Relational Learning with Variational Bayes

1 INTRODUCTION . American Psychological Association defines relational learning as ( VandenBos & APA , 2007 ) : Definition 1.1 ( Relational learning ) . Learning to differentiate among stimuli on the basis of relational properties rather than absolute properties . In other words , relational learning refers to the ability to recognize and respond to relationship ( called relational property ) among objects irrespective of the nature of those objects ( called absolute property ) . For example ( attributed to Doumas & Hummel ( 2013 ) ) , how do we come to understand that two circles are the same-shape in the same way that two squares are ? In this example , “ same-shape ” is the relational property and object shape is the absolute property . Relational learning has long been recognized as a hallmark of human cognition with strong implications for both human-like learning capabilities and generalization capacity ( Biederman , 1987 ; Medin et al. , 1993 ; Gentner , 2003 ; Penn et al. , 2008 ; Holyoak , 2012 ; Gentner & Smith , 2012 ) . We refer the interested readers to the provided references for a comprehensive discussion on this subject . Contemporaneously , the research on learning data relationships—also commonly called “ relational learning ” —has flourished in the machine learning community where the overarching goal is learning in a context where there may be relationships between learning examples , or where these examples may have a complex internal structure ( i.e. , consist of multiple components and there may be relationships between these components ) ( Getoor & Taskar , 2007 ; De Raedt et al. , 2016 ) . We argue that the key difference between the two “ relational learning ” definitions and their learning objectives is that Definition 1.1 takes the relationship learning problem one step further by requiring the data relationships be learned only on the basis of relational properties rather than absolute properties . To the best of our knowledge , this important distinction—learning relationships irrespective of the absolute properties—has not been rigorously studied in the unsupervised learning community , where most existing methods either encourage or do not constrain the relationships learning through absolute properties . In this work , we propose an unsupervised learning method—variational relational learning ( VRL ) — for addressing the relational learning problem as defined by Definition 1.1 . At its core , VRL encapsulates the relational learning problem with a probabilistic graphical model ( PGM ) in which we perform inference to learn about relational property and other relational processing tasks . Our contribution in this paper is threefold : First , we propose a probabilistic formulation for the relational learning problem defined by Definition 1.1 . Second , we encapsulate the relational learning problem with a PGM in which we perform learning and inference . Third , we propose an efficient and effective learning algorithm that can be trained end-to-end and completely unsupervised . ∗The author completed this research while working at ExxonMobil . Corresponding author e-mail : liu.kuanghung @ gmail.com 2 PROBLEM DEFINITION . We focus on a canonical form of the relational learning problem where we observed a paired dataset X = { ( a ( i ) , b ( i ) ) | i∈ [ 1 .. N ] } consisting of N i.i.d . samples generated from a joint distribution p ( a∈A , b∈B ) . We dissect the information in X into absolute property and relational property where absolute property represents specific features that describe individual a and b , and relational property represents the relationship between a and b irrespective of their absolute property . In this work , we interpret the absolute property of a and b as any information that characterizes ( even if only partially ) the marginal distribution p ( a ) and p ( b ) . We propose to represent the relational property as a latent random variable ( r.v . ) z that satisfies the following constraints : ( i ) p ( a , z ) = p ( a ) p ( z ) , ( ii ) p ( b , z ) = p ( b ) p ( z ) , ( iii ) p ( a , z | b ) 6= p ( a | b ) p ( z | b ) , ( iv ) p ( b , z | a ) 6= p ( b | a ) p ( z | a ) , ( 1 ) where in Eq . 1 ( i ) and 1 ( ii ) we interpret the specification of relational property in Definition 1.1— learning relationships irrespective of the absolute properties—as meaning statistical independence , while in Eq . 1 ( iii ) and 1 ( iv ) we ensure r.v . z contains relevant ( relationship ) information that further informs a and b about one another , i.e. , H ( a | b , z ) < H ( a | b ) , H ( b | a , z ) < H ( b | a ) where H ( · | · ) is the conditional entropy . It is easy to see that the following conditions are necessary for r.v . z to exist : ( 1 ) H ( b | a ) > 0 and H ( a | b ) > 0 , i.e. , a and b can not be fully determined by each other ; ( 2 ) r.v . a , b , z are not mutually independent , i.e. , p ( a , b , z ) 6= p ( a ) p ( b ) p ( z ) . Our goal for relational learning is to learn about relational property z that satisfies Eq . 1 in a completely unsupervised fashion . A motivating example for Eq . 1 is provided in Appendix A.1 . In addition , we are interested in two related relational processing tasks : relational discrimination and relational mapping defined as ( VandenBos & APA , 2007 ) : Definition 2.1 ( Relational discrimination in condition ) . A discrimination based on the relationship between or among stimuli rather than on absolute features of the stimuli . Definition 2.2 ( Relational mapping ) . The ability to apply what one knows about one set of elements to a different set of elements . Relational discrimination allows us to differentiate ( a ( i ) , b ( i ) ) from ( a ( j ) , b ( j ) ) based on their relational properties . And relational mapping allows us to apply the relational property of ( a ( i ) , b ( i ) ) to a different set of data , for example , deduce that b ( j ) is related to a ( j ) in the same way that b ( i ) is related to a ( i ) . 3 METHOD . Learning and inference relational property z that satisfies all four constraints in Eq . 1 is a challenging problem due to the hard independence constraints in Eq . 1 ( i ) and 1 ( ii ) . To overcome this challenge , we first introduce VRL as a tractable learning method that satisfies 3 ( out of 4 ) constraints in Eq . 1— Eq . 1 ( i ) , 1 ( iii ) , 1 ( iv ) . We then discuss VRL ’ s unique optimization challenges , which are partially attributable to its relaxation of the independence requirement in Eq . 1 ( ii ) . 3.1 VARIATIONAL RELATIONAL LEARNING . The proposed VRL method consists of two parts : first , we encapsulate the relational learning problem with a PGM , called VRL-PGM ; we then formulate various relational processing tasks as performing inference and learning in VRL-PGM . The VRL-PGM model , shown in Fig . 1 , samples data a , z , and b from parametric families of distributions—pθ ( a ) , pθ ( z ) , pθ ( b |a , z ) —that are differentiable almost everywhere with respect to ( w.r.t . ) a , z , and θ . In practice , we observe only a set of independent realizations { ( a ( i ) , b ( i ) ) | i ∈ [ 1 .. N ] } while the true parameter θ∗ and the corresponding latent variables z ( i ) are unobserved . A well-known property of the PGM shown in Fig . 1 is that r.v . a and z are independent with no variables observed , but not conditionally independent when b is observed , i.e. , pθ ( a , z ) = pθ ( a ) pθ ( z ) , pθ ( a , z | b ) 6= pθ ( a | b ) pθ ( z | b ) ( Bishop , 2006 ) . Consequently , VRL-PGM can be viewed as a parametric relational learning model that satisfies 3 ( out of 4 ) constraints in Eq . 1—Eq . 1 ( i ) , 1 ( iii ) , 1 ( iv ) ( note that Eq . 1 ( iv ) is trivially satisfied in VRL-PGM ) . Further discussions on the connection between VRL-PGM and the relational learning problem is provided in Appendix A.2 . Having established VRL-PGM , our primary learning objective is to approximate the unknown true likelihood function pθ ( b | a , z ) and posterior pθ ( z | a , b ) . Learning pθ ( z | a , b ) provides us a way to infer ( a ( i ) , b ( i ) ) ’ s relational property z ( i ) ; moreover , it serves as a basis for performing relational discrimination where we compare relational properties between different pairs of data . Learning pθ ( b | a , z ) allows us to perform relational mapping where we use the relational property of ( a ( i ) , b ( i ) ) to map a ( j ) to b ( j ) , i.e. , b ( j ) ∼ pθ ( b | a ( j ) , z ( i ) ) where z ( i ) ∼ pθ ( z | a ( i ) , b ( i ) ) . We estimate the parameter for pθ ( b |a , z ) by following the maximum-likelihood ( ML ) principle , and approximate the true posterior pθ ( z | a , b ) with variational Bayesian approach . More specifically , we use a variational distribution qφ ( z | a , b ) , parameterized by φ , to approximate the unknown ( and often intractable ) true posterior . Both θ and φ are learned through maximizing a variational lower bound , L ( θ , φ ; a ( i ) , b ( i ) ) ( abbreviated as L ( i ) ) , for the conditional log-likelihood log pθ ( b ( i ) | a ( i ) ) ( derivation is provided in Appendix C ) : L ( i ) = Eqφ ( z|a ( i ) , b ( i ) ) [ log pθ ( b ( i ) |a ( i ) , z ) + log pθ ( z ) − log qφ ( z|a ( i ) , b ( i ) ) ] . ( 2 ) Recall that learning z independent of a is central to our relational learning goal . While this independence assumption is built into VRL-PGM , the learning objective L ( i ) does not explicitly force z to be independent of a nor penalize learning a dependent z . In practice , there may be numerous reasons that could break this independence assumption , e.g. , insufficient training data , failure to reach the global optimum , non-identifiability of the model , etc. , and it may be desirable to explicitly enforce independence between z and a . One way to achieve this is to introduce a non-positive function that measures the dependency between a and z with maximum attained when they are independent . For example , we can append the negative mutual information between z and a , −I ( z ; a ) = −DKL ( pθ ( z , a ) ‖ pθ ( z ) pθ ( a ) ) , to L ( i ) : L ( i ) = Eqφ ( z|a ( i ) , b ( i ) ) [ log pθ ( b ( i ) |a ( i ) , z ) + log pθ ( z ) − log qφ ( z|a ( i ) , b ( i ) ) ] − I ( z ; a ) . ( 3 ) Since I ( z ; a ) ≥ 0 and I ( z ; a ) = 0 if and only if z and a are independent , the addition of −I ( z ; a ) to L ( i ) not only maintain the validity of the lower bound , but also retain its quality ( z and a are independent in VRL-PGM ) .

Summary: The paper proposes variational relational learning by learning relations between two inputs via variational inference on a probabilistic graphical model (PGM). The PGM that they use factors as p(a)p(z)p(b|a,z) where a,b are the two inputs and z is the supposed relationship between them. The example shown in the experiments is rotational mnist, where b is a rotated version of a, and z should encode the degree of rotation. The paper learns both the forward network and in inference network in a VAE-like approach; the elbo derivations appear correct to me.