Unsupervised Learning of the Set of Local Maxima

This paper describes a new form of unsupervised learning , whose input is a set of unlabeled points that are assumed to be local maxima of an unknown value function v in an unknown subset of the vector space . Two functions are learned : ( i ) a set indicator c , which is a binary classifier , and ( ii ) a comparator function h that given two nearby samples , predicts which sample has the higher value of the unknown function v. Loss terms are used to ensure that all training samples x are a local maxima of v , according to h and satisfy c ( x ) = 1 . Therefore , c and h provide training signals to each other : a point x′ in the vicinity of x satisfies c ( x ) = −1 or is deemed by h to be lower in value than x . We present an algorithm , show an example where it is more efficient to use local maxima as an indicator function than to employ conventional classification , and derive a suitable generalization bound . Our experiments show that the method is able to outperform one-class classification algorithms in the task of anomaly detection and also provide an additional signal that is extracted in a completely unsupervised way . 1 INTRODUCTION . ... from so simple a beginning endless forms most beautiful and most wonderful have been , and are being , evolved . ( Darwin , 1859 ) When we observe the natural world , we see the “ most wonderful ” forms . We do not observe the even larger quantity of less spectacular forms and we can not see those forms that are incompatible with existence . In other words , each sample we observe is the result of optimizing some fitness or value function under a set of constraints : the alternative , lower-value , samples are removed and the samples that do not satisfy the constraints are also missing . The same principle also holds at the sub-cellular level . For example , a gene can have many forms . Some of them are completely synonymous , while others are viable alternatives . The gene forms that become most frequent are those which are not only viable , but which also minimize the energetic cost of their expression ( Farkas et al. , 2018 ) . For example , the genes that encode proteins comprised of amino acids of higher availability or that require lower expression levels to achieve the same outcome have an advantage . One can expect to observe most often the gene variants that : ( i ) adhere to a set of unknown constraints ( “ viable genes ” ) , and ( ii ) optimize an unknown value function that includes energetic efficiency considerations . The same idea , of mixing constraints with optimality , also holds for man-made objects . Consider , for example , the set of houses in a given neighborhood . Each architect optimizes the final built form to cope with various aspects , such as the maximal residential floor area , site accessibility , parking considerations , the energy efficiency of the built product , etc . What architects find most challenging , is that this optimization process needs to correspond to a comprehensive set of state and city regulations that regard , for example , the proximity of the built mass of the house to the lot ’ s boundaries , or the compliance of the egress sizes with current fire codes . In another instance , consider the weights of multiple neural networks trained to minimize the same loss on the same training data , each using a different random initialization . Considering the weights of each trained neural network as a single vector in a sample domain , also fits into the framework of local optimality under constraints . By the nature of the problem , the obtained weights are the local optimum of some loss optimization process . In addition , the weights are sometimes subject to constraints , e.g. , by using weight normalization . The task tackled in this paper is learning the value function and the constraints , by observing only the local maxima of the value function among points that satisfy the constraints . This is an unsupervised problem : no labels are given in addition to the samples . The closest computational problem in the literature is one-class classification ( Moya et al. , 1993 ) , where one learns a classifier c in order to model a set of unlabeled training samples , all from a single class . In our formulation , two functions are learned : a classifier and a separate value function . Splitting the modeling task between the two , a simpler classifier can be used ( we prove this for a specific case ) and we also observe improved empirical performance . In addition , we show that the value function , which is trained with different losses and structure from those of the classifier , models a different aspect of the training set . For example , if the samples are images from a certain class , the classifier would capture class membership and the value function would encode image quality . The emergence of a quality model makes sense , since the training images are often homogeneous in their quality . The classifier c and the value function v provide training signals to each other , in an unsupervised setting , somewhat similar to the way adversarial training is done in GANs ( Goodfellow et al. , 2014 ) , although the situation between c and v is not adversarial . Instead , both work collaboratively to minimize similar loss functions . Let S be the set of unlabeled training samples from a space X . Every x ∈ S satisfies c ( x ) = 1 for a classifier c : X → { ±1 } that models the adherence to the set of constraints ( satisfies or not ) . Alternatively , we can think of c as a class membership function that specifies , if a given input is within the class or not . In addition , we also consider a value function v , and for every point x′ , such that ‖x′−x‖ ≤ , for a sufficiently small > 0 , we have : v ( x′ ) < v ( x ) . This structure leads to a co-training of v and c , such that every point x′ in the vicinity of x can be used either to apply the constraint v ( x′ ) < v ( x ) on v , or as a negative training sample for c. Which constraint to apply , depends on the other function : if c ( x′ ) = 1 , then the first constraint applies ; if v ( x′ ) ≥ v ( x ) , then x′ is a negative sample for c. Since the only information we have on v pertains to its local maxima , we can only recover it up to an unknown monotonic transformation . We therefore do not learn it directly and instead learn a comparator function h , which given two inputs , returns an indication which input has the higher value in v. An alternative view of the learning problem we introduce considers the value function v ( or equivalently h ) as part of a density estimation problem , and not as part of a multi-network game . In this view , c is the characteristic function ( of belonging to the support ) and h is the comparator of the probability density function ( PDF ) . 2 RELATED WORK . The input to our method is a set of unlabeled points . The goal is to model this set . This form of input is shared with the family of methods called one-class classification . The main application of these methods is anomaly detection , i.e. , identifying an outlier , given a set of mostly normal ( the opposite of abnormal ) samples ( Chandola et al. , 2009 ) . The literature on one class classification can be roughly divided into three parts . The first includes the classical methods , mostly kernel-base methods , which were applying regularization in order to model the in-class samples in a tight way ( Schölkopf et al. , 2001 ) . The second group of methods , which follow the advent of neural representation learning , employ classical one-class methods to representations that are learned in an unsupervised way ( Hawkins et al. , 2002 ; Sakurada & Yairi , 2014 ; Xia et al. , 2015 ; Xu et al. , 2015 ; Erfani et al. , 2016 ) , e.g. , by using autoencoders . Lastly , a few methods have attempted to apply a suitable one-class loss , in order to learn a neural networkbased representation from scratch ( Ruff et al. , 2018 ) . This loss can be generic or specific to a data domain . Recently , Golan & El-Yaniv ( 2018 ) achieved state of the art one-class results for visual datasets by training a network to predict the predefined image transformation that is applied to each of the training images . A score is then used to evaluate the success of this classifier on test images , assuming that out of class images would be affected differently by the image transformations . Despite having the same structure of the input ( an unlabeled training set ) , our method stands out of the one-class classification and anomaly detection methods we are aware of , by optimizing a specific model that disentangles two aspects of the data : one aspect is captured by a class membership function , similar to many one-class approaches ; the other aspect compares pairs of samples . This dual modeling captures the notion that the samples are not nearly random samples from some class , but also the local maximum in this class . While “ the local maxima of in-class points ” is a class by itself , a classifier-based modeling of this class would require a higher complexity than a model that relies on the structure of the class as pertaining to local maxima , as is proved , for one example , in Sec . A . In addition to the characterization as local maxima , the factorization between the constraints and the values also assists modeling . This is reminiscent of many other cases in machine learning , where a divide and conquer approach reduces complexity . For example , using prior knowledge on the structure of the problem , helps to reduce the complexity in hierarchical models , such as LDA ( Blei et al. , 2003 ) . While we use the term “ value function ” , and this function is learned , we do not operate in a reinforcement learning setting , where the term value is often used . Specifically , our problem is not inverse reinforcement learning ( Ng & Russell , 2000 ) and we do not have actions , rewards , or policies . 3 METHOD . Recall that S is the set of unlabeled training samples , and that we learn two functions c and v such that for all x ∈ S it holds that : ( i ) c ( x ) = 1 , and ( ii ) x is a local maxima of v. For every monotonic function f , the setting we define can not distinguish between v , and f ◦ v. This ambiguity is eliminated , if we replace v by a binary function h that satisfies h ( x , x′ ) = 1 if v ( x ) ≥ v ( x′ ) and h ( x , x′ ) = −1 otherwise . We found that training h in lieu of v is considerably more stable . Note that we do not enforce transitivity , when training h , and , therefore , h can be such that no underlying v exists . 3.1 TRAINING c AND h When training c , the training samples in S = { xi } mi=1 are positive examples . Without additional constraints , the recovery of c is an ill-posed problem . For example , Ruff et al . ( 2018 ) add an additional constraint on the compactness of the representation space . Here , we rely on the ability to generate hard negative points1 . There are two generators Gc and Gh , each dedicated to generating negative training points to either c or h , as described in Sec . 3.2 below . The two generators are conditioned on a positive point x ∈ S and each generates one negative point per each x : x′ = Gc ( x ) and x′′ = Gh ( x ) . The constraints on the negative points are achieved by multiplying two losses : one pushing c ( x′ ) to be negative , and the other pushing h ( x′′ , x ) to be negative . Let ` ( p , y ) : = − 12 ( ( y+1 ) log ( p ) + ( 1−y ) log ( 1−p ) ) be the binary cross entropy loss for y ∈ { ±1 } . c and h are implemented as neural networks trained to minimize the following losses , respectively : LC : = 1 m ∑ x∈S ` ( c ( x ) , 1 ) + 1 m ∑ x∈S ` ( c ( Gc ( x ) ) , −1 ) · ` ( h ( Gc ( x ) , x ) , −1 ) ( 1 ) LH : = 1 m ∑ x∈S ` ( h ( x , x ) , 1 ) + 1 m ∑ x∈S ` ( c ( Gh ( x ) ) , −1 ) · ` ( h ( Gh ( x ) , x ) , −1 ) ( 2 ) The first sum in LC ensures that c classifies all positive points as positive . The second sum links the outcome of h and c for points generated by Gc . It is given as a multiplication of two losses . This multiplication encourages c to focus on the cases where h predicts with a higher probability that the point Gc ( x ) is more valued than x . 1 “ hard negative ” is a terminology often used in the object detection and boosting literature , which means negative points that challenge the training process . Algorithm 1 Training c and h Require : S : positive training points ; λ : a trade-off parameter ; T : number of epochs . 1 : Initialize c , h , Gc and Gh randomly . 2 : for i = 1 , ... , T do 3 : Train Gc for one epoch to minimize −LC 4 : Train c for one epoch to minimize LC 5 : Train Gh for one epoch to minimize λm ∑ x∈S ||x−Gh ( x ) || − LH 6 : Train h for one epoch to minimize LH 7 : return c , h The first term of LC ( respectively LH ) depends on c ’ s ( respectively h ’ s ) parameters only . The second term of LC ( respectively LH ) , however , depends on both h ’ s and c ’ s parameters as well as Gc ’ s ( respectively Gh ’ s ) parameters . The loss LH is mostly similar . It ensures that h has positive values when the two inputs are the same , at least at the training points . In addition , it ensures that for the generated negative points x′ , h ( x′ , x ) is −1 , especially when c ( x′ ) is high . One can alternatively use a symmetric LH , by including an additional term 1 m ∑ x∈S ` ( c ( Gh ( x ) ) , −1 ) · ` ( h ( x , Gh ( x ) ) , 1 ) . This , in our experiments , leads to very similar results , and we opt for the slightly simpler version .

Summary: In this paper, the authors focus on the task of learning the value function and the constraints in unsupervised case. Different from the conventional classification-based approach, the proposed algorithm uses local maxima as an indicator function. The functions c and h and two corresponding generators are trained in an adversarial way. Besides, the authors analyzed that the proposed algorithm is more efficient than the conventional classification-based approach, and a suitable generalization bound is given. Overall, this work is theoretically complete and experimentally sufficient.