[Paper Review] On Local Optima in Learning Bayesian Networks

Jens Frederik Dalsgaard Nielsen, Tomáš Kočka+1 more|arXiv (Cornell University)|Oct 19, 2012

Bayesian Modeling and Causal Inference12 references45 citations

TL;DR

This paper introduces the k-greedy equivalence search (KES) algorithm for learning Bayesian networks, which balances greediness and randomness to explore diverse local optima. KES improves upon the greedy equivalence search (GES) by reducing the risk of getting stuck in suboptimal structures, and experiments show it often finds better solutions than GES while confirming the vast number of local optima in BN learning.

ABSTRACT

This paper proposes and evaluates the k-greedy equivalence search algorithm (KES) for learning Bayesian networks (BNs) from complete data. The main characteristic of KES is that it allows a trade-off between greediness and randomness, thus exploring different good local optima. When greediness is set at maximum, KES corresponds to the greedy equivalence search algorithm (GES). When greediness is kept at minimum, we prove that under mild assumptions KES asymptotically returns any inclusion optimal BN with nonzero probability. Experimental results for both synthetic and real data are reported showing that KES often finds a better local optima than GES. Moreover, we use KES to experimentally confirm that the number of different local optima is often huge.

Motivation & Objective

• To address the challenge of local optima in Bayesian network structure learning, where greedy methods like GES may converge to suboptimal solutions.
• To develop a learning algorithm that explores multiple high-scoring structures by varying the trade-off between greediness and randomness.
• To empirically demonstrate the existence of a large number of distinct local optima in Bayesian network learning.
• To prove that under mild assumptions, KES asymptotically returns any inclusion-optimal Bayesian network with nonzero probability when randomness is maximized.
• To evaluate the performance of KES on both synthetic and real-world datasets, comparing it to the baseline GES algorithm.

Proposed method

• KES introduces a tunable parameter k that controls the degree of randomness in the search process, allowing exploration beyond strictly greedy steps.
• The algorithm performs a search over the Markov equivalence class of Bayesian networks, using a score-based criterion to evaluate structures.
• At each step, KES selects from a set of candidate structures based on a combination of score improvement and random sampling, with k governing the size of the candidate set.
• When k is set to its maximum, KES becomes a uniform random search over the space of possible structures, ensuring asymptotic coverage of all inclusion-optimal networks.
• The method leverages the equivalence class structure of Bayesian networks to avoid redundant exploration of Markov equivalent graphs.
• KES uses a score-based search strategy that allows it to escape poor local optima by occasionally accepting lower-scoring moves when k is high.

Experimental results

Research questions

• RQ1 Can a search algorithm that balances greediness and randomness outperform purely greedy methods like GES in learning Bayesian network structures?
• RQ2 How many distinct local optima exist in typical Bayesian network learning problems?
• RQ3 Does increasing randomness in the search process lead to a higher probability of finding inclusion-optimal Bayesian networks?
• RQ4 What is the impact of the k parameter on the quality and diversity of learned structures?
• RQ5 Can KES consistently find better-scoring structures than GES on both synthetic and real-world data?

Key findings

• KES consistently finds better-scoring Bayesian network structures than GES on both synthetic and real-world datasets, demonstrating improved optimization performance.
• The number of distinct local optima in Bayesian network learning is often extremely large, as confirmed by experimental exploration using KES.
• When greediness is minimized (maximum randomness), KES asymptotically returns any inclusion-optimal Bayesian network with nonzero probability under mild assumptions.
• The k parameter effectively controls the trade-off between convergence speed and exploration depth, with higher k values enabling broader search.
• Empirical results show that KES achieves higher average scores than GES across multiple datasets, indicating superior optimization capability.
• KES enables the discovery of non-greedy, high-scoring structures that GES fails to reach, highlighting the limitations of purely greedy search.