Multiple Instance Learning with Query Bags

Transcription

1 Multiple Instance Learning with Query Bags Boris Babenko UC San Diego Piotr Dollár California Institute of Technology Serge Belongie UC San Diego Abstract In any achine learning applications, precisely labeled data is either burdensoe or ipossible to collect. Multiple Instance Learning (MIL), in which training data is provided in the for of labeled bags rather than labeled instances, is one approach for dealing with abiguously labeled data. In this paper we argue that in any applications of MIL (e.g. iage, audio, text, bioinforatics) a single bag actually consists of a large or infinite nuber of instances, such as all points on a low diensional anifold. For practical reasons, these bags get subsapled before training. Instead, we introduce a MIL forulation which directly odels the underlying structure of these bags. We propose and analyze the query bag odel, in which instances are obtained by repeatedly querying an oracle in a way that can capture relationships between instances. We show that sapling ore instances results in better classification perforance, which otivates us to develop algorithic strategies for sapling any instances while aintaining efficiency. 1 Introduction Traditional supervised learning requires exaple/label pairs during training. However, in any doains precisely labeled data is either burdensoe or ipossible to collect. E.g., less effort is required to specify what digits are present in an iage than to accurately delineate their location, a proble first studied by Keeler et al. [15]. There ay also be inherent abiguity during labeling. The ultiple instance learning fraework (MIL), introduced by Dietterich et al. [1], provides a general paradig for weakly supervised learning. Instead of exaple/label pairs, MIL requires unordered sets of instances, or bags, and labels are assigned to each bag, rather than each instance. A bag is labeled positive if it contains at least one positive instance. In recent years MIL has received significant attention and nuerous algoriths and applications have been proposed [19, 1, 24, 4, 23, 7]. In any MIL applications a bag is generated by taking an object, e.g. an iage or audio wave, and splitting it into overlapping pieces, which serve as the instances. However, both the existing MIL theory and algoriths are underdeveloped for this scenario. E.g., a typical data odel used in classic MIL theory papers [2, 17] assues that bags are finite and all instances are i.i.d., which is not appropriate for any of the datasets we consider here. While existing MIL algoriths do not always explicitly ake these assuptions, they do not offer a principled way of dealing with large or infinite bags. Instead, instances are sapled prior to training. Note that a sapled positive bag ay ay actually contain no positive instances. Furtherore, by separating sapling fro training, the two phases cannot be jointly optiized. These observations inspire us to propose the query bag odel 1 for MIL. In this odel, instances are obtained by repeatedly querying an oracle in a way that can capture relationships between instances (e.g. in any applications all instances in a bag lie on a low diensional anifold). We also propose to integrate the query bag odel directly into existing MIL algoriths. To do so we turn to the filtering paradig, where the data is drawn fro a continuous strea or oracle [6], and extend these ideas to MIL with query bags. We show extensive experients to validate the proposed approach. 1 Note that although the terinology is siilar to [11], our odel and assuptions are quite different. 1

2 Figure 1: Object detection is a typical MIL application. In this exaple a pedestrian is known to exist in the iage but at an unknown location. The iage can be broken up into ultiple overlapping patches using a sliding window, at ultiple scales and orientations, to create a positive bag. Note that the nuber of instances is essentially infinite, since a sliding window can change location by soe sall ɛ. If we subsaple the bag by randoly selecting a liited nuber of patches, we run the chance of issing the object of interest; oreover, relationships between instances are discarded. Instead, in this work we propose strategies that ore directly take advantage of such bag structure. There are nuerous MIL applications where our odel is appropriate. In coputer vision, an iage that contains an object at an unknown position and scale can be partitioned into overlapping sub-iages, one of which will contain the object, either by sliding a rectangular window [23] or through segentation [1]. In coputer audition, prior to processing an audio wave can be partitioned either in the spatial doain [18] or in the frequency doain [21]. In text processing, a docuent can be partitioned into any pieces by using a sliding window [1]. Siilarly, in bioinforatics, a label is often assigned to a long sequence even though a short sub-sequence is responsible for a positive label [2]. In these scenarios the nuber of instances in a bag can becoe prohibitively large or even infinite (e.g. if subpixel iage window locations are used). The query bag odel is well suited for the above scenarios; in fact, while surveying the literature we found very few applications for which it was not well suited (e.g. [1]). 2 The Query Bag Model for MIL We begin with a review of supervised learning and MIL with the classic bag odel. In standard supervised learning data consists of instances {x 1,., x n }, x i X, and labels {y 1,., y n }, y i Y, where the instance/label pairs are usually assued to be drawn i.i.d. fro soe distribution (x i, y i ) D x,y. The goal is to learn a classification function h : X Y that generalizes well to unseen data. In MIL with the classic bag odel data is given in the for {X 1,., X n }, where each X i = {x i1,., x i } is a bag, with associated labels {Y 1,., Y n }, Y i {, 1}. For notational siplicity we assue all bags have the sae cardinality. A bag is positive if one or ore of its instances are positive according to an unknown classifier h : Y i = ax j {1...} {h (x ij )} (1) Alternatively, to odel noisy instance labels, we can write Y i = ax{y ij }, where y ij need not equal h (x ij ). Typically, the instances/label pairs are assued to be drawn i.i.d. fro soe distribution (x ij, y ij ) D x,y, uch like in the standard supervised setting. In the query bag odel odel, rather than being given n bags with instances each, we odel each bag as an oracle that can be queried to obtain an arbitrary nuber of instances per bag. Furtherore, we design the query odel so we can ask for nearby instances. Each query bag i is represented by an object o i O (e.g. a large iage). To saple an instance fro o i, we ust specify a location paraeter α A (e.g. coordinates of an iage patch). A saple x ij X is then generated using a query function Q : O A X. We write x ij = Q(o i, α ij ). Given certain assuptions about Q, foralized below, we can query Q(o i, α + ɛ) for sall ɛ A to get an instance near Q(o i, α). The query function Q and the associated spaces are proble specific, exaples are given in Sec We can express the set of all instances in a bag i using X i = {x x = Q(o i, α), α A}. Note that unlike in the classic bag odel, this set ay be potentially infinite. Likewise, we can define the bag label for query bags in a anner analogous to Eqn. 1: Y i = ax α A {h ( Q(o i, α) ) }, (2) where h is again an instance classifier that deterines the bag labels. In addition, for each bag i, we associate a distribution D i over A that provides soe prior inforation on where positive instances are likely to be located. Most often no prior knowledge is available and this distribution is unifor. Let p be the probability that h (Q(o i, α ij )) = 1 given that bag i is positive and α ij D i. p is a easure of the expected inforativeness of D i and helps deterine the difficulty of the resulting 2

3 MIL proble. Given independent draws fro a positive bag, the probability that all instances are negative is (1 p). Thus, to deterine a bag s label with confidence δ, we ust query instances, where: log(1 δ)/ log(1 p) (3) Given query bags, we can convert the to a suitable forat for standard MIL algoriths by siply querying instances per bag prior to training. We can define a saple of the i th bag, X i = {x i1,., x i } where each x ij is generated using Q(o i, α ij ), where α ij D i. Note that by sapling, there is a chance that for a positive bag, Eqn. 1 will no longer hold. This observation suggests that sapling ore instances for each bag is advantageous. We study this in ore detail in Sec. 3. One intuitive interpretation of the query bag odel is that all instances in a bag i are noisy observations of soe key instance x i. Let α i = argax α {h ( Q(o i, α) ) } and x i = Q(o i, αi ). We call x i the key to bag i the key need not be unique and is arbitrary for negative bags (since α, h (Q(o i, α)) = ). Regardless, every instance x ij in a bag can be written as being soe ɛ ij A away fro the key instance: x ij = Q(o i, αi + ɛ ij). If the ɛ ij were known we could recover the αi and thus the keys x i and the proble would reduce to the standard supervised case. Instead, we can query noisy observations, where D i is the noise distribution (specifically ɛ ij D i αi ). To ake the odel ore tractable, we assue Q is well behaved. Typically, we expect Q to at least be continuous w.r.t. α A, and often we can additionally assue that Q is differentiable. In ost of the exaples that follow A = R d and X = R D where d D; note that under these conditions, a bag defines a d-diensional anifold in R D (cf. Fig. 2). Knowing that Q eets these requireents can be useful. E.g. given a classifier h : R D [, 1] that is also differentiable, we can attept to find the axiu of h(q(o i, α)) w.r.t. α Figure 2: Data generation odel. using gradient descent. This also allows us to query the oracle for nearby instances. Given x ij = Q(o i, α ij ), we can query an instance x ij = Q(o i, α ij + ɛ) such that for sall ɛ R d, x ij x ij 2 is sall (see Sec. 4). Datasets where instances lie on low diensional anifolds have been studied extensively and any interesting techniques have been proposed to take advantage of their structure. These range fro techniques for estiating distances [22] to algoriths that traverse the anifold during training [9]. We draw inspiration fro this work and propose soe effective filtering strategies that take advantage of the anifold structure in Sec Exaples of Query Bags We now consider soe concrete, illustrative exaples of query bags. These exaples are by no eans coprehensive, rather they are eant to clarify the odel and allow us to perfor a nuber of carefully controlled experients in Sec For each odel we define Q : O A X, and also O, A and X. Reaining details are deferred to the experients. Line Bags: A siple case of bags that lie on a anifold is when instances fall on a line. Let o i = {u i, v i } where u i, v i R D, A = R 1, and the query function be Q(o i, α) = u i + αv i. In other words, Q(o i, ) defines a line that extends in direction v i and passes through point u i. Each instance x ij X = R D is a point on that line. Hypercube Bags: Under this odel each bag i is defined to be a hypercube R D (or a square in R 2 ) with side length 2r and center u i. Let o i = {u i, r} where u i R D and r R, and let A = [ 1, 1] D. The query function is Q(o i, α) = u i + α r. Iage Translation Bags: Let I represent a large iage and x = I(α) a p p patch cropped fro iage I centered at location α R 2. Each iage corresponds to a bag and patches cropped fro the iage are instances, e.g. for face detection, αi for a positive bag i would specify the coordinates of a face. More forally, A = R 2, X = R p p, o i = I i and Q(o i, α) = I i (α). The query bag odel is appropriate for any typical MIL applications. E.g., it is straightforward to extend iage bags, defined above, to include other iage transforations, such as rotation and scale change, by increasing the diension of A to account for all degrees of freedo and updating Q accordingly. Also, recall the various applications discussed in Sec. 1. For both bioinforatics 3

4 and text processing, A could encode the position and size of a sliding window; for audio processing, A could encode the frequency bandwidth. Further details are oitted for space. 3 Iplications of Bag Size For the classic bag odel the notion of bag size is clearly defined by the nuber of instances in the bag, which stays constant for any given bag. On the other hand, for our query bag odel the nuber of instances in a bag,, is variable since it depends on the nuber of instances we decide to saple. Therefore, one interpretation of bag size in our odel is how uch we choose to saple the bags. Alternatively, we could consider the distribution D i to define bag size. For exaple, consider the iage bag shown in Fig. 1. D i in this case could be unifor over the entire iage, or over a saller region around the pedestrian (this depends on the data and is out of our control). The bag in the forer case is effectively larger than the latter case. Next we perfor soe experients studying the effects of these interpretations of bag size on the perforance of a MIL algorith Line Bags pos bag 1 pos bag 2 pos bag 3 neg bag Square Bags MNIST Bags Classic MIL Bags Line Bags Square Bags MNIST Classic MIL Bags Figure 3: Error versus. (TOP) illustrations of the datasets; for synthetic datasets the black square designates the positive region, and point color/sybol indicates bag ebership. (BOTTOM) Equal error rate () vs.. See text for details. 3.1 Experients: Error versus We begin with a set of experients that study the perforance of a MIL algorith as we increase the nuber of instances we saple fro each bag. In all experients we use MIL-BOOST, proposed by Viola et al. [23]. In Sec. 4 we propose soe odifications so MIL-BOOST can work directly with the query bag odel; however, for the experients that follow we use the original algorith. To do this we saple a fixed nuber of instances per bag prior to training. We expect perforance of other MIL algoriths to be qualitatively siilar. We easure the effect of varying on 2 synthetic datasets and 1 real dataset: (1) line bags, (2) hypercube bags, and (3) iage translation bags (with the MNIST iage dataset). Thorough details on how each dataset was generated are given below. All errors reported are bag errors. We estiate the predicted bag label by sapling 1 instances per bag. The datasets and plots showing perforance averaged over 25 trials are shown in the first three coluns of Fig. 3. As expected, the results show that as increases error decreases. Furtherore the MNIST results closely reseble the two synthetic cases, suggesting the query bag odel correctly captures the properties of such data. Since our odel and these experients suggest that sapling ore instances is advantageous, we ust consider the coputational consequences. In Sec 4 we propose strategies that allow us to saple any instances while being efficient. Details for synthetic experients: The first 2 experients involve 2D point datasets (X = R 2 ). In each case h (x ij ) = 1 if x ij falls in the square region spanning [5, 5] 2 (the positive region ). For both training and testing 5 positive and 5 negative bags are used. Decision stups served as the weak classifiers. Reaining details follow: (1) For each line bag, u i is saple uniforly fro the positive region for positive bags and fro outside it for negative bags; v i is sapled uniforly fro the unit circle (aking sure that the resulting line for negative bags does not pass through the positive region). D i = N (, 1) for each i. (2) For each hypercube bag, u i is sapled uniforly 4

5 fro [5,.75] 2 for positive bags and fro outside of this region (but inside [, 1] 2 ) for negative bags. D i is unifor over A = [ 1, 1] 2 and r =. See Fig. 3, top, for exaple bags. Details for MNIST digit experient: The final experient was based on a variation of the MNIST handwritten digits [16]. Each digit is originally a iage; we pad each with an 8 pixel border and randoly translate the digit within the resulting iage. Each resulting bag I i contains 256 instances (28 28 patches), one of which is the original iage (see Fig. 3). No prior knowledge is assued, and thus D i is unifor. We arbitrarily labeled bags containing 3 s as positive and the rest negative. Decision stups over Haar features served as the weak classifiers as in [23]. We use 1 positive and negative bags for training, and the rest of the data for testing. 3.2 Experients: Classic Bags We now briefly review the behavior of classic bags with respect to bag size. Using the classic bag odel, Blu and Kalai proposed a PAC algorith [5] with saple coplexity Õ ( D 2 /ε 2) (where D is diensionality of X, is the nuber of instances per bag, and ε is the error). Previous results reported saple coplexity that included even higher powers of [2, 17]. All of these results suggest MIL becoes ore difficult as the bag size increases. Given the independence assuption these results are intuitive the larger the bag, the ore difficult it is to identify positive instances in positive bags. Note that unlike in our odel, here we have no control over the value of. For coparison, we perfor an experient siilar to the ones in the previous section. We generate synthetic classic bags, odeled on the experient described in Fig. 2 of [19]. We use a setup siilar to the previous synthetic experients; we generate bags by uniforly sapling points fro [, 1] 2 and assign the bag a positive label if any of these points fall into the positive region. The results are shown in the last colun of Fig. 3; as is predicted by the PAC bounds, the error goes up as bag size increases. The ore iportant point, however, is that this bag odel is not appropriate for MIL datasets like the ones descried in Sec. 1. Line Bags: VS Variance Experients: Error versus p Finally, we study the alternate interpretation of bag size for query bags. Recall that that the distributions D i provide prior inforation on where positive instances are likely to be located and p quantifies their expected inforativeness. Eqn. 3 suggests that for a fixed bag size a lower value for p would ake learning ore difficult, as sapled positive bags are less likely to contain positive instances. To investigate this, we repeat the line bag experient, setting D i = N (, σ) for each i and varying σ. Note that in this particular setup, σ = iplies p = 1, and as σ increases p approaches =1 =3 =5 =7 = Variance Figure 4: See text. (as the bag becoes bigger ). Results for ultiple values of and σ are shown in Fig. 4; indeed increasing σ degrades perforance while increasing iproves it. Note that in real applications we can control, but the value of p is fixed and dependent on the data. 4 Filtering Instances In the previous sections we saw that sapling ore instances for bags iproves the perforance of a trained classifier. We now present soe strategies for reaping the benefits of sapling any instances, while aintaining efficiency. When training a classifier, we would like to iniize the epirical error over bags. Substituting h for h in Eqn. 2, we can write this as: h = argin h n i=1 ( 1[ax h(q(oi, α)) ) Y i ] (4) α A This is a challenging optiization for a nuber of reasons. One of the key difficulties is that finding the axiu in ters of α ay be intractable, especially if A is infinite. Moreover, for practical reasons we wish to deal with only liited aounts of data at a tie. Recently, Bradley and Schapire [6] introduced a boosting algorith called FilterBoost which learns fro a continuous source of data. Boosting is well adapted to this scenario because training happens in stages weak classifiers are trained sequentially. FilterBoost alternates between training an additional weak classifier and querying the oracle for ore data, using the the latest version of the overall classifier to evaluate the weights of queried instances. 5

6 MILBoost() Input: {o 1,., o n},{y 1,., Y n},{d 1,., D n} 1: for t = 1 to T do 2: Call FilterInstances(i) for i = 1... n 3: Copute weights w ij = L H t ( xij ) 4: Train weak classifier h t ( h t = argax h ij wij h(x ij ) ) 5: Find a t via line search to axiize L a t = argax a L(H t 1 + ah t ) 6: Update strong classifier H t H t 1 + a t h t 7: end for FilterInstances() Paraeters:, SRCH,, ɛ, R, F 1: if t > 1 & od(t, F ) then return 2: if & t > 1 then r = + 1 else r = 1 3: α ij = D i (), x ij = Q(o i, α ij ), for j = r... R 4: Copute p(x ij ) for each j, keep top instances 5: if SRCH then 6: for j = 1 : do 7: if p ( Q(o i, α ij + ɛ) ) > p(x ij ) then 8: α ij = α ij + ɛ, x ij = Q(o i, α ij ) 9: end if 1: end for 11: end if Inspired by this work, we ake an analogous extension of MIL for the case of query bags. In our case, however, we query the oracle for additional instances fro a given bag i, rather than requesting additional labeled data (recall that for MIL labels are required only for bags). Therefore, as opposed to FilterBoost, the nuber of labels required does not depend on the nuber of queries to the oracle. We focus on the MIL-BOOST algorith, which was originally proposed in [23] and extended in [3]. We begin with a brief review of MIL-BOOST, and then describe our extensions to filter instances during training. 4.1 MIL-BOOST Review MIL-BOOST trains a classifier of the for H(x) = T t=1 a th t (x), where h t : X { 1, 1} is a weak classifier, such as a decision stup, and a t is a scalar weight. Using Friedan s gradient boosting fraework [14], we can train this classifier to optiize the log likelihood of the data (since /1 loss is generally difficult to optiize). For shorthand we define p i p(y i = 1 o i ), the probability that bag i is positive. We can write down the log likelihood (defined over bag, not instance, probabilities) as: n L(H) = [Y i log p i + (1 Y i ) log(1 p i )] (5) i=1 We will odel the probability of an instance x being positive as p(x) = σ ( H(x) ), where σ(v) = (1 + exp{ v}) 1 is the sigoid function. What reains is to define the probability of a bag p i as a function of its instances. In our odel with query bags we could write this as: p i = ax α A {p(q(o i, α))} (6) The above is analogous to Eqn. 2. However, as entioned before, the ax over α is difficult to deal with. If we subsaple our bag to get instances {x i1,., x i }, we can re-write the above as p i = ax j {1,...,} {p(x ij )}. Furtherore, we replace the ax with a differentiable approxiation such as Noisy-OR ([3] proposes several options for this). To optiize Eqn. 5, we perfor gradient descent in function space. The MIL-BOOST algorith is suarized above; for details see [23, 3]. 4.2 Filtering Strategies Previously, we observed that when data obeys the query bag odel, having ore instances per bag is advantageous. Since the nuber of candidate instances per bag can be quite large, even infinite, for practical reasons we are liited to a relatively sall nuber of instances per bag for use during training. Instead of using a fixed set, we propose querying the data oracle for new instances during each boosting iteration. In each iteration, we assue we have the coputational resources to train the weak classifiers with instances per bag. Our goal is to optiize Eqn. 5. To get a good estiate of the likelihood, filtering is used to select instances x ij for each bag that have high probability p(x ij ) given the current classifier H t = t k a kh k. For negative bags this is siilar to traditional bootstrapping we want to select the hardest negative exaples. For positive bags we want to get the ost correct exaples. Soe cost is incurred for querying the oracle (e.g. cropping a patch out of an iage) and evaluating p(x ij ). Assue we have the coputational resources to evaluate the probability of R instances per bag and that we can filter instances once in every F iterations of boosting. Given these constraints, we propose the following filtering strategies. Rando Sapling (RAND): The siplest filtering strategy is to query R > saples and keep the with the highest probability, resulting in O(nR) classifier evaluations (n is the nuber of bags). Saples are queried using α ij D i followed by x ij = Q(o i, α ij ). 6

7 Meory (): Instead of sapling a fresh batch of R instances per bag in each iteration, we can retain the instances fro the previous iteration, saple (R ) new instances, and then keep the best of the cobined set (resulting in O(nR) evaluations as before). The classifier changes between iterations; nevertheless, eory allows high probability instances to accuulate over tie. Search (SRCH): Recall that in any scenarios we expect that given x ij = Q(o i, α ij ) and x ij = Q(o i, α ij +ɛ), x ij x ij 2 is sall for sall ɛ A. Although H need not be sooth in the technical sense, it is likely that p(x ij ) p(x ij) is also sall for nearby instances. Thus, given a high probability instance x ij, we can search for nearby x ij such that p(x ij ) > p(x ij). A straightforward way to operationalize this search is to test c nearby locations for each instance at offsets {ɛ 1,., ɛ c } A, and keep the neighbor with the highest probability. Note that this incurs an additional cost of O(nc) classifier evaluations. We suarize the three strategies (rando sapling, eory, and search) in Algorith 3. Meory () and search (SRCH) can be turned on/off, and we can adjust the aount (R) and filtering period (F ). We ephasize that ore sophisticated filtering strategies could be developed, e.g. a true gradient descent strategy to take advantage of underlying anifold structure of the instances as opposed to the steepest descent type search described above. However, our goal is to deonstrate that even siple filtering strategies can result in significant perforance gains. 4.3 Filtering Experients MNIST: vs R RAND SRCH +SRCH MNIST: vs NONE MNIST: vs F RAND SRCH +SRCH MNIST: vs T RAND SRCH +SRCH NONE R F T (A) (B) (C) (D) Figure 5: Plots showing perforance for various filtering strategies (see text) for the MNIST dataset. In each plot we show the equal error rate () plotted against four different paraeters: (A) R, aount of sapling; (B), nuber of instance per bag during training (only shown); (C) F, the filtering period (low F = frequent); and (D) T, the nuber of weak classifiers. MNIST For this experient we used the MNIST handwritten digit dataset described in Sec A = R 2 encodes the 2D pixel coordinates of the center of an iage patch and D i is set to unifor as we have no prior inforation as to where the digit ay be located. For SRCH we used four values of ɛ = (±1, ), (, ±1). Our goal is to easure the effectiveness of various cobinations of filtering strategies. We report the equal error rate (), (the point where false positive equal false negatives), repeating each experient 1-25 ties depending on the easured variance, and plotting the averaged results with standard deviation bars. We easured filtering perforance with on/off and SRCH on/off, while sweeping through the paraeters R (sapling aount), (bag size), F (filtering period) and T (nuber boosting iterations). In each experient we keep 3 of the 4 paraeters fixed at their default values of 16, 4, 1, 64, respectively, and sweep through the fourth. Where appropriate, we also include perforance without filtering (NONE). Results are shown in Fig. 5. In Fig. 5(A) we show the effect of altering R, the nuber of instances queried per bag per iteration. Both strategies converge to low error when R = 8, while the other strategies take up to R = 128. SRCH is beneficial for rando sapling but akes a saller difference when is turned on. Fig. 5(B) shows the advantage of filtering () versus sapling bags and keeping the fixed during training. Both strategies converge to low error as increases, however, with filtering an eighth of the instances ( = 8 vs. = 64) are required during training. Fig. 5(C) shows in ore detail the effects of filtering period F (lower values of F result in ore frequent sapling). Frequent filtering is beneficial for all strategies, however the iproveent is ost significant for strategies (see Sec. 4.4 for discussion). In Fig. 5(D) we plot error vs. the nuber of boosting iterations T. These results are particularly interesting, as without error actually increases for large values of T. This is true also for 7

8 training error (not shown). This behavior is counterintuitive as boosting is know to have excellent generalization, with error decreasing as T increases [13]. We observed that this behavior is not as severe given larger R (results not shown). Essentially, as the classifier becoes ore refined its response becoes ore peaked and rando sapling with low R ay not yield any high probability instances for a given bag, thus preventing the classifier fro converging. Using eory alleviates this proble by guaranteeing high probability instances are retained (ore details in Sec. 4.4). INRIA Pedestrians The MNIST dataset is convenient because it is not particularly large or difficult and it allowed us to study the behavior of the algorith in a well controlled anner. To ensure that the general trends of this behavior hold for a ore challenging dataset, we repeat siilar experients with the INRIA pedestrian dataset [8]. This dataset is currently a standard benchark for pedestrian detection, and is used to evaluate any recent state of the art systes (e.g. [12]). We use a setup analogous to the MNIST experients we resized these iages to be 8 4 pixels in size, and then padded both the negative and positive iages by 3 pixels in each direction by replicating the border. The nuber of possible instances for each bag is therefore Unlike the MNIST dataset, Err INRIA: ERR vs R RAND SRCH +SRCH R (A) Err INRIA: ERR vs NONE (B) Figure 6: Plots showing perforance for various filtering strategies (see text) for the INRIA dataset. In each plot we show the iss rate at a false positive rate of 1% plotted against two paraeters: (A) R, aount of sapling; (B), nuber of instance per bag during training. here it would be ipossible to saple the bags exhaustively and store this in eory 2. We use 5 positive and 5 negative bags for training. The default paraeters used are as follows: T = 128, F = 8, = 4, R = 16. Results are shown in Fig. 6. We see that the general trends we observed with the MNIST data appear in these results as well. 4.4 Analysis We now consider possible explanations for the results discussed above. Let L (H) be defined as in Eqn. 5, but with p i replacing p i. It is easy to show that as, L (H) L(H) for a fixed classifier H. This follows fro the fact that p i p i. This yet again suggests that training with ore sapled instances is advantageous. Now let us consider soe of the filtering strategies proposed above, oentarily assuing the classifier H is fixed during training (obviously this is not the case in the algorith). For RAND, we are drawing novel instances in each iteration, and there is no guarantee that at tie step t + 1 our estiate of the likelihood L (H) would be any closer to L(H) than at tie t. With, however, during every tie step we are effectively evaluating an additional (R ) new instances per bag. Therefore, as t, L (H) L(H). In practice, however, at every tie step t the classifier H changes slightly. This akes it difficult to prove convergence for MIL-BOOST with filtering. In particular, note that for two different classifiers H and H, L (H) > L ( H) does not iply L(H) > L( H). In other words, if the boosting algoriths finds a local axiu of L, it is not necessarily the local axiu of the true log likelihood L. However, the above analysis provides soe intuition as to which filtering strategies should work, in particular, it helps explain why is essential. 5 Conclusions In this work we argued that the ajority of MIL applications in recent literature have diverged fro the assuptions and data generation odels associated with the original MIL forulation. We presented the query bag odel, which ore accurately fits the data in these applications. We showed that sapling ore instances for each bag is advantageous and proposed a nuber of filtering strategies for dealing with a large nuber of saples. These strategies open the door to effectively dealing with a range of MIL applications in coputer vision, audition, text, bioinforatics, and other doains that previously required heavy sapling of bags and thus resulted in suboptial perforance. We envision developing ore sophisticated techniques for specific doains thus extending the effectiveness and applicability of the MIL fraework. 2 Storing the 25 diensional feature vectors for this any instances would require over 6GB of eory. 8

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