Multiple instance learning

Background

Keeler et al., in his work in early 1990s was the first one to explore the area of MIL. The actual term multi-instance learning was introduced in the middle of the 1990s, by Dietterich et al. while they were investigating the problem of drug activity prediction. They tried to create a learning systems that could predict whether new molecule was qualified to make some drug, or not, through analyzing a collection of known molecules. Molecules can have many alternative low-energy states, but only one, or some of them, are qualified to make a drug. The problem arose because scientists could only determine if molecule is qualified, or not, but they couldn’t say exactly which of its low-energy shapes are responsible for that.

One of the proposed ways to solve this problem was to use supervised learning, and regard all the low-energy shapes of the qualified molecule as positive training instances, while all of the low-energy shapes of unqualified molecules as negative instances. Dietterich et al. showed that such method would have a high false positive noise, from all low-energy shapes that are mislabeled as positive, and thus wasn’t really useful. Their approach was to regard each molecule as a labeled bag, and all the alternative low-energy shapes of that molecule as instances in the bag, without individual labels. Thus formulating multiple-instance learning.

Solution to the multiple instance learning problem that Dietterich et al. proposed is three axis-parallel rectangle (APR) algorithm. It attempts to search for appropriate axis-parallel rectangles constructed by the conjunction of the features. They tested the algorithm on Musk dataset, which is a concrete test data of drug activity prediction and the most popularly used benchmark in multiple-instance learning. APR algorithm achieved the best result, but it should be noted that APR was designed with Musk data in mind.

Problem of multi-instance learning is not unique to drug finding. In 1998, Maron and Ratan found another application of multiple instance learning to scene classification in machine vision, and devised Diverse Density framework. Given an image, an instance is taken to be one or more fixed-size subimages, and the bag of instances is taken to be the entire image. An image is labeled positive if it contains the target scene - a waterfall, for example - and negative otherwise. Multiple instance learning can be used to learn the properties of the subimages which characterize the target scene. From there on, these frameworks have been applied to a wide spectrum of applications, ranging from image concept learning and text categorization, to stock market prediction.

Definitions

If the space of instances is X , then the set of bags is the set of functions N X = { B : X → N } , which is isomorphic to the set of multi-subsets of X . For each bag B ∈ N X and each instance x ∈ X , B ( x ) is viewed as the number of times x occurs in B . Let Y be the space of labels, then a "multiple instance concept" is a map c : N X → Y . The goal of MIL is to learn such a concept. The remainder of the article will focus on binary classification, where Y = { 0 , 1 } .

Assumptions

Most of the work on Multiple instance learning, including Dietterich et al. (1997) and Maron & Lozano-P´erez (1997) early papers, make the assumption regarding the relationship between the instances within a bag and the class label of the bag. Because of its importance, that assumption is often called standard MI assumption.

Standard assumption

The standard assumption takes each instance x ∈ X to have an associated label y ∈ { 0 , 1 } which is hidden to the learner. The pair ( x , y ) is called an"instance-level concept". A bag is now viewed as a multiset of instance-level concepts, and is labeled positive if at least one of its instances has a positive label, and negative if all of its instances have negative labels. Formally, let B = { ( x 1 , y 1 ) , … , ( x n , y n ) } be a bag. The label of B is then c ( B ) = 1 − ∏ i = 1 n ( 1 − y i ) . Standard MI assumption is asymmetric, which means that if the positive and negative labels are reversed, the assumption has a different meaning. Because of that, when we use this assumption, we need to be clear which label should be the positive one.

Standard assumption might be viewed as too strict, and therefore in the recent years, researchers tried to relax that position, which gave rise to other more loose assumptions. Reason for this is the belief that standard MI assumption is appropriate for the Musk dataset, but since MLI can be applied to numerous other problems, some different assumptions could probably be more appropriate. Guided by that idea, Weidmann formulated a hierarchy of generalized instance-based assumptions for MIL. It consists of the standard MI assumption and three types of generalized MI assumptions, each more general than the last, s t a n d a r d ⊂ p r e s e n c e - b a s e d ⊂ t h r e s h o l d - b a s e d ⊂ c o u n t - b a s e d , with the count-based assumption being the most general and the standard assumption being the least general. One would expect an algorithm which performs well under one of these assumptions to perform at least as well under the less general assumptions.

Presence-, threshold-, and count-based assumptions

The presence-based assumption is a generalization of the standard assumption, wherein a bag must contain one or more instances that belong to a set of required instance-level concepts in order to be labeled positive. Formally, let C R ⊆ X × Y be the set of required instance-level concepts, and let # ( B , c i ) denote the number of times the instance-level concept c i occurs in the bag B . Then c ( B ) = 1 ⇔ # ( B , c i ) ≥ 1 for all c i ∈ C R . Note that, by taking C R to contain only one instance-level concept, the presence-based assumption reduces to the standard assumption.

A further generalization comes with the threshold-based assumption, where each required instance-level concept must occur not only once in a bag, but some minimum (threshold) number of times in order for the bag to be labeled positive. With the notation above, to each required instance-level concept c i ∈ C R is associated a threshold l i ∈ N . For a bag B , c ( B ) = 1 ⇔ # ( B , c i ) ≥ l i for all c i ∈ C R .

The count-based assumption is a final generalization which enforces both lower and upper bounds for the number of times a required concept can occur in a positively labeled bag. Each required instance-level concept c i ∈ C R has a lower threshold l i ∈ N and upper threshold u i ∈ N with l i ≤ u i . A bag B is labeled according to c ( B ) = 1 ⇔ l i ≤ # ( B , c i ) ≤ u i for all c i ∈ C R .

GMIL assumption

Scott, Zhang, and Brown (2005) describe another generalization of the standard model, which they call "generalized multiple instance learning" (GMIL). The GMIL assumption specifies a set of required instances Q ⊆ X . A bag X is labeled positive if it contains instances which are sufficiently close to at least r of the required instances Q . Under only this condition, the GMIL assumption is equivalent the to the presence-based assumption. However, Scott et. al. describe a further generalization in which there is a set of attraction points Q ⊆ X and a set of repulsion points Q ¯ ⊆ X . A bag is labeled positive if and only if it contains instances which are sufficiently close to at least r of the attraction points and are sufficiently close to at most s of the repulsion points. This condition is strictly more general than the presence-based, though it does not fall within the above hierarchy.

Collective assumption

In contrast to the previous assumptions where the bags were viewed as fixed, the collective assumption views a bag B as a distribution p ( x | B ) over instances X , and similarly view labels as a distribution p ( y | x ) over instances. The goal of an algorithm operating under the collective assumption is then to model the distribution p ( y | B ) = ∫ X p ( y | x ) p ( x | B ) d x .

Since p ( x | B ) is typically considered fixed but unknown, algorithms instead focus on computing the empirical version: p ^ ( y | B ) = 1 n B ∑ i = 1 n B p ( y | x i ) , where n B is the number of instances in bag B . Since p ( y | x ) is also typically taken to be fixed but unknown, most collective-assumption based methods focus on learning this distribution, as in the single-instance version.

While the collective assumption weights every instance with equal importance, Foulds extended the collective assumption to incorporate instance weights. The weighted collective assumption is then that p ^ ( y | B ) = 1 w B ∑ i = 1 n B w ( x i ) p ( y | x i ) , where w : X → R + is a weight function over instances and w B = ∑ x ∈ B w ( x ) .

Algorithms

There are two major flavors of algorithms for Multiple Instance Learning: instance-based and metadata-based,or embedding-based algorithms. The term "instance-based" denotes that the algorithm attempts to find a set of representative instances based on an MI assumption and classify future bags from these representatives. By contrast, metadata-based algorithms make no assumptions about the relationship between instances and bag labels, and instead try to extract instance-independent information (or metadata) about the bags in order to learn the concept. For a survey of some of the modern MI algorithms see Foulds and Frank

Instance-based algorithms

The earliest proposed MI algorithms were a set of "iterated-discrimination" algorithms developed by Dietterich et. al, and Diverse Density developed by Maron and Lozano-Pérez. Both of these algorithms operated under the standard assumption.

Iterated-discrimination

Broadly, all of the iterated-discrimination algorithms consist of two phases. The first phase is to grow an axis parallel rectangle (APR) which contains at least one instance from each positive bag and no instances from any negative bags. This is done iteratively: starting from a random instance x 1 ∈ B 1 in a positive bag, the APR is expanded to the smallest APR covering any instance x 2 in a new positive bag B 2 . This process is repeated until the APR covers at least one instance from each positive bag. Then, each instance x i contained in the APR is given a "relevance", corresponding to how many negative points it excludes from the APR if removed. The algorithm then selects candidate representative instances in order of decreasing relevance, until no instance contained in a negative bag is also contained in the APR. The algorithm repeats these growth and representative selection steps until convergence, where APR size at each iteration is taken to be only along candidate representatives.

After the first phase, the APR is thought to tightly contain only the representative attributes. The second phase expands this tight APR as follows: a Gaussian distribution is centered at each attribute and a looser APR is drawn such that positive instances will fall outside the tight APR with fixed probability. Though iterated discrimination techniques work well with the standard assumption, they do not generalize well to other MI assumptions.

Diverse Density

In its simplest form, Diverse Density (DD) assumes a single representative instance t ∗ as the concept. This representative instance must be "dense" in that it is much closer to instances from positive bags than from negative bags, as well as "diverse" in that it is close to at least one instance from each positive bag.

Let B + = { B i + } 1 m be the set of positively labeled bags and let B − = { B i − } 1 n be the set of negatively labeled bags, then the best candidate for the representative instance is given by t ^ = arg ⁡ max t D D ( t ) , where the diverse density D D ( t ) = P r ( t | B + , B − ) = arg ⁡ max t ∏ i = 1 m P r ( t | B i + ) ∏ i = 1 n P r ( t | B i − ) under the assumption that bags are independently distributed given the concept t ∗ . Letting B i j denote the jth instance of bag i, the noisy-or model gives:

P ( t | B i j ) is taken to be the scaled distance P ( t | B i j ) ∝ exp ⁡ ( − ∑ k s k 2 ( x k − ( B i j ) k ) 2 ) where s = ( s k ) is the scaling vector. This way, if every positive bag has an instance close to t , then P r ( t | B i + ) will be high for each i , but if any negative bag B i − has an instance close to t , P r ( t | B i − ) will be low. Hence, D D ( t ) is high only if every positive bag has an instance close to t and no negative bags have an instance close to t . The candidate concept t ^ can be obtained through gradient methods. Classification of new bags can then be done by evaluating proximity to t ^ . Though Diverse Density was originally proposed by Maron et. al. in 1998, more recent MIL algorithms use the DD framework, such as EM-DD in 2001 and DD-SVM in 2004, and MILES in 2006

A number of single-instance algorithms have also been adapted to a multiple-instance context under the standard assumption, including

Post 2000, there was a movement away from the standard assumption and the development of algorithms designed to tackle the more general assumptions listed above.

Because of the high dimensionality of the new feature space and the cost of explicitly enumerating all APRs of the original instance space, GMIL-1 is inefficient both in terms of computation and memory. GMIL-2 was developed as a refinement of GMIL-1 in an effort to improve efficiency. GMIL-2 pre-processes the instances to find a set of candidate representative instances. GMIL-2 then maps each bag to a Boolean vector, as in GMIL-1, but only considers APRs corresponding to unique subsets of the candidate representative instances. This significantly reduces the memory and computational requirements.

Metadata-based (or embedding-based) algorithms

By mapping each bag to a feature vector of metadata, metadata-based algorithms allow the flexibility of using an arbitrary single-instance algorithm to perform the actual classification task. Future bags are simply mapped (embedded) into the feature space of metadata and labeled by the chosen classifier. Therefore, much of the focus for metadata-based algorithms is on what features or what type of embedding leads to effective classification. Note that some of the previously mentioned algorithms, such as TLC and GMIL could be considered metadata-based.

They define two variations of kNN, Bayesian-kNN and citation-kNN, as adaptations of the traditional nearest-neighbor problem to the multiple-instance setting.

Generalizations

So far this article has considered multiple instance learning exclusively in the context of binary classifiers. However, the generalizations of single-instance binary classifiers can carry over to the multiple-instance case.