Efficient Detection of Exchangeable Factors in Factor Graphs

TL;DR

The paper tackles the computational bottleneck of detecting exchangeable factors in factor graphs, a prerequisite for lifted probabilistic inference. It introduces DEFT, a bucket-based algorithm that prunes the permutation search by leveraging subset-valued ranges and a degree-of-freedom metric, upper-bounding the necessary table comparisons by $d$, the minimum bucket degree of freedom. The authors provide theoretical guarantees and show through experiments that DEFT scales to much larger factor arities ($n$ up to 16) than naive or prior approaches, enabling practical detection of symmetries in real-world FG applications. This yields significant speedups for constructing lifted representations and performing efficient inference under domain-size variations. Overall, DEFT advances symmetry detection in probabilistic graphical models, facilitating scalable lifted inference in diverse domains.

Abstract

To allow for tractable probabilistic inference with respect to domain sizes, lifted probabilistic inference exploits symmetries in probabilistic graphical models. However, checking whether two factors encode equivalent semantics and hence are exchangeable is computationally expensive. In this paper, we efficiently solve the problem of detecting exchangeable factors in a factor graph. In particular, we introduce the detection of exchangeable factors (DEFT) algorithm, which allows us to drastically reduce the computational effort for checking whether two factors are exchangeable in practice. While previous approaches iterate all $O(n!)$ permutations of a factor's argument list in the worst case (where $n$ is the number of arguments of the factor), we prove that DEFT efficiently identifies restrictions to drastically reduce the number of permutations and validate the efficiency of DEFT in our empirical evaluation.

Figures (8)

• Figure 1: A toy example for an fg consisting of three Boolean rv $A$, $B$, and $C$ as well as two factors $\phi_1$ and $\phi_2$. The mappings of $\phi_1$ and $\phi_2$ are given in the respective tables on the right with $\varphi_1$, …, $\varphi_4 \in \mathbb{R}^+$.
• Figure 2: Another toy example for an fg consisting of three Boolean rv $A$, $B$, and $C$ as well as two factors $\phi_1$ and $\phi_2$. Observe that the full joint probability distribution encoded by the illustrated fg is the same as the probability distribution encoded by the fg depicted in \ref{['fig:example_fg']}.
• Figure 3: Two exchangeable factors $\phi_1(R_1, R_2, R_3)$ (abbreviated as $\phi_1$) and $\phi_2(R_4, R_5, R_6)$ (abbreviated as $\phi_2$) and their corresponding buckets. Rearranging, for example, the arguments of $\phi_2$ such that they appear in order $R_5$, $R_6$, $R_4$ results in identical tables of potential mappings for $\phi_1$ and $\phi_2$.
• Figure 4: Average run times of the "naive" approach, the approach used in cpr, and deft for different numbers of arguments $n$. For each choice of $n$, the proportion of identical potentials is varied between $0.0$ and $1.0$ and both exchangeable as well as non-exchangeable factors are considered.
• Figure 5: Possible rearrangements of the argument positions $1$, $2$, and $3$ for the sets of possible rearrangements $1 \mapsto \{2,3\}$, $2 \mapsto \{1\}$, and $3 \mapsto \{2,3\}$.

Theorems & Definitions (25)

• Definition 1: Factor Graph
• Example 1
• Definition 2: Exchangeable Factors
• Example 2
• Definition 3: Bucket
• Example 3
• Example 4
• Theorem 1
• proof
• Proposition 1: ?, ?