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Combining Local Symmetry Exploitation and Reinforcement Learning for Optimised Probabilistic Inference -- A Work In Progress

Sagad Hamid, Tanya Braun

TL;DR

This work tackles efficient probabilistic inference in factor graphs by recognizing that the elimination order governs intermediate-factor sizes, a problem that is NP-hard. It adapts a reinforcement-learning approach from tensor networks to PGMs via a PGMs–TN duality and uses this to learn contraction orders that minimize the cumulative cost $c(\rho)=\sum_{j=1}^{n-1} c(X_j)$. A key contribution is the introduction of local-symmetry–based compact encodings, where the encoding size satisfies $|\text{Dom}(S_{\boldsymbol{\bar{X}}})|=\binom{n'+d'-1}{d'-1}$, enabling the RL agent to discover more efficient elimination orders. Experimental and methodological results indicate substantial reductions in cumulative VE costs for models with local symmetries, and the framework admits extensions to other encodings and to evidence-driven queries.

Abstract

Efficient probabilistic inference by variable elimination in graphical models requires an optimal elimination order. However, finding an optimal order is a challenging combinatorial optimisation problem for models with a large number of random variables. Most recently, a reinforcement learning approach has been proposed to find efficient contraction orders in tensor networks. Due to the duality between graphical models and tensor networks, we adapt this approach to probabilistic inference in graphical models. Furthermore, we incorporate structure exploitation into the process of finding an optimal order. Currently, the agent's cost function is formulated in terms of intermediate result sizes which are exponential in the number of indices (i.e., random variables). We show that leveraging specific structures during inference allows for introducing compact encodings of intermediate results which can be significantly smaller. By considering the compact encoding sizes for the cost function instead, we enable the agent to explore more efficient contraction orders. The structure we consider in this work is the presence of local symmetries (i.e., symmetries within a model's factors).

Combining Local Symmetry Exploitation and Reinforcement Learning for Optimised Probabilistic Inference -- A Work In Progress

TL;DR

This work tackles efficient probabilistic inference in factor graphs by recognizing that the elimination order governs intermediate-factor sizes, a problem that is NP-hard. It adapts a reinforcement-learning approach from tensor networks to PGMs via a PGMs–TN duality and uses this to learn contraction orders that minimize the cumulative cost . A key contribution is the introduction of local-symmetry–based compact encodings, where the encoding size satisfies , enabling the RL agent to discover more efficient elimination orders. Experimental and methodological results indicate substantial reductions in cumulative VE costs for models with local symmetries, and the framework admits extensions to other encodings and to evidence-driven queries.

Abstract

Efficient probabilistic inference by variable elimination in graphical models requires an optimal elimination order. However, finding an optimal order is a challenging combinatorial optimisation problem for models with a large number of random variables. Most recently, a reinforcement learning approach has been proposed to find efficient contraction orders in tensor networks. Due to the duality between graphical models and tensor networks, we adapt this approach to probabilistic inference in graphical models. Furthermore, we incorporate structure exploitation into the process of finding an optimal order. Currently, the agent's cost function is formulated in terms of intermediate result sizes which are exponential in the number of indices (i.e., random variables). We show that leveraging specific structures during inference allows for introducing compact encodings of intermediate results which can be significantly smaller. By considering the compact encoding sizes for the cost function instead, we enable the agent to explore more efficient contraction orders. The structure we consider in this work is the presence of local symmetries (i.e., symmetries within a model's factors).

Paper Structure

This paper contains 8 sections, 4 theorems, 10 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Contributions
  3. Structure Exploitation & Efficient Inference
  4. Preliminaries
  5. Duality between PGMs and TNs
  6. MDP Formulation & RL Setting
  7. Exploitation of Local Symmetries
  8. Conclusion

Key Result

Theorem 2.1

For factor $\phi(\boldsymbol{Y})$ with $\boldsymbol{Y}_1$,$\boldsymbol{Y}_2 \subseteq \boldsymbol{Y}$, and $\boldsymbol{Y}_1 \cap \boldsymbol{Y}_2 = \emptyset$, we have $\sum_{\boldsymbol{Y}_1}\sum_{\boldsymbol{Y}_2}\phi(\boldsymbol{Y}) = \sum_{\boldsymbol{Y}_2}\sum_{\boldsymbol{Y}_1}\phi(\boldsymbo

Figures (3)

  • Figure 1: (a) A factor graph representing a joint distribution $P(X_1,X_2,X_3)$. (b) A TN with three shared indices and three tensors (labels for indices are omitted). (c) A factor graph for which there is no TN with a one-to-one correspondence.
  • Figure 2: A symmetric factor and its compact encoding.
  • Figure 3: Cumulative intermediate result sizes for VE without (solid line) and with (dashed line) compact encodings based on the presence of local symmetries.

Theorems & Definitions (6)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 5.1: Sum-Out with Symmetries
  • proof
  • Theorem 5.2: Multiplication with Symmetries
  • proof