Fast Lifted MAP Inference via Partitioning
Recently, there has been growing interest in lifting MAP inference algorithms for Markov logic networks (MLNs). A key advantage of these lifted algorithms is that they have much smaller computational complexity than propositional algorithms when symmetries are present in the MLN and these symmetries can be detected using lifted inference rules. Unfortunately, lifted inference rules are sound but not complete and can often miss many symmetries. This is problematic because when symmetries cannot be exploited, lifted inference algorithms ground the MLN, and search for solutions in the much larger propositional space. In this paper, we present a novel approach, which cleverly introduces new symmetries at the time of grounding. Our main idea is to partition the ground atoms and force the inference algorithm to treat all atoms in each part as indistinguishable. We show that by systematically and carefully refining (and growing) the partitions, we can build advanced any-time and any-space MAP inference algorithms. Our experiments on several real-world datasets clearly show that our new algorithm is superior to previous approaches and often finds useful symmetries in the search space that existing lifted inference rules are unable to detect.
Markov logic networks (MLNs) [5] allow application designers to compactly represent and reason about relational and probabilistic knowledge in a large number of application domains including computer vision and natural language understanding using a few weighted first-order logic formulas. These formulas act as templates for generating large Markov networks -the undirected probabilistic graphical model. A key reasoning task over MLNs is maximum a posteriori (MAP) inference, which is defined as the task of finding an assignment of values to all random variables in the Markov network that has the maximum probability. This task can be solved using propositional (graphical model) inference techniques. Unfortunately, these techniques are often impractical because the Markov networks can be quite large, having millions of variables and features.
Recently, there has been growing interest in developing lifted inference algorithms [4,6,17,22] for solving the MAP inference task [1,2,3,7,13,14,16,18,19]. These algorithms work, as much as possible, on the much smaller first-order specification, grounding or propositionalizing only as necessary and can yield significant complexity reductions in practice. At a high level, lifted algorithms can be understood as algorithms that identify symmetries in the first-order specification using lifted inference rules [9,13,19], and then use these symmetries to simultaneously infer over multiple symmetric objects. Unfortunately, in a vast majority of cases, the inference rules are unable to identify several useful symmetries (the rules are sound but not complete), either because the symmetries are approximate or because the symmetries are domain-specific and do not belong to a known type. In such cases, lifted inference algorithms partially ground some atoms in the MLN and search for a solution in this much larger partially propositionalized space.
In this paper, we propose the following straight-forward yet principled approach for solving this partial grounding problem [21,23]: partition the ground atoms into groups and force the inference algorithm to treat all atoms in each group as indistinguishable (symmetric). For example, consider a first-order atom R(x) and assume that x can be instantiated to the following set of constants: {1, 2, 3, 4, 5}. If the atom possesses the so-called non-shared or single-occurrence symmetry [13,19], then the lifted inference algorithm will search over only two assignments: all five groundings of R(x) are either all true or all false, in order to find the MAP solution. When no identifiable symmetries exist, the lifted algorithm will inefficiently search over all possible 32 truth assignments to the 5 1
Recently, there has been growing interest in developing lifted inference algorithms [4,6,17,22] for solving the MAP inference task [1,2,3,7,13,14,16,18,19]. These algorithms work, as much as possible, on the much smaller first-order specification, grounding or propositionalizing only as necessary and can yield significant complexity reductions in practice. At a high level, lifted algorithms can be understood as algorithms that identify symmetries in the first-order specification using lifted inference rules [9,13,19], and then use these symmetries to simultaneously infer over multiple symmetric objects. Unfortunately, in a vast majority of cases, the inference rules are unable to identify several useful symmetries (the rules are sound but not complete), either because the symmetries are approximate or because the symmetries are domain-specific and do not belong to a known type. In such cases, lifted inference algorithms partially ground some atoms in the MLN and search for a solution in this much larger partially propositionalized space.
In this paper, we propose the following straight-forward yet principled approach for solving this partial grounding problem [21,23]: partition the ground atoms into groups and force the inference algorithm to treat all atoms in each group as indistinguishable (symmetric). For example, consider a first-order atom R(x) and assume that x can be instantiated to the following set of constants: {1, 2, 3, 4, 5}. If the atom possesses the so-called non-shared or single-occurrence symmetry [13,19], then the lifted inference algorithm will search over only two assignments: all five groundings of R(x) are either all true or all false, in order to find the MAP solution. When no identifiable symmetries exist, the lifted algorithm will inefficiently search over all possible 32 truth assignments to the 5 ground atoms and will be equivalent in terms of (worst-case) complexity to a propositional algorithm. In our approach, we would partition the domain, say as {{1, 3}, {2, 4, 5}}, and search over only the following 4 assignments: all groundings in each part can be either all true or all false. Thus, if we are lucky and the MAP solution is one of the 4 assignments, our approach will yield significant reductions in complexity even though no identifiable symmetries exist in the problem.
Our approach is quite general and includes the fully lifted and fully propositional approaches as special cases. For instance, setting the partition size k to 1 and n respectively where n is the number of constants will yield exactly the same solution as the one output by the fully lifted and fully propositional approach. Setting k to values other than 1 and n yields a family of inference schemes that systematically explores the regime between these two extremes. Moreover, by controlling the size k of each partition we can control the size of the ground theory, and thus the space and time complexity of our algorithm.
We prove properties and improve upon our basic idea in several ways. First, we prove that our proposed approach yields a consistent assignment that is a lower-bound on the MAP value. Second, we show how to improve the lower bound and thus the quality of the MAP solution by systematically refining the partitions. Third, we show how to further improve the complexity of our refinement procedure by exploiting the exchangeability property of successive refinements. Specifically, we show that the exchangeable refinements can be arranged on a lattice, which can then be searched via a heuristic search procedure to yield an efficient any-time, any-space algorithm for MAP inference. Finally, we demonstrate experimentally that our method is highly scalable and yields close to optimal solutions in a fraction of the time as compared to existing approaches. In particular, our results show that for even small values of k (k bounds the partition size), our algorithm yields close to optimal MAP solutions, clearly demonstrating the power of our approach.
The paper considers lifted MAP inference for relational models.More precisely, it proposes to make use of partitions and partition lattices for symmetry-aware MAP inference. Overall, I very much like the direction of the paper. And the experimental results shows that approach works well; some of theresults are actually quite impressive. However, there are also some downsides:(1) The ideas exploited are floating in the literature and this should be acknowledged (2) The baselines in the experimental evaluation are rather weak; at least the rocket system should be used as another baselineLet me clarify this in more details. The paper is too short on related work. The discussion of the related work has tobe extended. At the current stage, it is not clear where thepresented work goes beyond what is known, in particular when it comes to algebraic lifted MAP approaches. The long list of references on the first page is not very informative there. More importantly,it is not even clear what "in a vast majority of cases, the inferencerules are unable to identify several useful symmetries" actually means running the risk of providing false statements about some of the other approaches. That is, the connections are not formally defined so far in the literature and without this, the claim is not proven at all. Moreover, many of the positive claims made about the presented workalso holds for the algebraic approaches as presented in [2, 13, 15]and also Martin Mladenov, Amir Globerson, Kristian Kersting: Lifted Message Passing as Reparametrization of Graphical ModelsUAI 2014Martin Mladenov, Kristian Kersting:Equitable Partitions of Concave Free Energies.UAI 2015All of them show that we can solve the relaxed MAP LPefficiently using (fractional) automorphisms. Plugging them intoa brunch-and-cut/bound ILP solver would result in a very similar if not identical approach, modulo the efficient recompilation of the lifted network and partial grounding. They could becombined with SA hierarchies as e.g. done inUdi Apsel, Kristian Kersting, Martin Mladenov: Lifting Relational MAP-LPs Using Cluster Signatures.AAAI 2014Moreover, the domain approach presented in the present paper appears to be similar if not identical to the domain graph approach presented in some of the above mentioned related approaches. The authors should clarify the differences. Indeed, one of the big differences is the refinement of the lifted during inference. However, much of the lattice approach is alreadypresented in the above mentioned work, indeed from an algebraicperspective and not from a logical perspective, but this is a detail.Moreover, the idea of relifting during inference has been presented already in [7].That is, the paper misses a discussion of related work. This should carefully acknowledge the origins of ideas and in turn showecase the originality of the presented idea. Without, it reads more like putting well-known ideas together. Nevertheless, putting the ideas together into a single system is really nice and scales inference very well.