istne omprison 1D mein omprison Comprison funtions: Cn not express similrity (only orer) Distne-omprison funtions: Cn not mximize or minimize (istnes re ll internl to set)

34 Distne-omprison-se hoie Distne-omprison-se funtions re omprison funtions on the set of pirwise istnes. Pper poses mny questions out istne-omprison, few nswers. Relte to open lerning questions for: Crow mein lgorithm [Heikinheimo-Ukkonen 2013] Stohsti triplet emeing [Vn Der Mten-Weinerger 2012] Crowsoure lustering [Vinyk-Hssii 2016] Metri emeing [Shultz-Johims 2004].

35 Summry Inferene for omprison-se funtions generlly not more iffiult thn sorting. Ative vs. pssive, fixe vs. mixe query omplexity frmeworks. Open questions: Results for high-im (EBA?), istne-omprison, RUMs. Lerning/non-stti gents? Other reent work: [Benson et l. WWW 16] On the relevne of irrelevnt lterntives [Ugner-Rgin, NIPS 16] Mrkov hin moel generlizing BTL/MNL, n violte IIA. [Mystre-Grossgluser ICML 17] For BTL with ~uniform qulity, log 5 (n) inepenent Quiksorts reover ext rnk for lmost ll items. [Peyskhovih-Ugner NetEon 17] Mhine lerning pttion of the Simonson- Tversky moel for ontextul utility.