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Tandem Inference: an Out-Of-Core Streaming Algorithm for Very Large-Scale Relational Inference

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Tandem Inference: an Out-Of-Core Streaming Algorithm for Very Large-Scale Relational Inference The Thirty-Fourth AAAI Conference on Artificial Intelligence (AAAI-20) Tandem Inference: An Out-of-Core Streaming Algorithm for Very Large-Scale Relational Inference Sriram Srinivasan∗ Eriq Augustine∗ Lise Getoor UC Santa Cruz UC Santa Cruz UC Santa Cruz [email protected] [email protected] [email protected] Abstract knowledge graphs (Pujara et al. 2013), diagnosis of physi- cal systems (Chen, Chen, and Qian 2014), and information Statistical relational learning (SRL) frameworks allow users retrieval (Srinivasan et al. 2019b). to create large, complex graphical models using a compact, rule-based representation. However, these models can quickly While SRL methods have seen a great deal of success, become prohibitively large and not fit into machine memory. they face a major challenge when scaling to large datasets. In this work we address this issue by introducing a novel tech- The benefit of easily generating large graphical models from nique called tandem inference (TI). The primary idea of TI is a few template rules can turn into a curse as the graphical to combine grounding and inference such that both processes models can quickly grow to intractable sizes that do not fit happen in tandem. TI uses an out-of-core streaming approach in memory. Consider a simple transitive rule: Link(A, B) ∧ to overcome memory limitations. Even when memory is not Link(B,C) → Link(A, C), commonly used in conjunc- an issue, we show that our proposed approach is able to do tion with link prediction tasks. When this rule is instantiated inference faster while using less memory than existing ap- (grounded) with a simple dataset of 1000 entries, a graphical proaches. To show the effectiveness of TI, we use a popular SRL framework called Probabilistic Soft Logic (PSL). We model with a billion potentials is generated. implement TI for PSL by proposing a gradient-based infer- To address this issue, several approaches have been pro- ence engine and a streaming approach to grounding. We show posed. Lifted inference (de Salvo Braz, Amir, and Roth that we are able to run an SRL model with over 1B cliques in 2005; Singla and Domingos 2008; den Broeck et al. 2011; under nine hours and using only 10 GB of RAM; previous ap- Kersting 2012; Sarkhel et al. 2014; Kimmig, Mihalkova, and proaches required more than 800 GB for this model and are Getoor 2015; Srinivasan et al. 2019a) is a commonly em- infeasible on common hardware. To the best of our knowl- ployed and effective approach that exploits symmetry in the edge, this is the largest SRL model ever run. data to generate a more compact model on which to perform inference. While effective in many settings, a key draw- 1 Introduction back of these approaches is that evidence or noisy data can break symmetries, making lifting less effective. To address Statistical relational learning (SRL) (Richardson and this issue, approximate lifting approaches have also been Domingos 2006; Getoor and Taskar 2007; Raedt and Ker- proposed (Sen, Deshpande, and Getoor 2009; den Broeck, sting 2011) is an effective method of combining weighted Choi, and Darwiche 2012; Venugopal and Gogate 2014; first-order logic with probabilistic inference to make high- Das et al. 2019) which exploit approximate symmetries, al- quality, structured predictions. A characterizing trait of SRL lowing for greater and more robust compression. However, is the ability to generate large graphical models from a if the ground model lacks significant symmetry, approximate small set of logical templates (rules). Several different SRL lifting may improve tractability only at the cost of correct- frameworks have been proposed, each exploring different ness. types of graphical models, inference algorithms, or hyperpa- rameter learning methods (Richardson and Domingos 2006; Several approaches orthogonal to lifted inference have Bach et al. 2017; Venugopal, Sarkhel, and Gogate 2016). also been proposed that attempt to perform efficient ground- SRL methods have achieved state-of-the-art results in a var- ing by utilizing hybrid database approaches (Niu et al. ied set of domains such as image classification (Aditya, 2011), exploiting the structure of rules (Augustine and Yang, and Baral 2018), activity recognition (London et al. Getoor 2018), perform efficient approximate counting for 2013), natural language processing (Ebrahimi, Dou, and faster inference (Venugopal, Sarkhel, and Gogate 2015; Lowd 2016), recommender systems (Kouki et al. 2015), Sarkhel et al. 2016; Das et al. 2016), or distributing models across multiple machines (Magliacane et al. 2015). However ∗These authors contributed equally to this work. these methods, while quite useful, only provide partial solu- Copyright c 2020, Association for the Advancement of Artificial tions to grounding and efficient inference at scale. The hy- Intelligence (www.aaai.org). All rights reserved. brid database approach increases runtime substantially, ex- 10259 ploiting rule structure requires large amounts of memory to the template rules, called ground rules, which form a HL- store the ground model, approximating counting methods MRF. Each ground rule corresponds to a clique potential in are applicable to discrete graphical models only, and dis- the HL-MRF. A HL-MRF can be formally defined as: tributing across several machines does not reduce the overall Definition 1 (Hinge-loss Markov random field). Let memory required to run a large program. y = {y1,y2, ..., yn} be n RVs, x = {x1,x2, ..., xm} In this paper, we propose an alternate approach to scal- be m observed variables or evidence, and φ = ing which performs inference in tandem with grounding. {φ1,φ2, ..., φK } be K potential functions such that d d This enables us to scale SRL systems to large, previously φi(x, y)=max(i(x, y), 0) i or i(x, y) i . Where i is intractable, models. Our approach, which we refer to as tan- a linear function and di ∈{1, 2} provides a choice of two dem inference (TI), uses a novel streaming grounding archi- different loss functions, di =1(i.e., linear) and di =2(i.e, tecture and an out-of-core inference algorithm that utilizes quadratic). For weights w ∈{w1,w2, ..., wK } a hinge-loss a disk cache in order to consume a fixed amount of mem- energy function can be defined as: ory. This allows TI to scale unbounded by a machine’s main K memory. Furthermore, even with increased I/O overhead, E y|x w φ x, y y ∈ , x ∈ , our approach runs the entire process of grounding and in- ( )= i i( ) ; s.t., [0 1] ; [0 1] (1) ference in a fraction of the runtime required by traditional i=1 approaches. Since TI is orthogonal to lifting and some of the and the HL-MRF is defined as: other strategies, it can be combined with them for further 1 improvements. P (y|x, w)= exp(−E(y|x)) (2) Z(y) The TI concept is general and can potentially be applied to several different SRL frameworks. In this paper, we show Z y exp −E y|x where ( )= y ( ( )). how it can be implemented in probabilistic soft logic (PSL) (Bach et al. 2017). PSL is a SRL framework that generates The template rules in PSL define the interaction between a special kind of undirected graphical model called a hinge- different observed and unobserved RVs. Each predicate of a x, y ∈ , loss Markov random field (HL-MRF). A key distinguishing rule in PSL generates continuous RVs [0 1]. There- factor of a HL-MRF is that it makes a continuous relaxation fore, a rule can be interpreted as continuous relaxations of a∨b on random variables (RVs) which transforms the inference Boolean logical connectives. For example, corresponds ,a b a ∧ b problem into a convex optimization problem. This allows to the hinge potential min(1 + ), and corresponds ,a b − PSL to use optimizers such as alternating direction method to max(0 + 1) (see (Bach et al. 2017) for full details). of multipliers (ADMM) (Boyd et al. 2011) to perform effi- The combination of all ground rules define the hinge-loss cient inference. energy function. Once the energy function is generated the Our key contributions are as follows: 1) we propose a task of inference is to find the maximum aposteriori estimate (MAP) of the RVs y given evidence x. This is performed by general framework, TI, which uses streaming grounding and out-of-core streaming inference to perform memory effi- maximizing the density function or minimizing the energy cient, large-scale inference in SRL frameworks; 2) we derive function in (1). MAP inference is expressed as: a stochastic gradient descent-based inference method (SGD) argmax P (y|x) = argmin E(y|x) (3) and show that the SGD-based method can outperform the tra- y y ditionally used ADMM-based method; 3) we develop an effi- The above expression is a convex optimization problem and cient streaming grounding architecture and SGD-based out- is efficiently solved by a convex optimizer such as ADMM. of-core inference system that runs faster than previous state- We illustrate how HL-MRFs are instantiated using PSL of-the-art systems; 4) through experiments on two large with the following example: models, FRIENDSHIP-500M and FRIENDSHIP-1B, which Example 1. Consider a social network with users U ∈ require over 400GB and 800GB of memory respectively, {U ,...,U } F ∈ , u×u we show that, using just 10GB of memory, we can perform 1 u and friendship links [0 1] , where Friend(U1,U2) denotes the degree of friendship between inference on FRIENDSHIP-500M in under four hours and U1 and U2, near 0 if they are not friends and near 1 if FRIENDSHIP-1B in under nine hours; and 5) we perform an F ⊂ F empirical evaluation on eight realworld datasets to validate they are.
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