Bayesian Combinatorial Lottery Ticket Machine: Bayes Meets Extremal Combinatorics

ICLR 2026 Conference Submission18617 Authors

Keywords: Extremal combinatorics, Lottery Ticket hypothesis, Bayesian nonparametrics

Abstract: This paper proposes a fundamentally new framework for Bayesian machine learning (BML) by introducing redundant objects inspired by \emph{extremal combinatorics}.} In general, BML offers the advantage of quantifying uncertainty across all possible models (hypotheses) that explain data (observable phenomena in nature and society), making it particularly useful in applications that require transparency in data analysis, such as biomedical health and finance. However, its practical use is hindered by challenges in (1) model construction (often involving measure theory), (2) inference algorithm design (such as Markov chain Monte Carlo (MCMC) tailored to individual case studies), and (3) theoretical analysis (such as the mixing time of MCMC). This paper aims to mitigate the aforementioned obstacles, particularly for BML that infers combinatorial structures (permutations, partitions, binary sequences, binary trees, acyclic directed graphs, Cambrian trees, rectangular partitions, etc.) including consensus ranking, classification, factor analysis, hierarchical clustering, causal inference, phylogenetic analysis, and relational data analysis. Our key insight is to apply redundant universal objects, inspired by extremal combinatorics, as general-purpose generative probabilistic models independent of specific application scenarios. Our contributions include (1) a unified model construction method that expresses random target objects by randomly selecting a substructure from the universal objects, (2) a simple MCMC algorithm for Bayesian inference that is naturally derived from this model representation, and (3) a clue to the theoretical upper bound on the MCMC mixing time.

Primary Area: probabilistic methods (Bayesian methods, variational inference, sampling, UQ, etc.)

Submission Number: 18617