Scalable Sampling for Nonsymmetric Determinantal Point Processes

1 INTRODUCTION . A determinantal point process ( DPP ) on M items is a model , parameterized by a symmetric kernel matrix , that assigns a probability to every subset of those items . DPPs have been applied to a wide range of machine learning tasks , including stochastic gradient descent ( SGD ) ( Zhang et al. , 2017 ) , reinforcement learning ( Osogami & Raymond , 2019 ; Yang et al. , 2020 ) , text summarization ( Dupuy & Bach , 2018 ) , coresets ( Tremblay et al. , 2019 ) , and more . However , a symmetric kernel can only capture negative correlations between items . Recent works ( Brunel , 2018 ; Gartrell et al. , 2019 ) have shown that using a nonsymmetric DPP ( NDPP ) allows modeling of positive correlations as well , which can lead to significant predictive performance gains . Gartrell et al . ( 2021 ) provides scalable NDPP kernel learning and MAP inference algorithms , but leaves open the question of scalable sampling . The only known sampling algorithm for NDPPs is the Cholesky-based approach described in Poulson ( 2019 ) , which has a runtime of O ( M3 ) and thus does not scale to large item collections . There is a rich body of work on efficient sampling algorithms for ( symmetric ) DPPs , including recent works such as Derezinski et al . ( 2019 ) ; Poulson ( 2019 ) ; Calandriello et al . ( 2020 ) . Key distinctions between existing sampling algorithms include whether they are for exact or approximate sampling , whether they assume the DPP kernel has some low-rank K M , and whether they sample from the space of all 2M subsets or from the restricted space of size-k subsets , so-called k-DPPs . In this work we focus on exact sampling for low-rank kernels , and provide scalable algorithms for NDPPs . Our contributions are as follows , with runtime and memory details summarized in Table 1 : • Linear-time sampling ( Section 3 ) : We show how to transform the O ( M3 ) Choleskydecomposition-based sampler from Poulson ( 2019 ) into an O ( MK2 ) sampler for rank-K kernels . • Sublinear-time sampling ( Section 4 ) : Using rejection sampling , we show how to leverage existing sublinear-time samplers for symmetric DPPs to implement a sublinear-time sampler for a subclass of NDPPs that we call orthogonal NDPPs ( ONDPPs ) . • Learning with orthogonality constraints ( Section 5 ) : We show that the scalable NDPP kernel learning of Gartrell et al . ( 2021 ) can be slightly modified to impose an orthogonality constraint , yielding the ONDPP subclass . The constraint allows us to control the rejection sampling algorithm ’ s rejection rate , ensuring its scalability . Experiments suggest that the predictive performance of the kernels is not degraded by this change . For a common large-scale setting where M is 1 million , our sublinear-time sampler results in runtime that is hundreds of times faster than the linear-time sampler . In the same setting , our linear-time sampler provides runtime that is millions of times faster than the only previously known NDPP sampling algorithm , which has cubic time complexity and is thus impractical in this scenario . 2 BACKGROUND . Notation . We use [ M ] : = { 1 , . . . , M } to denote the set of items 1 through M . We use IK to denote the K-by-K identity matrix , and often write I : = IM when the dimensionality should be clear from context . Given L ∈ RM×M , we use Li , j to denote the entry in the i-th row and j-th column , and LA , B ∈ R|A|×|B| for the submatrix formed by taking rows A and columns B . We also slightly abuse notation to denote principal submatrices with a single subscript , LA : = LA , A . Kernels . As discussed earlier , both ( symmetric ) DPPs and NDPPs define a probability distribution over all 2M subsets of a ground set [ M ] . The distribution is parameterized by a kernel matrix L ∈ RM×M and the probability of a subset Y ⊆ [ M ] is defined to be Pr ( Y ) ∝ det ( LY ) . For this to define a valid distribution , it must be the case that det ( LY ) ≥ 0 for all Y . For symmetric DPPs , the non-negativity requirement is identical to a requirement that L be positive semi-definite ( PSD ) . For nonsymmetric DPPs , there is no such simple correspondence , but prior work such as Gartrell et al . ( 2019 ; 2021 ) has focused on PSD matrices for simplicity . Normalizing and marginalizing . The normalizer of a DPP or NDPP distribution can also be written as a single determinant : ∑ Y⊆ [ M ] det ( LY ) = det ( L+ I ) ( Kulesza & Taskar , 2012 , Theorem 2.1 ) . Additionally , the marginal probability of a subset can be written as a determinant : Pr ( A ⊆ Y ) = det ( KA ) , for K : = I − ( L+ I ) −1 ( Kulesza & Taskar , 2012 , Theorem 2.2 ) * , where K is typically called the marginal kernel . Intuition . The diagonal element Ki , i is the probability that item i is included in a set sampled from the model . The 2-by-2 determinant det ( K { i , j } ) = Ki , iKj , j −Ki , jKj , j is the probability that both i and j are included in the sample . A symmetric DPP has a symmetric marginal kernel , meaning Ki , j = Kj , i , and hence Ki , iKj , j −Ki , jKj , i ≤ Ki , iKj , j . This implies that the probability of including both i and j in the sampled set can not be greater than the product of their individual inclusion probabilities . Hence , symmetric DPPs can only encode negative correlations . In contrast , NDPPs can have Ki , j and Kj , i with differing signs , allowing them to also capture positive correlations . 2.1 RELATED WORK . Learning . Gartrell et al . ( 2021 ) proposes a low-rank kernel decomposition for NDPPs that admits linear-time learning . The decomposition takes the form L : = V V > + B ( D − D > ) B > for V , B ∈ RM×K , and D ∈ RK×K . The V V > component is a rank-K symmetric matrix , which can model negative correlations between items . The B ( D −D > ) B > component is a rank-K skewsymmetric matrix , which can model positive correlations between items . For compactness of notation , we will write L = ZXZ > , where Z = [ V B ] ∈ RM×2K , and X = [ IK 0 0 D−D > ] ∈ R2K×2K . The marginal kernel in this case also has a rank-2K decomposition , as can be shown via application of the Woodbury matrix identity : K : = I − ( I +L ) −1 = ZX ( I2K +Z > ZX ) −1 Z > . ( 1 ) * The proofs in Kulesza & Taskar ( 2012 ) typically assume a symmetric kernel , but this particular one does not rely on the symmetry . Algorithm 1 Cholesky-based NDPP sampling ( Poulson , 2019 , Algorithm 1 ) 1 : procedure SAMPLECHOLESKY ( K ) . marginal kernel factorization Z , W 2 : Y ← ∅ Q←W 3 : for i = 1 to M do 4 : pi ←Ki , i pi ← z > i Qzi 5 : u← uniform ( 0 , 1 ) 6 : if u ≤ pi then Y ← Y ∪ { i } 7 : else pi ← pi − 1 8 : KA ←KA − KA , iKi , Api for A : = { i+ 1 , . . . , M } Q← Q− Qziz > i Q pi 9 : return Y Note that the matrix to be inverted can be computed from Z and X in O ( MK2 ) time , and the inverse itself takes O ( K3 ) time . Thus , K can be computed from L in time O ( MK2 ) . We will develop sampling algorithms for this decomposition , as well as an orthogonality-constrained version of it . We use W : = X ( I2K +Z > ZX ) −1 in what follows so that we can compactly write K = ZWZ > . Sampling . While there are a number of exact sampling algorithms for DPPs with symmetric kernels , the only published algorithm that clearly can directly apply to NDPPs is from Poulson ( 2019 ) ( see Theorem 2 therein ) . This algorithm begins with an empty set Y = ∅ and iterates through the M items , deciding for each whether or not to include it in Y based on all of the previous inclusion/exclusion decisions . Poulson ( 2019 ) shows , via the Cholesky decomposition , that the necessary conditional probabilities can be computed as follows : Pr ( j ∈ Y | i ∈ Y ) = Pr ( { i , j } ⊆ Y ) Pr ( i ∈ Y ) = Kj , j − ( Kj , iKi , j ) /Ki , i , ( 2 ) Pr ( j ∈ Y | i /∈ Y ) = Pr ( j ∈ Y ) − Pr ( { i , j } ⊆ Y ) Pr ( i /∈ Y ) = Kj , j − ( Kj , iKi , j ) / ( Ki , i − 1 ) . ( 3 ) Algorithm 1 ( left-hand side ) gives pseudocode for this Cholesky-based sampling algorithm† . There has also been some recent work on approximate sampling for fixed-size k-NDPPs : Alimohammadi et al . ( 2021 ) provide a Markov chain Monte Carlo ( MCMC ) algorithm and prove that the overall runtime to approximate ε-close total variation distance is bounded by O ( M2k3 log ( 1/ ( εPr ( Y0 ) ) ) , where Pr ( Y0 ) is probability of an initial state Y0 . Improving this runtime is an interesting avenue for future work , but for this paper we focus on exact sampling . 3 LINEAR-TIME CHOLESKY-BASED SAMPLING . In this section , we show that the O ( M3 ) runtime of the Cholesky-based sampler from Poulson ( 2019 ) can be significantly improved when using the low-rank kernel decomposition of Gartrell et al . ( 2021 ) . First , note that Line 8 of Algorithm 1 , where all marginal probabilities are updated via an ( M − i ) -by- ( M − i ) matrix subtraction , is the most costly part of the algorithm , making overall time and memory complexities O ( M3 ) and O ( M2 ) , respectively . However , when the DPP kernel is given by a low-rank decomposition , we observe that marginal probabilities can be updated by matrix-vector multiplications of dimension 2K , regardless of M . In more detail , suppose we have the marginal kernel K = ZWZ > as in Eq . ( 1 ) and let zj be the j-th row vector in Z . Then , for i 6= j : Pr ( j ∈ Y | i ∈ Y ) = Kj , j − ( Kj , iKi , j ) /Ki , i = z > j ( W − ( Wzi ) ( z > i W ) z > i Wzi ) zj , ( 4 ) Pr ( j ∈ Y | i /∈ Y ) = z > j ( W − ( Wzi ) ( z > i W ) z > i Wzi − 1 ) zj . ( 5 ) The conditional probabilities in Eqs . ( 4 ) and ( 5 ) are of bilinear form , and the zj do not change during sampling . Hence , it is enough to update the 2K-by-2K inner matrix at each iteration , and obtain the marginal probability by multiplying this matrix by zi . The details are shown on the right-hand side of Algorithm 1 . The overall time and memory complexities are O ( MK2 ) and O ( MK ) , respectively . †Cholesky decomposition is defined only for a symmetric positive definite matrix . However , we use the term “ Cholesky ” from Poulson ( 2019 ) to maintain consistency with this work , although Algorithm 1 is valid for nonsymmetric matrices .

Summary: This paper gives an exact sampling algorithm to sample from DPPs based on non-symmetrical, low-rank, kernel matrices of the form $L=VV^\top + B(D-D^\top)B^\top$ where $V$ and $B$ are of size $M$ by $K$ and $D$ is square, non-symmetric and of size $K$ by $K$. The marginal kernel of such a DPP may be written in the form $K=ZWZ^\top$ with $Z$ a $M$ by $2K$ matrix and $W$ a square matrix of size $2K$ that is not symmetric. The authors then: - discuss in Section 2 how Poulson's algorithm is adapted to sample from such NDPPS - in Section 3: show that, as L is low-rank, the generic $O(M^3)$ cost of Poulson's algorithm can be reduced to $O(MK^2$ as updates of the marginal kernel can be efficiently done by only updating the inner $W$ matrix - in Section 4: give a sublinear-time rejection-sampling based algorithm adapting the tree-based algorithm of Gillenwater et al. In this section, they: 	* propose a well-adapted, DPP based on a symmetric kernel $\hat{L}$ that is easy-to-sample (it is a DPP based on a symmetric PSD kernel and can be sampled from many different fast existing algorithms), and for which any set $Y$ verifies $det(L_Y)\leq det(\hat{L}_Y)$ where the upper-bound is actually attained (Thm 1). In order to exactly control the rejection rate of their proposed algorithm (that is equal to $det(\hat{L}+I) / det(L+I)$, the authors simplify the model by supposing that $V$ and $B$ are orthogonal to each other (Thm 2) 	* in passing, the authors observe that the complexity of Gillenwater et al's algorithm can in fact be easily improved by a factor $k$ where $k$ is the size of the sampled set  - in Sections 5 and 6, experiments are provided comparing their samping method with the state-of-the-art