{"title": "A Framework for Multiple-Instance Learning", "book": "Advances in Neural Information Processing Systems", "page_first": 570, "page_last": 576, "abstract": "", "full_text": "A Framework for Multiple-Instance Learning \n\nOded Maron \n\nNE43-755 \n\nAI Lab, M.I. T. \n\nCambridge, MA 02139 \n\noded@ai.mit.edu \n\nTomas Lozano-Perez \n\nNE43-836a \nAI Lab, M.I.T. \n\nCambridge, MA 02139 \n\ntlp@ai.mit.edu \n\nAbstract \n\nMultiple-instance learning is a variation on supervised learning, where the \ntask  is  to  learn  a  concept  given  positive and  negative  bags  of instances. \nEach  bag  may  contain many  instances,  but a  bag  is  labeled  positive even \nif only one of the instances in it falls  within the concept.  A  bag is  labeled \nnegative  only  if all  the  instances  in  it are  negative.  We  describe  a  new \ngeneral  framework,  called Diverse  Density,  for  solving multiple-instance \nlearning problems.  We  apply this framework to learn a simple description \nof a person from a series of images (bags) containing that person, to a stock \nselection problem, and to the drug activity prediction problem. \n\n1  Introduction \n\nOne ofthe drawbacks of applying the supervised learning model is that it is not always possible \nfor a teacher to provide labeled examples for training.  Multiple-instance learning provides a \nnew  way  of modeling the teacher's weakness.  Instead of receiving a set of instances which \nare labeled positive or negative, the learner receives a set of bags that are labeled positive or \nnegative.  Each bag contains many instances.  A bag is labeled negative if all the instances in \nit are negative.  On the other hand, a bag is labeled positive if there is at least one instance in it \nwhich is positive. From a collection of labeled bags, the learner tries to induce a concept that \nwill label  individual instances correctly.  This problem is  harder than even  noisy supervised \nlearning since the ratio of negative to positive instances in a positively-labeled bag (the noise \nratio) can be arbitrarily high. \n\nThe  first  application  of multiple-instance  learning  was  to  drug  activity  prediction.  In  the \nactivity prediction application, one objective is to predict whether a candidate drug molecule \nwill  bind strongly to a target protein known to be involved in some disease state.  Typically, \n\n\fA Frameworkfor Multiple-Instance Learning \n\n571 \n\none has examples of molecules that bind well  to the target protein and also of molecules that \ndo not bind well.  Much as in a lock and key, shape is the most important factor in determining \nwhether  a  drug  molecule  and  the  target  protein  will  bind.  However,  drug  molecules  are \nflexible, so they can adopt a wide range of shapes.  A positive example does not convey what \nshape the molecule took in order to bind - only that one of the shapes that the molecule can \n~ake was the right one.  However,  a negative example means that none of the shapes that the \nmolecule can achieve was the right key. \n\nThe multiple-instance learning model was only recently formalized by [Dietterich et ai., 1997]. \nThey assume a hypothesis class of axis-parallel rectangles, and develop algorithms for dealing \nwith the drug activity prediction problem described above.  This work was followed by [Long \nand  Tan,  1996],  where  a  high-degree polynomial  PAC  bound was given for the  number of \nexamples needed to learn in the multiple-instance learning model.  [Auer,  1997] gives a more \nefficient algorithm,  and  [Blum and  Kalai,  1998] shows that learning from multiple-instance \nexamples is reducible to PAC-learning with two sided noise and to the Statistical Query model. \nUnfortunately, the last three papers make the restrictive assumption that all instances from all \nbags are generated independently. \n\nIn this paper,  we  describe a framework called Diverse Density for  solving multiple-instance \nproblems.  Diverse Density is a measure of the intersection of the positive bags minus the union \nof the negative bags.  By maximizing Diverse  Density we  can  find  the point of intersection \n(the desired  concept),  and  also  the  set of feature  weights  that lead to the  best  intersection. \nWe  show results  of applying this algorithm to a difficult synthetic training set as  well as  the \n\"musk\" data set  from  [Dietterich et ai.,  1997].  We  then  use  Diverse  Density  in  two  novel \napplications:  one is to learn a simple description of a person from a series of images that are \nlabeled positive if the person is somewhere in the image and negative otherwise.  The other is \nto deal with a high amount of noise in a stock selection problem. \n\n2  Diverse Density \n\nWe  motivate the  idea of Diverse  Density  through  a  molecular  example.  Suppose that  the \nshape of a candidate molecule can be adequately described by a feature vector.  One instance \nof the  molecule  is  therefore  represented  as  a  point in  n-dimensional feature  space.  As  the \nmolecule changes its shape (through both rigid and non-rigid transformations), it will trace out \na manifold through this n-dimensional space l .  Figure  l(a) shows the paths of four molecules \nthrough a 2-dimensional feature space. \n\nIf a  candidate  molecule  is  labeled  positive,  we  know  that  in  at  least  one  place  along  the \nmanifold, it took on the right shape for it to fit into the target protein. If the molecule is labeled \nnegative, we know that none of the conformations along its manifold will allow binding with \nthe target protein. If we assume that there is only one shape that will bind to the target protein, \nwhat do  the positive and negative manifolds tell  us  about  the  location of the  correct shape \nin  feature  space?  The  answer:  it  is  where  all  positive feature-manifolds  intersect  without \nintersecting any negative feature-manifolds.  For example, in Figure  lea) it is point A. \n\nUnfortunately,  a  multiple-instance bag  does  not  give  us  complete  distribution information, \nbut  only  some  arbitrary  sample  from  that  distribution.  In  fact,  in  applications  other  than \ndrug discovery, there is not even a  notion of an underlying continuous manifold.  Therefore, \nFigure  l(a)  becomes  Figure \nl(b).  The  problem  of trying  to  find  an  intersection changes \n\nI In  practice, one needs to restrict consideration to  shapes of the molecule that have sufficiently low \n\npotential energy.  But, we ignore this restriction in this simple illustration. \n\n\f572 \n\nO.  Maron and T.  Lozano-Perez \n\nposttlVe \n~g.1 \nx \n\nx \n\nnegattv. bag \n0 \n\no \n\npositive \nbag 112 \n\nX \nX \n\npoint A \n\nCl  ~A  A \n\nCl  ~Cl Cl \n\nCl \n\n~Itlv. \nbag #13 \n\n(a) \n\nA \n\nA \n\nx \nx \n\nX \n\nCl \n\nCl \n\nA \nA \nA \n\npositive \nbag 113 \n\nXx \n\nCl \n\no \n\nX \n\n(b) \n\no \n\no \n\nThe  different  shapes  that  a  molecule  can \ntake on are represented as a path.  The inter(cid:173)\nsection point of positive paths is where they \ntook on the same shape. \n\nSamples  taken  along  the  paths.  Section B \nis a high density area,  but point A is a high \nDiverse Density area. \n\nFigure  1:  A motivating example for Diverse Density \n\nto  a  problem  of trying  to  find  an  area  where  there  is  both  high  density  of positive points \nand  low density of negative points.  The difficulty with using regular density is  illustrated in \nFigure  1 (b), Section B.  We  are not just looking for high density, but high \"Diverse Density\". \nWe define Diverse Density at a point to be a measure of how many different positive bags have \ninstances near that point, and how far the negative instances are from that point. \n\n2.1  Algorithms for multiple-instance learning \n\nIn this section, we  derive a probabilistic measure of Diverse Density, and test it on a difficult \nartificial data set.  We  denote positive bags as  Bt, the ph  point in that bag  as  Bt, and the \nvalue  of the  kth  feature  of that  point as  Bt k '  Likewise,  BiJ  represents  a negative  point. \nAssuming  for  now  that  the  true  concept  is  a  single point t,  we  can  find  it  by  maximizing \nPr(x  = t  I Bt, ... , B;;, B], ... , B;) over all  points x  in feature  space.  If we use Bayes' \nrule and  an  uninformative prior over the  concept location,  this is  equivalent to  maximizing \nthe likelihood Pr( Bt , . .. , B;;, B] , ... ,B; I x  =  t).  By making the additional assumption \nthat the  bags are conditionally independent given the  target concept t, the best hypothesis is \nargmax x  TIi Pr(Bt  I x  = t) TIi Pr(B;  I x  = t).  Using Bayes'  rule once more (and again \nassuming a uniform prior over concept location), this is equivalent to \n\nargm:x II Pr(x = t  I Bn II Pr(x = t I B i-) \u00b7 \n\ni \n\ni \n\n(1) \n\nThis is a general definition of maximum Diverse Density, but we need to define the terms in the \nproducts to instantiate it.  One possibility is a noisy-or model:  the probability that not all points \nmissed the target is Pr(x =  t  I Bt) =  Pr(x =  t  I Bi1, B:!i, . .. ) =  1- TIj(I-Pr(x = tIBt\u00bb, \nand likewise Pre x  = t  I Bj- )  =  TIj  (1 - Pr( x  =  t  I Bij\u00bb.  We model the causal probability of \nan  individual instance on a potential target as related to the distance between them.  Namely, \nPre x  = t  I  Bij) = exp( -\nII  Bij  - x  11 2 ).  Intuitively, if one of the instances in a positive bag \nis close to x, then  Pre x  = t  I Bt) is  high.  Likewise,  if every  positive bag  has  an  instance \nclose to x and no negative bags are close to x, then x  will have high Diverse Density.  Diverse \nDensity at an intersection of n  bags is exponentially higher than it is at an intersection of n - 1 \nbags, yet all it takes is one well placed negative instance to drive the Diverse Density down. \n\n\fA Frameworkfor Multiple-Instance Learning \n\n573 \n\nFigure 2:  Negative and positive bags drawn from the same distribution, but labeled according \nto their intersection with the middle square.  Negative instances are dots, positive are numbers. \nThe square contains at least one instance from every positive bag and no negatives, \n\nThe  Euclidean  distance  metric  used  to  measure  \"closeness\"  depends  on  the  features  that \ndescribe the instances.  It is likely that some of the features are irrelevant, or that some should \nbe weighted to  be more  important than others.  Luckily,  we can  use the same framework  to \nfind  not only the best location in feature  space,  but also the  best  weighting of the features. \nOnce  again,  we  find  the  best scaling  of the  individual features  by  finding  the  scalings  that \nmaximize  Diverse Density.  The  algorithm returns both a location x  and  a scaling vector s, \nwhere 1/  Bij  - X W=  Lk ShBijk - Xk)2 . \nNote  that the  assumption  that  all  bags  intersect  at  a  single point is  not  necessary.  We  can \nassume more complicated concepts, such as for example a disjunctive concept ta  V to . In this \ncase,  we  maximize  over a  pair of locations  Xa  and  Xo  and  define  Pr(xa  =  ta  V Xb  =  to  I \nBij) =  maXXa ,Xb(Pr(Xa  = ta  I Bij ), Pr(xo  =  to  I Bij )). \nTo test the algorithm, we created an artificial data set:  5 positive and 5 negative bags, each with \n50 instances. Each instance was chosen uniformly at randomly from a [0 , 100] x [0,  100]  E n2 \ndomain, and the concept was a 5  x  5 square in the middle of the domain.  A bag was labeled \npositive if at least one of its instances fell  within the square, and negative if none did, as shown \nin Figure 2.  The square in the middle contains at least one instance from every positive bag \nand no negative instances.  This is a difficult data set because both positive and negative bags \nare drawn from the same distribution. They only differ in a small area of the domain. \n\nUsing regular density (adding up the contribution of every positive bag and subtracting negative \nbags; this is roughly what a supervised learning algorithm such as n~arest neighbor performs), \nwe  can  plot the  density  surface  across  the  domain.  Figure  3(a)  shows  this  surface  for  the \ndata set in Figure 2,  and it is clear that finding  the peak (a candidate hypothesis) is difficult. \nHowever, when we plotthe Diverse Density surface (using the noisy-or model) in Figure 3(b), \nit  is easy  to  pick out  the  global  maximum  which  is  within  the  desired  concept.  The  other \n\n\f574 \n\nO.  Maron and T.  Ulzano-Perez \n\n(a) Surface using regular density \n\n(b) Surface using Diverse Density \n\nFigure 3:  Density surfaces over the example data of Figure 3 \n\nmajor peaks in Figure 3(b) are the result of a chance concentration of instances from different \nbags.  With a bit more bad luck, one of those peaks could have eclipsed the one in the middle. \nHowever, the chance of this decreases  as the number of bags (training examples) increases. \n\nOne remaining issue is how to find the maximum Diverse Density.  In general, we are searching \nan  arbitrary density landscape and the number of local  maxima and  size of the search  space \ncould prohibit any  efficient exploration.  In  this paper,  we  use gradient ascent  with multiple \nstarting points.  This has worked succesfully in every test case because we know what starting \npoints to  use.  Th'e  maximum  Diverse Density  peak is made of contributions from  some set \nof positive points.  If we  start an  ascent  from  every  positive point,  one of them  is likely  to \nbe  closest to  the  maximum,  contribute the most to it and  have  a climb directly  to  it.  While \nthis heuristic is sensible for maximizing with respect to location, maximizing with respect to \nscaling of feature weights may  still lead to local maxima. \n\n3  Applications of Diverse Density \n\nBy way  of benchmarking,  we  tested the  Diverse  Density approach  on  the  \"musk\" data sets \nfrom  [Dietterich et ai.,  1997],  which  were  also  used  in  [Auer,  1997].  We  also  have  begun \ninvestigating two  new  applications of multiple-instance learning.  We  describe preliminary \nresults on all of these below.  The musk data sets contain feature vectors describing the surfaces \nof a  variety  of low-energy  shapes  from  approximately  100 molecules.  Each  feature  vector \nhas  166 dimensions.  Approximately half ofthese molecules are known to smell \"musky,\" the \nremainder are  very similar molecules that do not smell musky.  There are two musk data sets; \nthe Musk-l data set is smaller, both in having fewer molecules and many fewer instances per \nmolecule.  Many  (72) of the molecules are shared  between the two data sets,  but the second \nset includes more instances for the shared molecules. \n\nWe  approached  the  problem  as  follows:  for  each  run,  we  held  out  a  randomly  selected \n1/10 of the  data  set  as  a  test  set.  We  computed  the  maximum  Diverse  Density  on  the \ntraining set by  multiple gradient ascents,  starting at each positi ve instance.  This produces a \n\n\fA Frameworkfor Multiple-Instance Learning \n\n575 \n\nmaximum feature  point as  well  as  the best feature  weights corresponding to that point.  We \nnote that typically less  than  half of the  166 features  receive  non-zero  weighting.  We  then \ncomputed a distance threshold that optimized classification performance under leave-one-out \ncross validation within the training set.  We used the feature weights and distance threshold to \nclassify the examples of the test set; an example was deemed positive if the weighted distance \nfrom the maximum density point to any of its instances was below the threshold. \n\nThe  table below lists the average  accuracy  of twenty runs,  compared  with the performance \nof the two principal algorithms reported in [Dietterich et aI.,  1997]  (i tera ted-discrim \nAPR and GFS  elim-kde  APR),  as  well  as  the  MULTINST  algorithm from  [Auer,  1997l. \nWe  note that the performances reported for i  terated-discrim  APR  involves choosing \nparameters to maximize test set performance and so probably represents an  upper bound for \naccuracy  on  this  data set.  The  MULTINST  algorithm assumes  that all  instances  from  all \nbags are generated independently.  The Diverse Density results, which required no tuning, are \ncomparable or better than those ofGFS  elim-kde  APR and MULTINST. \n\nMusk Data Set  1 \n\nalgorithm \niterated-discrim APR \nGFS elim-kde APR \nDiverse Density \nMULTINST \n\naccuracy \n\n92.4 \n91.3 \n88.9 \n76.7 \n\nMusk Data Set 2 \n\nalgorithm \niterated-discrim APR \nMULTINST \nDi verse Densi ty \nGFS elim-kde APR \n\naccuracy \n\n89.2 \n84.0 \n82.5 \n80.4 \n\nWe also investigated two new  applications of multiple-instance learning.  The first of these is \nto learn  a simple description of a person from a series  of images that are  labeled positive if \nthey contain the person and negative otherwise.  For a positively labeled image we only know \nthat the person is somewhere in it,  but we  do not know where.  We  sample 54 subimages of \nvarying centers  and sizes  and  declare  them  to  be instances  in one positive bag  since one of \nthem contains the person.  This is repeated for every positive and negative image. \n\nWe use a very simple representation for the instances.  Each subimage is divided into three parts \nwhich roughly correspond to where the head, torso and legs of the person would be. The three \ndominant colors (one  for each  subsection) are  used to  represent  the  image.  Figure 4 shows \na training set where every  bag  included two people,  yet the algorithm learned a description \nof the  person  who  appears  in  all  the  images.  This  technique  is  expanded  in  [Maron  and \nLakshmiRatan,  1998] to learn descriptions of natural  images and use  the learned concept to \nretrieve similar images from a large image database. \n\nAnother new application uses Diverse Density in  the stock selection problem.  Every month, \nthere are stocks that perform well for fundamental reasons and stocks that perform well because \nof flukes;  there are many  more of the latter,  but  we  are  interested in the former.  For every \nmonth, we take the  100 stocks  with the highest return and  put them in a positive bag, hoping \nthat at  least  one of them did well  for  fundamental  reasons.  Negative bags  are created  from \nthe  bottom 5  stocks  in  every  month.  A  stock  instance  is  described  by  17  features  such as \nmomentum, price to fair-value,  etc.  Grantham, Mayo,  Van  Otterloo &  Co.  kindly provided \nus  with data on  the 600 largest US  stocks since  1978.  We  tested the algorithm through five \nruns of training for ten years, then testing on the next year.  In each run, the algorithm returned \nthe stock description (location in feature  space and a scaling of the features) that maximized \nDiverse  Density.  The  test  stocks  were  then  ranked  and decilized  by  distance (in  weighted \nfeature  space) to the max-DD point.  Figure 5 shows the average return of every decile.  The \nreturn  in  the top  decile  (stocks  that are  most  like  the  \"fundamental  stock\")  is  positive and \n\n\f576 \n\nO.  Maron and T.  Lozano-Perez \n\nreturn \n\na. 1 \n\n\u00b70. 2 \n\nn \u2022 \n6  f  8 \n\n5 \n\n4 \n\n9 \n\n10 \n\ndeclle \n\nperson in common. \n\n\u00b70. 4 \nFigure  6:  Black  bars  show  Diverse  Den(cid:173)\nsity's  average  return  on  a  decile,  and  the \nwhite bars show GMO's predictor's return. \n\nhigher than the average return of a GMO predictor.  Likewise, the return in the bottom decile \nis negative and below that of a GMO predictor. \n\n4  Conclusion \n\nIn this paper,  we have shown that Diverse Density is a general tool with which to learn from \nMultiple-Instance examples.  In  addition,  we  have  shown  that  Multiple-Instance problems \noccur  in  a  wide  variety  of domains.  We  attempted  to  show  the  various  ways  in  which \nambiguity can  lead to the  Multiple-Instance framework:  through lack of knowledge  in  the \ndrug  discovery .example,  through  ambiguity  of representation  in  the  vision  example,  and \nthrough a high degree of noise in the stock example. \n\nAcknowledgements \nWe  thank Peter Dayan and Paul Viola at MIT and Tom  Hancock and Chris Darnell at  GMO \nfor helpful discussions and the AFOSR ASSERT program, Parent Grant#:F49620-93-1-0263 \nfor their support of this research. \nReferences \n\n[Auer,  1997]  P.  Auer. On Learning from Multi-Instance Examples:  Empirical Evaluation of a \ntheoretical Approach.  NeuroCOLT Technical  Report Series, NC-TR-97-025, March  1997. \n\n[Blum and  Kalai,  1998]  A.  Blum and A.  Kalai.  A Note on Learning from Multiple-Instance \n\nExamples.  To  appear in Machine Learning,  1998. \n\n[Dietterich et aI.,  1997]  T.  G.  Dietterich,  R.  H.  Lathrop,  and  T.  Lozano-Perez.  Solving the \nMultiple-Instance Problem with Axis-Parallel  Rectangles.  Artificial Intelligence Journal, \n89, 1997. \n\n[Long and Tan,  1996]  P.  M.  Long  and  L.  Tan.  PAC-learning  axis  alligned  rectangles  with \nrespect  to  product distributions from  multiple-instance examples.  In  Proceedings  of the \nJ 996 Conference on Computational Learning Theory,  1996. \n\n[Maron and LakshmiRatan,  1998]  O. Maron and A. LakshmiRatan. Multiple-Instance Learn(cid:173)\n\ning for Natural Scene Classification. In Submitted to CVPR-98, 1998. \n\n\f", "award": [], "sourceid": 1346, "authors": [{"given_name": "Oded", "family_name": "Maron", "institution": null}, {"given_name": "Tom\u00e1s", "family_name": "Lozano-P\u00e9rez", "institution": null}]}