Scalable Learning and MAP Inference for Nonsymmetric Determinantal Point Processes

1 INTRODUCTION . Determinantal point processes ( DPPs ) have proven useful for numerous machine learning tasks . For example , recent uses include summarization ( Sharghi et al. , 2018 ) , recommender systems ( Wilhelm et al. , 2018 ) , neural network compression ( Mariet & Sra , 2016 ) , kernel approximation ( Li et al. , 2016 ) , multi-modal output generation ( Elfeki et al. , 2019 ) , and batch selection , both for stochastic optimization ( Zhang et al. , 2017 ) and for active learning ( Bıyık et al. , 2019 ) . For subset selection problems where the ground set of items to select from has cardinality M , the typical DPP is parameterized by an M ×M kernel matrix . Most prior work has been concerned with symmetric DPPs , where the kernel must equal its transpose . However , recent work has considered the more general class of nonsymmetric DPPs ( NDPPs ) and shown that these have additional useful modeling power ( Brunel , 2018 ; Gartrell et al. , 2019 ) . In particular , unlike symmetric DPPs , which can only model negative correlations between items , NDPPs allow modeling of positive correlations , where the presence of item i in the selected set increases the probability that some other item j will also be selected . There are many intuitive examples of how positive correlations can be of practical importance . For example , consider a product recommendation task for a retail website , where a camera is found in a user ’ s shopping cart , and the goal is to display several other items that might be purchased . Relative to an empty cart , the presence of the camera probably increases the probability of buying an accessory like a tripod . Although NDPPs can theoretically model such behavior , the existing approach for NDPP learning and inference ( Gartrell et al. , 2019 ) is often impractical in terms of both storage and runtime requirements . These algorithms require memory quadratic in M and time quadratic ( for inference ) or cubic ( for learning ) in M ; for the not-unusual M of 1 million , this requires storing 8TB-size objects in memory , with runtime millions or billions of times slower than that of a linear-complexity method . In this work , we make the following contributions : Learning : We propose a new decomposition of the NDPP kernel which reduces the storage and runtime requirements of learning and inference to linear in M . Fortuitously , the modified decomposition retains all of the previous decomposition ’ s modeling power , as it covers the same part of the NDPP kernel space . The algebraic manipulations we apply to get linear complexity for this decomposition can not be applied to prior work , meaning that our new decomposition is crucial for scalability . Inference : After learning , prior NDPP work applies a DPP conditioning algorithm to do subset expansion ( Gartrell et al. , 2019 ) , with quadratic runtime in M . However , prior work does not examine the general problem of MAP inference for NDPPs , i.e. , solving the problem of finding the highestprobability subset under a DPP . For symmetric DPPs , there exists a standard greedy MAP inference algorithm that is linear in M . In this work , we develop a version of this algorithm that is also linear for low-rank NDPPs . The low-rank requirement is unique to NDPPs , and highlights the fact that the transformation of the algorithm from the symmetric to the nonsymmetric space is non-trivial . To the best of our knowledge , this is the first MAP algorithm proposed for NDPPs . We combine the above contributions through experiments that involve learning NDPP kernels and applying MAP inference to these kernels to do subset selection for several real-world datasets . These experiments demonstrate that our algorithms are much more scalable , and that the new kernel decomposition matches the predictive performance of the decomposition from prior work . 2 BACKGROUND . Consider a finite set Y = { 1 , 2 , . . . , M } of cardinalityM , which we will also denote by [ [ M ] ] . A DPP on [ [ M ] ] defines a probability distribution over all of its 2M subsets . It is parameterized by a matrix L ∈ RM×M , called the kernel , such that the probability of each subset Y ⊆ [ [ M ] ] is proportional to the determinant of its corresponding principal submatrix : Pr ( Y ) ∝ det ( LY ) . The normalization constant for this distribution can be expressed as a single M ×M determinant : ∑ Y⊆ [ [ M ] ] det ( LY ) = det ( L + I ) ( Kulesza et al. , 2012 , Theorem 2.1 ) . Hence , Pr ( Y ) = det ( LY ) / det ( L + I ) . We will use PL to denote this distribution . For intuition about the kernel parameters , notice that the probabilities of singletons { i } and { j } are proportional to Lii and Ljj , respectively . Hence , it is common to think of L ’ s diagonal as representing item qualities . The probability of a pair { i , j } is proportional to det ( L { i , j } ) = LiiLjj − LijLji . Thus , if −LijLji < 0 , this indicates i and j interact negatively . Similarly , if −LijLji > 0 , then i and j interact positively . Therefore , off-diagonal terms determine item interactions . ( The vague term “ interactions ” can be replaced by the more precise term “ correlations ” if we consider the DPP ’ s marginal kernel instead ; see Gartrell et al . ( 2019 , Section 2.1 ) for an extensive discussion . ) In order to ensure that PL defines a probability distribution , all principal minors of L must be non-negative : det ( LY ) ≥ 0 . Matrices that satisfy this property are called P0-matrices ( Fang , 1989 , Definition 1 ) . There is no known generative method or matrix decomposition that fully covers the space of all P0 matrices , although there are many that partially cover the space ( Tsatsomeros , 2004 ) . One common partial solution is to use a decomposition that covers the space of symmetric P0 matrices . By restricting to the space of symmetric matrices , one can exploit the fact that L ∈ P0 if L is positive semidefinite ( PSD ) * ( Prussing , 1986 ) . Any symmetric PSD matrix can be written as the Gramian matrix of some set of vectors : L : = V V > , where V ∈ RM×K . Hence , the V V > decomposition provides an easy means of generating the entire space of symmetric P0 matrices . It also has a nice intuitive interpretation : we can view the i-th row of V as a length-K feature vector describing item i . Unfortunately , the symmetry requirement limits the types of correlations that a DPP can capture . A symmetric model is able to capture only nonpositive interactions between items , since LijLji = L2ij ≥ 0 , whereas a nonsymmetric L can also capture positive correlations . ( Again , see Gartrell et al . ( 2019 , Section 2.1 ) for more intuition . ) To expand coverage to nonsymmetric matrices in P0 , it is natural to consider nonsymmetric PSD matrices . In what follows , we denote by P+0 the set of all nonsymmetric ( and symmetric ) PSD matrices . Any nonsymmetric PSD matrix is in P0 ( Gartrell et al. , 2019 , Lemma 1 ) , so P+0 ⊆ P0 . However , unlike in the symmetric case , the set of nonsymmetric PSD * Recall that a matrix L ∈ RM×M is defined to be PSD if and only if x > Lx ≥ 0 , for all x ∈ RM . matrices does not fully cover the set of nonsymmetric P0 matrices . For example , consider L = ( 1 5/3 1/2 1 ) with det ( L { 1 } ) , det ( L { 2 } ) , det ( L { 1,2 } ) ≥ 0 , but x > Lx < 0 for x = ( −1 1 ) . Still , nonsymmetric PSD matrices cover a large enough portion of the P0 space to be useful in practice , as evidenced by the experiments of Gartrell et al . ( 2019 ) . This work covered the P+0 space by using the following decomposition : L : = S + A , with S : = V V > for V ∈ RM×K , and A : = BC > −CB > for B , C ∈ RM×K . This decomposition makes use of the fact that any matrix L can be decomposed uniquely as the sum of a symmetric matrix S = ( L + LT ) /2 and a skew-symmetric matrix A = ( L−LT ) /2 . All skew-symmetric matrices A are trivially PSD , since x > Ax = 0 for all x ∈ RM . Hence , the L here is guaranteed to be PSD simply because its S uses the standard Gramian decomposition V V > . In this work we will also only consider P+0 , and leave to future work the problem of finding tractable ways to cover the rest of P0 . We propose a new decomposition of L that also covers the P+0 space , but allows for more scalable learning . As in prior work , our decomposition has inner dimension K that could be as large as M , but is usually much smaller in practice . Our algorithms work well for modest values of K. In cases where the natural K is larger ( e.g. , natural language processing ) , random projections can often be used to significantly reduce K ( Gillenwater et al. , 2012a ) . 3 NEW KERNEL DECOMPOSITION AND SCALABLE LEARNING . Prior work on NDPPs proposed a maximum likelihood estimation ( MLE ) algorithm ( Gartrell et al. , 2019 ) . Due to that work ’ s particular kernel decomposition , this algorithm had complexity cubic in the number of items M . Here , we propose a kernel decomposition that reduces this to linear in M . We begin by showing that our new decomposition covers the space of P+0 matrices . Before diving in , let us define Σi : = ( 0 λi −λi 0 ) as shorthand for a 2× 2 block matrix with zeros on-diagonal and opposite values off-diagonal . Then , our proposed decomposition is as follows : L : = S + A , with S : = V V > and A : = BCB > , ( 1 ) where V , B ∈ RM×K , and C ∈ RK×K is a block-diagonal matrix with some diagonal blocks of the form Σi , with λi > 0 , and zeros elsewhere . The following lemma shows that this decomposition covers the space of P+0 matrices . Lemma 1 . Let A ∈ RM×M be a skew-symmetric matrix with rank ` ≤ M . Then , there exist B ∈ RM× ` and positive numbers λ1 , . . . , λb ` /2c , such that A = BCB > , where C ∈ R ` × ` is the block-diagonal matrix with b ` /2c diagonal blocks of size 2 given by Σi , i = 1 , . . . , b ` /2c and zero elsewhere . The proof of Lemma 1 and all subsequent results can be found in Appendix F. With this decomposition in hand , we now proceed to show that it can be used for linear-time MLE learning . To do so , we must show that corresponding NDPP log-likelihood objective and gradient can be computed in time linear in M . Given a collection of n observed subsets { Y1 , ... , Yn } composed of items from Y = [ [ M ] ] , the full formulation of the regularized log-likelihood is : φ ( V , B , C ) = 1 n n∑ i=1 log det ( VYiV > Yi +BYiCB > Yi ) − log det ( V V > +BCB > + I ) −R ( V , B ) , ( 2 ) where VYi ∈ R|Yi|×K denotes a matrix composed of the rows of V that correspond to the items in Yi . The regularization term , R ( V , B ) , is defined as follows : R ( V , B ) = α M∑ i=1 1 µi ‖vi‖22 + β M∑ i=1 1 µi ‖bi‖22 , ( 3 ) where µi counts the number of occurrences of item i in the training set , vi and bi are rows of V and B , respectively , and α , β > 0 are tunable hyperparameters . This regularization is similar to that of prior works ( Gartrell et al. , 2017 ; 2019 ) . We omit regularization for C. Theorem 1 shows that computing the regularized log-likelihood and its gradient both have time complexity linear in M . The complexities also depend on K , the rank of the NDPP , and K ′ , the size of the largest observed subset in the data . For many real-world datasets we observe that K ′ M , and we set K = K ′ . Hence , linearity in M means that we can efficiently perform learning for datasets with very large ground sets , which is impossible with the cubic-complexity L decomposition in prior work ( Gartrell et al. , 2019 ) . Theorem 1 . Given an NDPP with kernel L = V V > +BCB > , parameterized by V of rank K , B of rank K , and a K ×K matrix C , we can compute the regularized log-likelihood ( Eq . 2 ) and its gradient in O ( MK2 +K3 +nK ′3 ) time , where K ′ is the size of the largest of the n training subsets .

Summary: Nonsymmetric determinantal point processes (NDPPs) received some attention recently because they allow modeling of both negative and positive correlations between items. This paper developed scalable learning and MAP inference algorithms with space and time complexity linear in ground set size, which is a huge improvement compared to previous approaches. Experimental results show that the algorithms scale significantly better, and can roughly match the predictive performance of prior work.