# The Diverse Cohort Selection Problem-Books Pdf

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4 for t n n 1 do, with output Mt which is correct with probability 1 as we show. 5 Mt Oracle u t , in Theorems 4 2 and 4 4 If w Mt and w M t differ SWAP looks at a. 6 for ai 1 nr, set of candidate arms in the symmetric difference of Mt and M t and. 7 rad t ai 2 log Tt ai chooses the arm pt with the largest uncertainty bound rad t pt . 8 if ai Mt then SWAP then chooses to either strong or weak pull the selected. 9 u t ai u t ai rad t ai arm pt using a strong pull policy depending on parameters s and j . 10 else A strong pull policy is defined as spp R 1 R 1 0 1 . 11 u t ai u t ai rad t ai For example in the experiments in Section 5 we use the following. pull policy ,12 M t Oracle u t s j,13 if w M t w Mt then spp s j 3 . 14 Out Mt This policy tries to balance information gain and cost When. 15 return Out the strong pull gain is high relative to cost then many more strong. 16 pt arg maxa M M M M rad t a pulls will be performed When the weak pull gain is low relative. t t t t, 17 spp s j to cost then fewer strong pulls will be performed as discussed in.
18 with probability do Example 4 1 , 19 Strong pull pt Once an arm is pulled the empirical mean u t 1 pt and the. 20 Tt 1 pt Tt pt s information gain Tt 1 pt is updated A reward from a strong arm. 21 Cost t 1 Cost t j is counted s times more than a weak pull . 22 else, Example 4 1 Suppose we wish to find a cohort of size K 2 from. 23 Weak pull pt, three arms A a 1 a 2 a 3 Run SWAP for t iterations Figure 1. 24 Tt 1 pt Tt pt 1, shows that SWAP maintains empirical utilities u t and uncertainty. 25 Cost t 1 Cost t 1, bounds rad t In this case M a 1 a 2 and M a 1 a 3 Arm a 3 .
26 Update empirical mean u t 1 using observed reward. therefore is the arm in the symmetric difference a 2 a 3 with the. 27 Tt 1 a Tt a a pt, highest uncertainty which therefore needs to be pulled Further . assume that a 3 needs x information gain for SWAP to end When. Algorithm 1 gives pseudocode for SWAP It starts by weak pulling j 1 and s 1 the best pulling strategy would be to weak pull a 3. all arms once to initialize an empirical estimate of the true under for x times When j 1 and s y where y 1 the best pulling. lying utility of each arm It then iteratively pulls arms chooses strategy would be to strong pull a 3 for ceil yx times Finally when. to weak or strong pull based on a general strategy updates em j z and s y where y z 1 the best pulling strategy would be. pirical estimates of arms and terminates with the optimal i e to strong pull a 3 for floor yx 1 z x mod y times and weak. objective maximizing subset of arms with probability 1 for pull a 3 for 1 z x mod y x mod y times where 1 a 1. some user supplied parameter when a 0 and 0 otherwise In reality we do not know how many. During each iteration t SWAP starts by finding the set of arms times an arm needs to be pulled which is why we introduce a. Mt that according to current empirical estimates of their means probabilistic strong pull policy like that in Equation 3 . maximizes the objective function via an oracle It then computes a. confidence radius rad t a for each arm a and estimates the worst Analysis We now formally analyze SWAP We define X Cost . case utility of that arm with the corresponding bound If an arm a E Cost as the expected cost or expected j value and X Gain . Session 2F Agent Societies and Societal Issues 2 AAMAS 2019 May 13 17 2019 Montr al Canada. E Gain as the expected gain or the expected s value Assume. that each arm a n has mean u a withr an sub Gaussian tail T vs Hardness. Theoretical bound, 4nCost 3t 10 9, Following Chen et al set rad t a 2 log Tt a for Actual cost. all t 0 107, Notice that if we use strong pull policy spp s j 0 then we. only perform weak arm pulls and SWAP reduces to Chen et al s 105. CLUCB We call this reduction the weak only pull problem Chen et. al proved that CLUCB returns the optimal set M and uses at most 103 105 107. O width M 2 H samples Similarly if we set spp s j 1 then we H. only perform strong arm pulls dubbed the strong only pull problem . We show that this version of SWAP returns the optimal set M and Figure 2 Exploration of bounds in practice vs the theoret . costs at most O width M 2 H s ical bounds of Theorem 4 4 with respect to hardness note. that both axes are a log scale , Theorem 4 2 Given any 0 1 any decision class M . 2 n and any expected rewards u Rn assume that the reward. distribution a for each arm a n has mean u a with an sub It is nontrivial to determine where the general version of SWAP is. Gaussian tail Let arg maxM M w M denote the optimal set better than both the SWAP algorithm with only strong pulls and the. SWAP algorithm with only weak pulls given the non asymptotic. Set rad t a 2 log Tt a for all t 0 and a n Then nature of all three bounds Chen et al results and Theorems 4 2. with probability at least 1 the SWAP algorithm with only strong and 4 4 Based on our experiments 5 we conjecture that there is. pulls where j 1 and s j returns the optimal set Out M and a of s and j pairs where SWAP is the optimal algorithm even for. 2 relatively low numbers of arm pulls though it is problem specific . width M 2 H log nj 3 2 H This is discussed more in Section 7 3 . T O 4 , where T denotes the total cost used by the SWAP algorithm and H is.
5 TOP K EXPERIMENTS, defined in Eq 2 In this section we experimentally validate the SWAP algorithm. under a variety of arm pull strategies We first explore 5 1 the. Although s and j are problem specific it is important to know efficacy of our bounds in Theorem 4 4 and Corollary 4 3 in simu . when to use the strong only pull problem over the weak only pull lation Then we deploy SWAP on real data 5 2 drawn from one. problem Corollary 4 3 provides weak bounds for s and j for the of the largest computer science graduate programs in the United. strong only pull problem We also explore its ramifications experi States We show that SWAP provides a higher overall utility with. mentally in Figure 3a as discussed in Section 5 1 equivalent cost to the actual admissions process . Corollary 4 3 SWAP with only strong pulls is equally or more. efficient than SWAP with only weak pulls when s 0 and 0 j . 5 1 Gaussian Arm Experiment, C 3 3 where C 4n H We begin by validating the tightness of our theoretical results in a. simulation setting that mimics the assumptions made in Section 4 . We now address the general case of SWAP for any probabilistic We pull from a Gaussian distribution around each arm When arm. strong pull policy parameterized by s and j In Theorem 4 4 we a is weak pulled a reward is pulled from a Gaussian distribution. show that SWAP returns M in O width M 2 H X Gain samples . with mean ua the arm s true utility and standard deviation . Similarly when arm a is strong pulled the algorithm is charged j. Theorem 4 4 Given any 1 2 3 0 1 any decision class cost and a reward is pulled from a distribution with mean ua and. M 2 n and any expected rewards u Rn assume that the reward . standard deviation s This strong pull distribution is equivalent. distribution a for each arm a n has mean u a with an sub to pulling the arm s times and averaging the reward thus ensuring. r M arg max,Gaussian tail Let , M M w M denote the optimal set an information gain of s . Set rad t a 2 log Tt a for all t 0 and a n set We ran all three algorithms SWAP with the strong pull policy. r r defined in Equation 3 SWAP with only strong pulls and SWAP with. only weak pulls while varying s and j For each s and j pair we. , 1 2 log 12 2 T and set 2 2 log 21 3 n Then with. ran the algorithms at least 4 000 times with a randomly generated. probability at least 1 1 1 2 1 3 the SWAP algorithm set of arm values Random seeds were maintained across policies . Algorithm 1 returns the optimal set Out M and We then compared the cost of running each of the algorithms 1. 2 2, 2 3 To test Corollary 4 3 Figure 3a compares SWAP with only weak.
width M H log n X Cost 1 H 1 pulls to SWAP with only strong pulls We found that Corollary 4 3. T O 5 , X Gain 2 is a weak bound on the boundary value of j The general version. of SWAP should be used when it performs better costs less than. where T denotes the total cost used by Algorithm 1 and H is defined 1 Allcode to replicate this experiment can be found here https github com . in Eq 2 principledhiring SWAP , Session 2F Agent Societies and Societal Issues 2 AAMAS 2019 May 13 17 2019 Montr al Canada. w T, Optimal zone of SWAP, Strong vs Weak Pull, SWAP 80 1 0 5 1978 53 . 5 Actual 73 96 2000, Table 1 Graduate Admissions Simulation of SWAP Compar . 10 0 10 ison of top K utility w and cost T of SWAP with results of the. actual admissions process The values in parentheses are the. 15 15, standard deviations , 5 10 15 5 10 15, s values s values.
Example 6 1 Return to a similar setting to Example 4 1 with the theoretical bounds of Theorem 4 4 with respect to hard . three arms a 1 a 2 a 3 A and true utilities u a 1 0 6 u a 2 ness Note that both axes are a log scale . 0 5 and u a 3 0 3 Assume there exist L 2 classes and let arms. a 1 and a 2 belong to class 1 and arm a 3 belong to class 2 Then for a Gender Actual Gender SWAP. cohort of size K 2 w top will select cohort M top a a while. w div will select cohort M div a 1 a 3 Indeed w top M top. , 1 1 0 9 w top M div while w div M top 1 1 1 05 1 3 . , , 0 6 0 3 w div M div , Maximizing a general submodular function is computationally. difficult Nemhauser, et al 24 proved that a close to optimal that is F. w div M 1 e1 OPT greedy algorithm exists for submodular . monotone functions that are subject to a cardinality constraint We. a Actual b SWAP, use that standard greedy packing algorithm in our implementation. of the oracle Region Actual Region SWAP, N America China.
6 2 Diverse Gaussian Arm Experiments N America, To determine if SWAP works in this submodular setting we ran. Africa India, Other Americas, simulations over a variety of hardness levels We instantiated the Africa. China Europe, problem similarly to that of Section 5 1 with the added complexity Middle Other. of dividing the arms into three partitions East East Europe. India Asia, Figure 4a shows the cost of running SWAP compared to the Asia. theoretical bounds of the linear model over increasing hardness. levels The results show that SWAP performs well for the majority c Actual d SWAP. of cases However for some cases the cost becomes very large To. deal with those situations we can use a probably approximately Figure 5 Comparison of true and SWAP simulated admis . PAC relaxation of Algorithm 1 where Line 13 becomes sions gender 5a 5b region 5c 5d . If w M t w Mt The results from this PAC relaxation. where 0 01 can be found in Figure 4b Note that the definition Gender Region of Origin. , of hardness found in Equation 2 does not quite fit this situation w top w div w top w div.
since the graphs in Figure 4 have higher costs for some lower hard . SWAP 8 5 0 03 12 1 0 06 8 0 0 03 22 1 0 03 , ness problems while having lower cost for some higher hardness. Actual 8 6 11 8 8 6 20 47, problems Given that the PAC relaxation performs well with low. costs over all of the tested hardness problems we propose that Table 2 SWAP s average gain in diversity over different. SWAP can be used with w div and perhaps other submodular and classes . monotone functions ,6 3 Diverse Graduate Admissions Experiment. Using the same setting as described in Section 5 2 we simulate a Results We compare two objective functions w top and w div . SWAP admissions process with the submodular function w div We w top treats all applicants as members of one global class This. partition groups by gender which is binary in our dataset and mimics a top K objective where applicants are valued based on. multi class region of origin We found that we did not have to resort individual merit alone w div promotes diversity using reported. to the PAC version of SWAP to tractably run the simulation over gender and region of origin for class memberships We use those. various partitions of the graduate admissions data classes as our objective during separate runs of SWAP . Session 2F Agent Societies and Societal Issues 2 AAMAS 2019 May 13 17 2019 Montr al Canada. allocation can negatively affect applicants from traditionally un . Random and Uniform derrepresented minority groups We suggest a formally structured. vs SWAP and Actual, process to help prevent disadvantaged people from falling through. wTOP Actual the cracks We discuss benefits Section 7 1 and limitations Sec . 8 SWAP, tion 7 2 to this approach as well as mechanism design suggestions.
15 Shweta Jain Sujit Gujar Onno Zoeter and Y Narahari 2014 A Quality Assur . 1479 1486 , ing Multi armed Bandit Crowdsourcing Mechanism with Incentive Compatible. 34 Laura Gollub Williamson James E Campion Stanley B Malos Mark V Roehling . Learning In International Conference on Autonomous Agents and Multi Agent. and Michael A Campion 1997 Employment interview on trial Linking interview. Systems AAMAS , structure with litigation outcomes Journal of Applied Psychology 82 6 1997 . 16 Matthew Joseph Michael Kearns Jamie H Morgenstern and Aaron Roth 2016 . Fairness in Learning Classic and Contextual Bandits In Proceedings of the Annual. 35 Yingce Xia Tao Qin Weidong Ma Nenghai Yu and Tie Yan Liu 2016 Budgeted. Conference on Neural Information Processing Systems NIPS 325 333 . multi armed bandits with multiple plays In Proceedings of the International Joint. 17 Kwang Sung Jun Kevin Jamieson Robert Nowak and Xiaojin Zhu 2016 Top. Conference on Artificial Intelligence IJCAI , Arm Identification in Multi Armed Bandits with Batch Arm Pulls In AISTATS . 36 Yisong Yue and Carlos Guestrin 2011 Linear submodular bandits and their. 18 Julia Levashina Christopher J Hartwell Frederick P Morgeson and Michael A. application to diversified retrieval In Proceedings of the Annual Conference on. Campion 2014 The structured employment interview Narrative and quantitative. Neural Information Processing Systems NIPS 2483 2491 . review of the research literature Personnel Psychology 67 1 2014 241 293 .

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