PRIOR ROBUST OPTIMIZATION. Balasubramanian Sivan. A dissertation submitted in partial fulfillment of the requirements for the degree of

Transcription

1 PRIOR ROBUST OPTIMIZATION By Balasubraanan Svan A dssertaton subtted n partal fulfllent of the requreents for the degree of Doctor of Phlosophy (Coputer Scences) at the UNIVERSITY OF WISCONSIN MADISON 2013 Date of fnal oral exanaton: July 2, 2013 Ths dssertaton s approved by the followng ebers of the Fnal Oral Cottee: Shuch Chawla (Advsor), Assstant Professor, Coputer Scences Erc Bach, Professor, Coputer Scences Jn-Y Ca, Professor, Coputer Scences Marzena Rostek, Assocate Professor, Econocs Jason D. Hartlne, Assocate Professor, Electrcal Engneerng and Coputer Scence, Northwestern Unversty

2 c Copyrght by Balasubraanan Svan 2013 All Rghts Reserved

3 I dedcate ths thess to y parents, Appadura Svan & Anandavall Svan.

4 Acknowledgents I a ndebted to y advsor Prof. Shuch Chawla for shapng y vews on research fro the ground up, and for beng understandng and patent n ths process. Her ablty to convncngly artculate the ert of a research drecton was very effectve n nsprng e to work on a proble, and also n gudng e on how to councate y research n a talk/paper. Shuch s crystal clear descrpton of a research proble and the core arguent to be constructed there n ade research look uch less coplcated than I agned. She was a great and convenent source of nforaton and nsght. Thanks for all the care and attenton you gave e n these past fve years Shuch! Many thanks to Prof. Jason Hartlne for hostng e at Northwestern Unversty several tes and also for vstng Wsconsn often. My ultple vsts to what we graduate students call as Jason Hartlne Suer Research Insttute wll be aong the fondest eores of y PhD lfe. Thanks for the long and frutful collaboraton Jason! I a grateful to Dr. Nkhl Devanur for hostng e as a suer ntern at Mcrosoft Research Redond, and also for several subsequent vsts. A large part of ths thess s based on jont work wth Nkhl whle I was at Redond. I have benefted ensely fro Nkhl s stress-free entorng style, both acadecally and otherwse. Thanks for beng such a joy to work wth Nkhl! Thanks also to Dr. Jennfer Chayes and Dr. Chrstan Borgs for hostng e as a suer nter at Mcrosoft Research New England, and for spendng an hour or ore n weekly eetngs n the dst of ther ja packed schedule! Both acadecally and socally, MSR New England was a real fun place to be, thanks to Dr. Madhu Sudan, Dr. Brendan Lucer, Vasls Syrgkans, Hu Fu, Nck Gravn and Mckey Brautbar. Thanks to all y PhD cottee ebers for ther valuable te: Prof. Erc Bach, Prof. Jn-Y Ca, Prof. Shuch Chawla, Prof. Jason D. Hartlne and Prof. Marzena Rostek, and y prel cottee eber Prof. Deter van Melkebeek. Aong the hghlghts of y stay at Wsconsn are y day long research dscussons wth Davd Malec. My stay at Wsconsn would have been uch less nterestng wthout you Davd. Thanks to all y fellow theory graduate students for both akng the place ore vbrant, and for all the fun stress-bustng dscussons. Thanks to all y coauthors. Specal thanks to Chrs Wlkens and Molly Wlkens for hostng e for one full week at ther resdence n Berkeley when Chrs and I were

5 workng on our Sngle-Call paper. It was really knd of the to have done ths, and the stay was totally fun! I a fortunate to have had Prof. C. Pandu Rangan as y undergraduate entor. Hs nvtng atttude towards research was alost entrely the reason for e decdng to do a PhD. The TCS lab at IIT Madras, whch he anages, could not have been ore vbrant! My chldhood frend and genune well-wsher Suresh, who I a outrageously lucky to have found, was really helpful n antanng y santy n PhD lfe. Our weekly phone calls exchangng suares of lfe n Chenna and Madson (and conference travels, and everythng under the Sun) was qute refreshng. For an nternatonal graduate student lke e n the US, t s qute crucal to have a support syste back n the hoe country wth people lke y aunt and uncle, Subha and Chandrasekharan. Ther extraordnary support durng y dad s surgery, whch happened whle I was n the US and couldn t go back to Inda, was very portant n reovng y restlessness and puttng e at ease. Thanks to the ost recent entrant to y faly, y wfe Varsha, for all the words of encourageent and nspraton, and for fllng y lfe wth joy. Thanks also to y n-laws for ther constant love and concern for y welfare. Thanks to y grandother Parvath for her one of a knd love towards e. The warth and affecton I receved fro her as a chld fored a crucal part of y eotonally secure chldhood. Most portantly, I thank y parents Appadura Svan and Anandavall Svan for supportng e heart and soul n all y endeavors. They have always put y welfare above everythng else n ther lves, and have ade contnued sacrfces n gvng e only the best n everythng. I affectonately dedcate ths thess to the.

6 v Contents Acknowledgents Contents Abstract v v 1 Introducton Pror Robust Optzaton Motvatng Exaples Our Contrbutons Pror Robust Optzaton n Internet Advertsng Pror Robust Revenue Maxzng Aucton Desgn Pror Robust Mechanss for Machne Schedulng Prerequstes and Dependences Bblographc Notes Pror Robust Optzaton n Internet Advertsng Introducton & Suary of Results Prelnares & Man Results Resource Allocaton Fraework Near-Optal Onlne Algorth for Resource Allocaton Asyptotcally Optal Onlne Algorth for Adwords Greedy Algorth for Adwords Fast Approxaton Algorths for Large Mxed Packng and Coverng Integer Progras Near-Optal Pror Robust Onlne Algorths for Resource Allocaton Copletely Known Dstrbutons Unknown Dstrbuton, Known W E Copletely Unknown Dstrbuton Approxate Estatons Adversaral Stochastc Input Proof of Near-Optalty of Onlne Algorth for Resource Allocaton 48

7 v 2.5 Asyptotcally Optal Pror Robust Onlne Algorths for Adwords Saturated Instances: Copletely Known Dstrbuton Saturated Instances: Copletely Unknown Dstrbuton General Instances: Copletely Known Dstrbuton General Instances: Partally Known Dstrbuton Approxate Estatons Proof of Asyptotc Optalty of Onlne Algorth for Adwords Greedy Algorth for Adwords Fast Approxaton Algorth for Large Mxed Packng & Coverng Integer Progras Specal Cases of the Resource Allocaton Fraework Network Routng and Load Balancng Cobnatoral Auctons Adwords and Dsplay Ads Probles Concluson Pror Robust Revenue Maxzng Aucton Desgn Introducton & Suary of Results Prelnares Targeted Advertsng and the Non-..d. Irregular Settng One Extra Bdder fro Every Populaton Group Just One Extra Bdder n Total for Hazard Rate Donant Dstrbutons Non-Targeted Advertsng and the..d. Irregular Settng Vckrey wth Sngle Reserve for Irregular Settngs Deferred Proofs Concluson Pror Robust Mechanss for Machne Schedulng Introducton & Suary of Results Prelnares & Man Results Man Results Probablstc Analyss The Bounded Overload Mechans The Seve and Bounded Overload Mechans

8 v 4.5 Deferred Proofs Concluson Bblography 124

9 v Abstract The focus of ths thess s optzaton n the presence of uncertan nputs. In a broad class of algorthc probles, uncertanty s odeled as nput beng drawn fro one aong a large known unverse of dstrbutons, however the specfc dstrbuton s unknown to the algorth. The goal then s to develop a sngle algorth that for every dstrbuton n ths unverse, perfors approxately as well as the optal algorth talored for that specfc dstrbuton. Such algorths are robust to assuptons on pror dstrbutons. Pror robust optzaton retans the robustness of worst-case analyss whle gong beyond the pessstc possblty results of worst-case analyss. Apart fro ths theoretcal appeal, the ablty to use the sae algorth for every pror dstrbuton akes pror robust algorths well-suted for deployent n real systes. Indeed, ost pror robust algorths n lterature are sple to pleent and soe of the have been observed to perfor well n large-scale systes. In ths thess, we desgn and analyze pror robust algorths n two dstnct areas of research: onlne algorths and echans desgn. In onlne algorths, we use a hybrd arguent to develop near optal onlne algorths for a general fraework of probles, called the resource allocaton fraework, wth several well otvated applcatons to Internet ad servng. In echans desgn, we use saplng and supply ltaton technques to develop pror robust truthful approxately revenue optal auctons, and the frst pror robust truthful echanss for approxate akespan nzaton n achne schedulng.

10 1 1 Introducton 1.1 Pror Robust Optzaton Two aong the fundaental challenges that an algorth desgner for optzaton probles contends wth are: 1. lted coputatonal resources: constraned to develop algorths that use te and space at ost a polynoal n nput sze; 2. lted access to nput: for any probles, obtanng the entre nput before the algorth begns workng on the s ether a costly atter or sply possble. The algorth, therefore, has to to ether bear the cost for fgurng out the nput accurately or ake ts decsons by knowng only parts of the nput. The frst of these challenges has been addressed extensvely n Coputer Scence lterature va the theory of approxaton algorths ( Vaz01, WS11]). The focus of ths thess s on the second challenge: n partcular, we focus on the challenges posed by: onlne nputs, where the nput arrves pece-by-pece and the algorth s requred to act edately after a pece arrves wthout knowng the future peces; nput beng dstrbuted across several selfsh agents who reveal ther pece of the nput only when the algorth ncentvzes the. Incentvzng partcpants restrcts what the algorth can do on the nput so obtaned. We begn by revewng how one forally odels nput uncertanty whch s at the core of the second challenge. Although there are several approaches to do ths, there are two approaches that have ganed currency n Coputer Scence lterature, naely, copettve analyss and stochastc analyss. Copettve Analyss. At one extree s copettve analyss. Here, the algorth desgner faces nput uncertanty whle the benchark s onscent,.e., t knows the entre nput ahead of te. The perforance s easured through what s called copettve rato: the worst aong all possble nputs of the rato of the

11 2 perforance of the algorth and that of the benchark, naely, worst I ALG(I) OPT I. Note that OPT has a subscrpt I denotng that the benchark s nstance-wse optal. For probles wth onlne nputs, ths eans that OPT knows the entre nput ahead of te, for probles wth selfsh partcpants ths agan eans that OPT has access to the entre nput ahead of te and s therefore not requred to ncentvze partcpants to get these nputs. Ths benchark s clearly a strong one snce t s explctly ore powerful than ALG, and thus any postve result wth ths benchark s rearkable. However because of ts strength, t often leads to pessstc bounds on what s achevable. An edate queston s why should the benchark not be the optal algorth that also faces nput uncertanty,.e., optal onlne algorth or the optal ncentve copatble echans. The trouble wth such a benchark s that the optal onlne algorth s not well defned: any onlne algorth can be odfed to perfor slghtly worse on one nput and slghtly better on soe other nput, and thus there s no such thng as an optal onlne algorth. Sae goes wth optal ncentve copatble algorths: perforance on one nput can often be coprosed to prove perforance on soe other nput. To crcuvent ths proble, the benchark s ade onscent and hence knows the entre nput ahead of te. Stochastc Analyss. At the other extree s stochastc analyss. Here, t s assued that the nput s drawn fro a known dstrbuton, and both the algorth and the benchark know ths dstrbuton, but not the entre nput. The perforance s easured through the rato of the expected perforance of the algorth and that of the benchark,.e., E I F ALG F (I) E I F OPT F (I). Note that both ALG and OPT have just F n ther subscrpt,.e, they both know only the dstrbuton, but not the entre nstance. For probles wth onlne nputs ths corresponds to the benchark of the expected optal onlne soluton, and for probles wth selfsh partcpants ths corresponds to the benchark of the expected

12 3 optal ncentve copatble soluton. The probles that occurred whle defnng these quanttes n copettve analyss vansh when we use expectatons n defnton: snce the relatve portance of each nput s ebedded n the dstrbuton, there s a well-defned optal way to tradeoff between dfferent nputs. Absence of such an optal tradeoff was precsely the dffculty n defnng ths benchark n copettve analyss. Snce OPT s not explctly ore powerful than ALG, one can shoot even for a perforance as good as OPT, thus crcuventng the pessstc bounds n copettve analyss. However, ths odel s subject to two ajor crtcss: one, nput dstrbutons ay be hard to obtan; two, we rsk over-fttng by desgnng algorths for specfc dstrbutons, and, estaton errors render such algorths suboptal. Pror Robust Analyss. Gven the above two extrees n odelng nput uncertanty, a possble ddle ground s as follows: the nput s drawn fro soe dstrbuton. But the algorth desgner s unaware of the specfc dstrbuton. He only knows the huge unverse of dstrbutons to whch the actual dstrbuton belongs. The benchark, on the other hand, knows the exact dstrbuton, and corresponds to the optal algorth talored specfcally for that dstrbuton. Ths corresponds to the expected optal onlne soluton for probles wth onlne nputs, and the expected optal ncentve copatble soluton for probles wth selfsh partcpants. The perforance s easured through the worst aong all dstrbutons n the unverse, of the rato of the expected perforance of the algorth and that of the benchark,.e., worst F U E I F ALG(I) E I F OPT F (I). (1.1) Note that the benchark has a subscrpt F denotng that the benchark s optally talored for a specfc dstrbuton, where as the algorth has to be the sae for every dstrbuton. So what we are askng for n pror robust optzaton s a sngle algorth whch, for every dstrbuton n the unverse, perfors approxately as well as the optal algorth desgned specfcally for that dstrbuton. It s n ths sense that these algorths are pror robust,.e., they are robust to the actual pror dstrbuton on nputs. Such algorths are also well suted for deployent n real world systes because they ake very lttle assuptons on nputs, and hence are

13 4 robust to nput fluctuatons. An Even Stronger Benchark. Soetes, we can gve our guarantees aganst an even stronger benchark, naely, we easure our perforance as: worst F U E I F ALG(I) E I F OPT I. (1.2) Note that the benchark s not just optal for a dstrbuton, but t s an nstance-wse optal benchark just as n copettve analyss. In other words, for onlne nputs the benchark s the expected optal offlne soluton, and for probles wth selfsh partcpants, the benchark s the expected optal soluton that s not constraned to be ncentve copatble. Ths s the strongest benchark possble n the presence of a dstrbuton. Our results n Chapters 2 and 4 use ths benchark. Pushng to the Two Extrees. Pror robust analyss can be pushed to the two extrees of copettve and stochastc analyss as follows: when the unverse U contans all possble dstrbutons, ncludng pont asses, pror robust analyss s the sae as copettve analyss. When the unverse contans exactly one dstrbuton, pror robust analyss becoes stochastc analyss. The goal n general s to develop pror robust algorths for as rch a unverse as possble. Approxaton Rato/Factor. We use the ter approxaton rato/factor to denote the ratos n expressons (1.1) and (1.2). Whch of these two s beng used wll be ade explct whle we use t. The focus of ths thess s the desgn and analyss of pror robust algorths for two dstnct areas of research: onlne algorths and echans desgn. Along the way, we hghlght the versatlty of broadly applcable technques n developng these algorths. 1.2 Motvatng Exaples Before statng our results forally, we provde two otvatng exaples for desgnng pror robust algorths.

14 5 Revenue Maxzaton n Internet Advertsng. Every popular webste you are aware of s lkely to dsplay one or ore banner advertseents alongsde the content t offers. These advertseents are called dsplay ads (dfferent fro sponsored search ads that you see n search engnes), and are used by frs as a eans to proote ther brand. In ters of revenue, the dsplay ads ndustry s a ult-bllon dollar ndustry and s projected to get ahead of even the ncredbly lucratve search ads ndustry n a few years. At a hgh level ths s the echans to get an dsplayed n a webste: the advertser sgns a contract wth the webste that reads soethng lke n the cong onth 5 llon pressons of y ad wll be served to ales n the age group 21 to 35 n the Chcago area for 10 cents per presson. The webste sgns such contracts wth several advertsers, and, has to pay a penalty per presson when t fals to honor the contract. When an presson arrves,.e., a user vsts the webste, the decson of whch ad to dsplay fro aong the any ads that have asked for beng shown to ths user has to be ade. Ths decson s clearly an onlne proble because the decson has to be ade wthout knowng whch users wll arrve n the future. The supply s relatvely constant,.e., the total nuber of user vsts to a webste doesn t vary uch and hence s known relatvely accurately. The webste ust therefore be judcous n usng ts lted supply of user vsts, so as to honor all of ts contract and stll dong well n ters of revenue. What consttutes a good algorth n ths settng? If we resort to copettve analyss to answer ths queston, we would be coparng wth the offlne optal algorth as our benchark. Even n the specal case where all the advertsers ask for the sae nuber of pressons, all of the have the sae prce per presson and the sae penalty per presson, nothng better than a 1 1/e ( 63%) approxaton s possble even wth a randozed algorth MSVV05]. Can we get approxatons close to 1 f we ove away fro copettve analyss? If we resort to stochastc analyss, the suggested algorth wll be very dependent on the dstrbuton. Gven ths, t would be qute useful to have a sngle algorth that perfors very well and obtans a close-to-1 approxaton regardless of the underlyng dstrbuton fro whch the pressons arrve. How do we go about desgnng algorths wth such strong guarantees? Answer n Chapter 2. Revenue Maxzaton n Auctons. Thnk of your neghbor Bob who sells antques n ebay va the standard second prce auctons that ebay perts: the hghest

15 6 bdder wns the te and pays the seller the second hghest bd. The seller s also allowed to put a nu bd (called as a reserve bd) so that the wnner pays the nu of the reserve bd and the second hghest bd. Naturally Bob lkes to axze hs revenue and s unable to sleep well untl he s convnced that he has done everythng fro hs sde to run an optal aucton. When can Bob be convnced that he s runnng a good aucton? Consder a sple nstance wth just two bdders: one has a value (the axu prce that a bdder s wllng to pay) of $100 and the other has a value of $10. Bob obvously doesn t know the values of hs bdders before the aucton begns. What consttutes a good aucton here? If Bob pessstcally uses copettve analyss to easure the goodness of hs aucton, he s out of luck: the optal aucton wll place a reserve of $100, and hence get a revenue of $100. Bob on the other hand s copletely n the dark about what reserve to place. Whatever reserve Bob places, there are bdder values for whch the obtaned revenue s a vanshngly sall fracton of the optal revenue. If Bob resorted to stochastc analyss, naely, assue that hs bdders values are drawn fro dstrbutons F 1 and F 2, then aucton theory Mye81] wll tell h the optal aucton. But, the aucton t suggests s qute coplex and often outsde what s pertted by ebay. Even n cases where the optal aucton turns out to be wthn the scope of ebay s pertted auctons, t requres Bob to know the dstrbuton well. The best opton for Bob therefore would be to ake no assuptons about dstrbutons, but stll be able to provde a provable guarantee of hs aucton s perforance. How exactly should Bob go about desgnng such an aucton? Answer n Chapter Our Contrbutons We now suarze our results n ths thess. For an n depth connecton to related work and surroundng dscussons, the reader s referred to the respectve chapters. As entoned earler, n ths thess we desgn and analyze pror robust algorths for two dstnct areas of research: probles wth onlne nputs and probles wth selfsh partcpants.

16 Pror Robust Optzaton n Internet Advertsng There has been an ncreasng nterest n onlne algorths otvated by applcatons to onlne advertsng. The ost well known proble s the Adwords proble ntroduced by Mehta et al. MSVV05], (defned forally n Chapter 2), that odels the algorthc proble nvolved n search engne revenue optzaton. Here the algorth needs to assgn keywords arrvng onlne to bdders to axze proft, subject to budget constrants for the bdders. The proble has been analyzed n the tradtonal fraework for onlne algorths: worst-case copettve analyss. As wth any onlne probles, worst case copettve analyss s not entrely satsfactory and there has been a drve n the last few years to go beyond the worst-case analyss. The predonant approach has been to assue that the nput satsfes soe stochastc property. For nstance the rando perutaton odel (ntroduced by Goel and Mehta GM08]) assues that the adversary pcks the set of keywords, but the order n whch the keywords arrve s chosen unforly at rando. We refer the reader to the dscusson n Chapter 2 for the coonly assued stochastc propertes and how they copare aganst each other. For ths work, we assue the closely related..d. odel: assue that the nput (e.g., keywords n the Adwords proble) are..d. saples fro a fxed dstrbuton, whch s unknown to the algorth. In ths odel, the unverse of dstrbutons U s the set of all possble..d. dstrbutons over the queres. Inforal Suary of Results n Chapter Near-optal onlne algorth for resource allocaton fraework. Frst, we defne a general fraework of onlne probles called the resource allocaton fraework, whch s a sgnfcant generalzaton of several nterestng specal cases ncludng dsplay ads, Adwords, onlne network routng, onlne cobnatoral auctons etc. Inforally, requests (thnk of keywords n Adwords) arrve onlne and have to be served usng n avalable resources. Each resource has a capacty (thnk of advertser budgets n Adwords) and cannot be used beyond capacty. There are dfferent optons to serve a request, and dfferent optons are assocated wth dfferent profts (thnk of a query s bd dfferng across advertsers n Adwords). In the..d. odel, the requests are..d saples fro a fxed dstrbutons that s unknown to the algorth. We desgn a near-optal pror robust algorth, whose approxaton rato

17 8 gets arbtrarly close to 1 when the revenue and capacty consupton of each request gets less sgnfcant copared to the total set of requests that arrve. Further, we show that the rate at whch the approxaton rato tends to 1 s alost optal: we show that no algorth, even those that have coplete knowledge of the dstrbuton, can get a faster convergence to 1. In contrast, n the copettve settng, no algorth can get beyond a 1 1/e approxaton even for the specal case of Adwords even f all the bds are nfntesally sall copared to the advertser budgets. 2. Near-optal onlne algorth for adwords. Due to ts sgnfcance, we sngle out the specal case of Adwords and desgn an onlne algorth that goes beyond what s possble for the general resource allocaton fraework. We desgn an algorth whose approxaton rato converges to 1 faster than our convergence rate for the general resource allocaton fraework, and, we show that ths s the fastest convergence rate any algorth can guarantee even f t knew the dstrbuton copletely. Unlke our algorth for resource allocaton, our Adwords algorth needs the knowledge of a few paraeters fro the dstrbuton n order to accoplsh ths faster convergence rato. An ntrgung open queston s whether ths proveent for Adwords s possble wthout any knowledge of dstrbuton at all. 3. Fast offlne approxaton algorth for large xed packng & coverng IPs. We consder a class of xed coverng and packng nteger progras (IPs) nspred by the resource allocaton fraework, and desgn a fast offlne approxaton algorth for these nteger progras based on our onlne algorth. Our algorth s randozed, and solves the offlne proble n an onlne anner va saplng. The basc dea s to conclude the feasblty of the orgnal nteger progra usng the feasblty of the sapled porton of the nteger progra (the latter s found by our onlne algorth). Probles n the resource allocaton fraework where the nstances are too large to use tradtonal algorths occur farly often, especally n the context of onlne advertsng. Often approxate but quck solutons are preferable. 4. Beyond..d.: The Adversaral Stochastc Input (ASI) Model. One

18 9 drawback of the..d. odel s that the dstrbuton does not vary over te,.e., at every te step, a request s drawn fro the sae dstrbuton. In settngs lke Adwords or dsplay ads, dfferent perods of the day have dfferent dstrbutons, and nspred by ths, we generalze the..d. odel to the adversaral stochastc nput odel. Inforally, dstrbutons are allowed to vary over te, and an adversary s allowed to pck, even adaptvely, whch dstrbuton wll get used at a partcular te step. We provde three dfferent generalzatons of the..d. odel, and extend our results to all these three odels wth strong guarantees slar to the..d. unknown odel. Our guarantees are aganst the worst dstrbuton the adversary pcks. If gven the help of a few dstrbutonal paraeters, the sae guarantee extends for the average of the dstrbutons whch the adversary pcks Pror Robust Revenue Maxzng Aucton Desgn Consder the sale of a sngle te, wth n nterested buyers. Buyer has a value v for the te. Assue that each v s drawn ndependently fro a dstrbuton F. For ths settng aucton theory Mye81] suggests the revenue optal aucton. Ths aucton however requres the knowledge of a dstrbuton dependent paraeter to be pleented. Can we soehow do wthout knowledge of the dstrbuton? Bulow and Kleperer BK96] showed that when the dstrbuton F satsfes a techncal regularty condton (see Chapter 3 for the defnton) recrutng one ore agent drawn fro the sae dstrbuton F, and just runnng the second prce aucton on the n + 1 agents wll gve at least as uch revenue as the revenue optal aucton on n agents. Inforally, advertsng and spreadng the word about the aucton to boost the nuber of bdders (whch s clearly pror robust) pays off better than perforng arket analyss to deterne the bdder dstrbutons. Whle the techncal regularty condton ncludes any natural dstrbutons, t s also volated by sple ones lke ultodal dstrbutons (n fact, even bodal ones). For nstance, the convex cobnaton of two regular dstrbutons s lkely to result n a bodal dstrbuton. Ths stuaton of a bdder s value beng a convex cobnaton of two regular dstrbutons s aong the ost coon reasons for regularty beng volated n an aucton stuaton: n Bob s ebay aucton n the otvatng exaple, t could well be that each bdder for Bob s a professonal antque collector wth probablty p p and an aateur wth probablty p a, wth respectve dstrbutons F p

19 10 and F a. Whle F p and F a could satsfy the techncal regularty condton, ther convex cobnaton p p F p + p a F a s often not regular. That s, a heterogeneous arket s qute lkely to result n bdders who volate the regularty condton. Is there a natural extenson of Bulow and Kleperer s result to ths wdely prevalent settng? Inforal Suary of Results n Chapter One extra bdder fro every regular dstrbuton. When the dstrbutons of each of the n agents can be expressed as convex cobnatons of k regular dstrbutons (possbly dfferent convex cobnatons,.e., the n agents could be fro dfferent dstrbutons), we prove that recrutng one agent fro each of these k regular dstrbutons, and runnng the second prce aucton on these n + k agents wthout any reserve prce wll gve at least half the revenue of the optal aucton on the orgnal n agents. Inforally, whle advertsng was the soluton for regular dstrbutons, a targeted advertsng capagn s the soluton for heterogeneous arkets wth rregular dstrbutons. In Bob s aucton wth professonals and aateurs, f Bob anages to brng n one addtonal professonal and aateur and run the second prce aucton, he gets at least half the optal revenue n the orgnal settng wth n bdders. 2. Just one extra bdder for hazard rate donant dstrbutons. If the k underlyng regular dstrbutons are such that one of the hazard rate donates the rest (see Chapter 3 for a defnton; for exaple f all the k underlyng dstrbutons are exponentals wth possbly dfferent rates, or power law dstrbutons wth possbly dfferent scales, or unfor dstrbutons over possbly dfferent ntervals, one of the s guaranteed to hazard rate donate the rest), recrutng just one addtonal agent fro the hazard rate donant dstrbuton gves at least half as uch revenue as the optal aucton run on the orgnal n agents Pror Robust Mechanss for Machne Schedulng Schedulng jobs n achnes to nze the copleton te of the last job (naely, akespan) s a central proble n Coputer Scence, partcularly n the context of resource allocaton. Even fro an Econoc vewpont, nzng akespan s

20 11 relevant because t can be nterpreted as axzng farness across achnes. For the feld of Algorthc Mechans Desgn, akespan nzaton s the senal proble and was ntroduced by Nsan and Ronen NR99]. In the proble they ntroduce, there are achnes and n jobs. The achnes are operated by selfsh agents who hold the runtes of jobs as prvate, and wll reveal the only f the schedulng algorth properly ncentvzes the. Nsan and Ronen NR99] desgn a truthful (ncentve copatble) echans that obtans a approxaton n the copettve settng,.e., the benchark s the optal akespan possble wthout the truthfulness constrant. Ashlag, Dobznsk and Lav ADL09] later showed that ths s alost the best one can hope for n the copettve settng: they showed that for a large class of natural echanss, nothng better than a approxaton s achevable. In ths work, just as for resource allocaton probles, we ask, whether we can get better approxaton guarantees f we go beyond copettve analyss. Whle the lterature on pror robust echanss has prarly focused on the lnear objectve of revenue axzaton HR09, DRY10, RTCY12], n ths work, wth approprate dstrbutonal assuptons, we gve the frst non-trval guarantees for the non-lnear objectve of akespan nzaton. Inforal Suary of Results n Chapter An O(n/) approxaton wth..d. achnes and non-..d. jobs. When the runtes of a job on dfferent achnes are..d. rando varables (but dfferent jobs could have dfferent dstrbutons), we desgn a pror robust truthful echans that obtans a O(n/) approxaton to OPT 1/2, whch s the optal akespan wthout ncentve copatblty constrants that can be obtaned by usng at ost /2 achnes. Ths result says that f we augent our resources by doublng the nuber of achnes and use our echans, we get a O(n/) approxaton to OPT. When n = O(), ths gves a constant factor approxaton. In contrast, n the copettve settng, even when n = the Ashlag et al. ADL09] hardness of approxaton of factor reans ntact. Further, for a large class of dstrbutons, ncludng those that satsfy the Monotone Hazard Rate (MHR) property (ntutvely, those that have tals no heaver than the exponental dstrbuton: ths ncludes the unfor, noral, exponental dstrbutons etc.), we show that OPT 1/2 and OPT are wthn a factor

21 12 of 4. That s, for ths class of dstrbutons, even wthout resource augentaton, we get a O(n/) approxaton. 2. Sublogarthc approxatons wth..d. achnes and jobs. The above result gves good approxatons when n s relatvely sall. When n log we desgn another echans, whch, wth the addtonal assupton that the jobs are also dentcally dstrbuted, obtans a O( log ) approxaton to OPT 1/3. Further, f the dstrbutons belong to the large class of dstrbutons entoned n the prevous paragraph (ncludng MHR dstrbutons), we obtan O(log log ) 2 approxaton to OPT. 1.4 Prerequstes and Dependences Prerequstes. All necessary techncal prerequstes are provded n the respectve chapters. Apart fro those provded n the chapters, falarty wth basc probablty, expectatons, and bg O notaton s all that s assued. Dependences. The chapters are self contaned, and can be read n any order. 1.5 Bblographc Notes The work presented n ths thess s contaned n the research papers DJSW11, DSA12, SS13, CHMS13]. 1. The results n Chapter 2 are based on two jont works DJSW11, DSA12]. Our results for general resource allocaton DJSW11] are based on jont work wth Nkhl Devanur, Kaal Jan and Chrs Wlkens. The proved results for the specal case of Adwords are based on jont work DSA12] wth Nkhl Devanur and Yoss Azar. 2. The results n Chapter 3 are based on jont work SS13] wth Vasls Syrgkans. 3. The results n Chapter 4 are based on jont work CHMS13] wth Shuch Chawla, Jason D. Hartlne and Davd Malec.

22 13 2 Pror Robust Optzaton n Internet Advertsng Organzaton. In ths chapter, we desgn and analyze pror robust algorths for revenue axzaton n Internet advertsng. The chapter s organzed as follows. In Secton 2.1 we nforally suarze our results, put the n context of related work wth soe addtonal dscusson. In Secton 2.2 we provde all the necessary prelnares and the foral stateents of all our results n ths chapter. In Secton 2.3 we desgn and analyze pror robust onlne algorths for the resource allocaton fraework, whch s a strct generalzaton of the Adwords proble, the dsplay ads proble and other nterestng specal cases. In Secton 2.4 we prove that the approxaton factor we get n Secton 2.3 s alost optal. In Secton 2.5, we consder the nterestng specal case of our resource allocaton fraework, naely, the Adwords proble, and desgn and analyze an onlne algorth that obtans an proved approxaton over what s possble for the general case n Secton 2.3, and we prove ts tghtness n Secton 2.6. In Secton 2.7 we analyze the greedy algorth and show that t obtans a 1 1/e approxaton for the Adwords proble wth a large bd-to-budget rato,.e., large γ, provng over the prevous best known trval factor of 1/2. In Secton 2.8 we use our onlne algorths to gve fast approxaton algorths for xed packng and coverng probles. In Secton 2.9 we dscuss soe specal cases of the resource allocaton fraework and conclude wth soe open questons n Secton Introducton & Suary of Results The results n ths chapter fall nto dstnct categores of pror robust algorths for onlne probles and fast approxaton algorths for offlne probles. However they all share coon technques. There has been an ncreasng nterest n onlne algorths otvated by applcatons to onlne advertsng. The ost well known s the Adwords proble ntroduced by Mehta et al. MSVV05], where the algorth needs to assgn keywords arrvng onlne to bdders to axze proft, subject to budget constrants for the bdders. The proble has been analyzed n the tradtonal fraework for onlne algorths: worst-

23 14 case copettve analyss. As wth any onlne probles, the worst-case copettve analyss s not entrely satsfactory and there has been a drve n the last few years to go beyond the worst-case analyss. The predonant approach has been to assue that the nput satsfes soe stochastc property. For nstance the rando perutaton odel (ntroduced by Goel and Mehta GM08]) assues that the adversary pcks the set of keywords, but the order n whch the keywords arrve s chosen unforly at rando. A closely related odel s the..d. odel: assue that the keywords are..d. saples fro a fxed dstrbuton, whch s unknown to the algorth. Stronger assuptons such as..d. saples fro a known dstrbuton have also been consdered. Frst Result: Near-Optal Pror Robust Onlne Algorths for Resource Allocaton Probles. A key paraeter on whch any of the algorths for Adwords depend s the bd to budget rato, whch easures how sgnfcant any sngle keyword/query s when copared to the total budget. For nstance Mehta et al. MSVV05] and Buchbnder, Jan and Naor BJN07] desgn an algorth that acheves a worst case copettve rato that tends to 1 1/e as the bd to budget rato (whch we denote by γ) tends to 0. (Note that γ approachng zero s the easest case. Even wth γ approachng zero, 1 1/e s the best copettve rato that any randozed algorth can acheve n the worst case.) Devanur and Hayes DH09] showed that n the rando perutaton odel, the copettve rato tends to 1 as γ tends to 0. Ths result showed that copettve rato of algorths n stochastc odels could be uch better than that of algorths n the worst case. The portant queston snce then has been to deterne the optal trade-off between γ and the copettve rato. DH09] showed how to get a 1- O(ɛ) copettve rato when γ s ɛ at ost O( 3 ) where n s the nuber of advertsers and s the nuber of n log(n/ɛ) keywords. Subsequently Agrawal, Wang and Ye AWY09] proved the bound on γ to O( ɛ 2 n log(/ɛ) ). The papers of Feldan et al. FHK+ 10] and Agrawal, Wang and Ye AWY09] have also shown that the technque of DH09] can be extended to other onlne probles. The frst an result n ths paper s the followng 3-fold proveent of prevous results: (Theores 2.2 and 2.3) 1. We gve an algorth whch guarantees a 1 ɛ approxaton factor when ɛ γ = O( 2 ) Ths s alost optal; we show that no algorth, even f log(n/ɛ)

24 15 t knew the dstrbuton, can guarantee a 1 ɛ approxaton factor when γ = ω( ɛ2 log(n) ). 2. The bound apples to a ore general odel of stochastc nput, called the adversaral stochastc nput odel. Ths s a generalzaton of the..d. odel wth unknown dstrbuton, n that the dstrbutons are allowed to change over te. We provde three dfferent generalzatons n Secton Our results generalze to a ore general class of onlne probles that we call the resource allocaton fraework. A foral defnton of the fraework s presented n Secton and a dscusson of any nterestng specal cases s presented n Secton 2.9. Sgnfcance. Regardng the bound on γ, the reoval of the factor of n s sgnfcant. Consder for nstance the Adwords proble and suppose that the bds are all n 0,1]. The earler bound ples that the budgets need to be of the order of n/ɛ 2 n order to get a 1 ɛ copettve algorth, where n s the nuber of advertsers. Wth realstc values for these paraeters, t sees unlkely that ths condton would be et. Whle wth the proved bounds presented n ths paper, we only need the budget to be of the order of log n/ɛ 2 and ths condton s et for reasonable values of the paraeters. Furtherore, n the ore general resource allocaton fraework, the current hghest upper bound on γ s fro Agrawal, Wang and Ye AWY09] and equals ɛ O( 2 ). Here K s the nuber of avalable optons (see Secton 2.2.2) and n n log(k/ɛ) typcal applcatons lke network routng, k could be exponental n n, and thus, the factor saved by our algorth becoes quadratc n n. I.I.D. vs Rando Perutatons. We note here that so far, all the algorths for the..d. odel (wth unknown dstrbuton) were actually desgned for the rando perutaton odel. It sees that any algorth that works for one should also work for the other. However we can only show that our algorth works n the..d. odel, so the natural queston s f our algorth works for the rando perutaton odel. It would be very surprsng f t ddn t. Adversaral Stochastc Input (ASI) Model. One drawback of the stochastc odels consdered so far s that they are te nvarant, that s the nput dstrbuton

25 16 reans the sae for every request. The adversaral stochastc nput odel allows the nput dstrbuton to change over te. The foral defntons of these three generalzatons are presented whle we use the n Secton Second Result: Iproved Onlne Algorths for Adwords usng a few Paraeters fro the Dstrbuton. Due to ts sgnfcance, we sngle out one specal case of the resource allocaton fraework, naely, the Adwords proble, for further study. Is t possble to desgn algorths that go beyond what s possble for the general resource allocaton fraework,.e., s t to possble to get a 1 ɛ approxaton algorth for the Adwords proble for γ = ω(ɛ 2 / log(n)). In our second result (Theore 2.4), we desgn an onlne algorth for adwords that obtans 1 ɛ approxaton for Adwords whenever γ = O(ɛ 2 ). We also show that ths s the best one can hope for: no algorth can guarantee, even f t knew the coplete dstrbuton, a 1 ɛ approxaton for γ = ω(ɛ 2 ). Ths algorth however s not copletely pror robust: the algorth we desgn requres n paraeters fro the dstrbuton. If we need a copletely pror robust algorth, then the slghtly worse result for the general resource allocaton fraework (Theore 2.2) s the best known currently. However we note here that requrng n paraeters fro the dstrbuton s nforaton theoretcally strctly weaker than askng for the knowledge of the entre dstrbuton whch could have an exponental (even nfnte) support. Open Queston. Desgn a pror robust algorth for adwords that guarantees a 1 ɛ approxaton for adwords when γ = O(ɛ 2 ). Thrd Result: Pror Robust 1 1/e approxaton Greedy Algorth for Adwords. A natural algorth for the Adwords proble that s wdely used for ts splcty s the greedy algorth: always atch an ncong query to the advertser that has the axu effectve bd (the nu of bd and reanng budget) for that query. Because of ts wde use, prevously the perforance of the greedy algorth has been analyzed by Goel and Mehta GM08] who showed that n the rando perutaton and the..d. odels, t has a copettve rato of 1 1/e wth an assupton whch s essentally that γ tends to 0.

26 17 It has been an portant open proble to analyze the perforance of greedy algorth n a stochastc settng for unbounded γ,.e., for all 0 γ 1. The best factor known so far s 1/2, and ths works for the worst case also. Nothng better was known, even n the stochastc odels. The thrd result n ths chapter s that for the Adwords proble n the..d. unknown dstrbutons odel, wth no assupton on γ (.e., γ could be as bg as 1), the greedy algorth gets an approxaton factor of 1 1/e aganst the optal fractonal soluton to the expected nstance (Theore 2.8). We note here that there are other algorths that acheve a 1 1/e approxaton for the Adwords proble wth unbounded γ, but the greedy algorth s the only pror robust algorth known, and t qute sple too. For exaple, our our second result (Theore 2.4) wll becoe a 1 1/e approxaton at γ = 1 (see Secton 2.5 for how we get 1 1/e at γ = 1), but t requres a few paraeters fro the dstrbuton. Slarly Alae et al. AHL12] desgn a randozed algorth that obtans a 1 1/e approxaton, but requres the knowledge of the entre dstrbuton. Fourth Result: Fast Approxaton Algorths for Mxed Packng and Coverng Integer Progras. Charles et al. CCD + 10] consdered the followng (offlne) proble: gven a lopsded bpartte graph G = (L, R, E), that s a bpartte graph where = L R = n, does there exst an assgnent M : L R wth (j, M(j)) E for all j L, and such that for every vertex R, M 1 () B for soe gven values B. Even though ths s a classc proble n cobnatoral optzaton wth well known polynoal te algorths, the nstances of nterest are too large to use tradtonal approaches to solve ths proble. (The value of n partcular s very large.) The approach used by CCD + 10] was to essentally desgn an onlne algorth n the..d. odel: choose vertces fro L unforly at rando and assgn the to vertces n R n an onlne fashon. The onlne algorth s guaranteed to be close to optal, as long as suffcently any saples are drawn. Therefore t can be used to solve the orgnal proble (approxately): the onlne algorth gets an alost satsfyng assgnent f and only f the orgnal graph has a satsfyng assgnent (wth hgh probablty). The fourth result n ths chapter s a generalzaton of ths result to get fast approxaton algorths for a wde class of xed packng and coverng nteger progras (IPs) nspred by probles n the resource allocaton fraework (Theore 2.9). Probles n the resource allocaton fraework where the nstances are too

27 18 large to use tradtonal algorths occur farly often, especally n the context of onlne advertsng. Foral stateents and a ore detaled dscusson are presented n Secton Hgh Level Descrpton of Technques. The underlyng dea used for all these results can be suarzed at a hgh level as thus: consder a hypothetcal algorth called Hypothetcal-Oblvous that knows the dstrbuton fro whch the nput s drawn and uses an optal soluton w.r.t. ths dstrbuton. Now suppose that we can analyze the perforance of Hypothetcal-Oblvous by consderng a potental functon and showng that t decreases by a certan aount n each step. Now we can desgn an algorth that does not know the dstrbuton as follows: consder the sae potental functon, and n every step choose the opton that nzes the potental functon. Snce the algorth nzes the potental n each step, the decrease n the potental for ths algorth s better than that for Hypothetcal-Oblvous and hence we obtan the sae guarantee as that for Hypothetcal-Oblvous. The choce of potental functon vares across the results; also, whether we nze or axze the the potental functon vares across the results. For nstance, our frst result (Theore 2.2), the perforance of Hypothetcal- Oblvous s analyzed usng Chernoff bounds. The Chernoff bounds are proven by showng bounds on the expectaton of the oent generatng functon of a rando varable. Thus the potental functon s the su of the oent generatng functons for all the rando varables that we apply the Chernoff bounds to. The proof shows that n each step ths potental functon decreases by soe ultplcatve factor. The algorth s then desgned to acheve the sae decrease n the potental functon. A partcularly pleasng aspect about ths technque s that we obtan very sple proofs. For nstance, the proof of Theore 2.8 s extreely sple: the potental functon n ths case s sply the total aount of unused budgets and we show that ths aount (n expectaton) decreases by a factor of 1 1/ n each step where there are steps n all. Multplcatve-Weght Updates. Our technques and the resultng algorths for our frst and fourth results (Theore 2.2 and Theore 2.9) bear a close reseblance to the algorths of Young You95, You01] for derandozng randozed roundng and the fast approxaton algorths for solvng coverng/packng LPs of Plotkn,

28 19 Shoys and Tardos PST91], Garg and Köneann GK98],Flescher Fle00]. In fact Arora, Hazan and Kale AHK05] showed that all these algorths are related to the ultplcatve weghts update ethod for solvng the experts proble and especally hghlghted the slarty between the potental functon used n the analyss of the ultplcatve update ethod and the oent generatng functon used n the proof of Chernoff bounds and Young s algorths. Hence t s no surprse that our algorth whch uses Chernoff bounds s also a ultplcatve update algorth. Our algorth s closer n sprt to Young s algorths than others. A basc dfference of our algorth fro ths prevous set of results s that our algorth uses the specal structure of the polytope k x j,k 1 (as aganst the ore general polytopes n these works) n gvng a ore effcent soluton both for the offlne and onlne versons of our proble. For nstance, for our offlne proble the nuber of oracle calls requred wll have a quadratc dependence on γ f we used the PST91] algorth, where as usng the specal structure of the polytope, we obtan a lnear dependence on γ. It s possble that our algorth can also be nterpreted as an algorth for the experts proble. In fact Mehta et al. MSVV05] asked f there s a 1 o(1) copettve algorth for Adwords n the..d odel wth sall bd to budget rato, and n partcular f the algorths for experts could be used. They also conjectured that such an algorth would teratvely adjust a budget dscount factor based on the rate at whch the budget s spent. Our algorths for resource allocaton proble when specalzed for Adwords look exactly lke that and wth the connectons to the experts fraework, we answer the questons n MSVV05] n the postve. 2.2 Prelnares & Man Results Resource Allocaton Fraework We consder the followng fraework of optzaton probles. There are n resources, wth resource havng a capacty of c. There are requests; each request j can be satsfed by a vector x j {0, 1} K, wth coordnates x j,k, such that k x j,k 1. Thnk of vector x j as pckng an opton to satsfy a request fro a total of K optons (We also overload K to denote the set of optons). The vector x j consues a,j x j aount of resource, and gves w,j x j aount of type proft. The a,j s and w,j s are non-negatve vectors of length K (and so are the x j s). The co-ordnates of the vectors

29 20 a,j and w,j wll be denoted by a jk and w jk respectvely,.e., the k th opton consues a jk aount of resource and gves a type proft of w jk. The objectve s to axze the nu aong all types of proft subject to the capacty constrants on the resources. The followng s the lnear progra relaxaton of the resource allocaton proble: Maxze n w,j x j s.t. j, j j, k a,j x j c x j,k 1 j, k, x j,k 0 Note that droppng a request by not pckng any opton at all s feasble too. For expostonal convenence, we wll denote not pckng any opton at all as havng pcked the opton ( ay not be n the set K) for whch a j = 0 and w j = 0 for all, j. We consder two versons of the above proble. The frst s an onlne verson wth stochastc nput: requests are drawn fro an unknown dstrbuton. The second s an offlne proble when the nuber of requests s uch larger than the nuber of resources, and our goal s to desgn a fast PTAS for the proble Near-Optal Onlne Algorth for Resource Allocaton We now consder an onlne verson of the resource allocaton fraework. Here requests arrve onlne. We consder the..d. odel, where each request s drawn ndependently fro a gven dstrbuton. The dstrbuton s unknown to the algorth. The algorth knows, the total nuber of requests. To defne our benchark, we now defne the expected nstance. Expected Instance. Consder the followng expected nstance of the proble, where everythng happens as per expectaton. It s a sngle offlne nstance whch s a functon of the gven dstrbuton over requests and the total nuber of requests. Every request n the support of the dstrbuton s also a request n ths nstance. The capactes of the resources n ths nstance are the sae as n the orgnal nstance.