When Do Credal Sets Stabilize? Fixed-Point Theorems for Credal Set Updates

TL;DR

This work addresses the stability of iterative updates on credal sets, i.e., nonempty, closed, convex subsets of probability measures, within imprecise probabilistic machine learning. It develops fixed-point theory for credal-set updates by leveraging Hausdorff-continuity of update rules and the weak$^\star$ topology, proving existence (nonempty, compact) of fixed points and, under stronger contraction-type conditions, uniqueness and convergence of iterates. The results are instantiated in Credal Bayesian Deep Learning (CBDL), with conditions on priors and likelihoods ensuring continuity and enabling geometric convergence rates, and they extend to pessimistic variants (PCBDL). An illustrative FGCS example demonstrates practical convergence behavior, providing a rigorous foundation for stable learning under imprecision and guiding the design of IPML algorithms with guaranteed convergence properties. Overall, the paper illuminates the dynamics of imprecise probabilistic learning and offers a framework for analyzing when credal-set updates stabilize, with potential impact on robust continual and active learning settings.

Abstract

Many machine learning algorithms rely on iterative updates of uncertainty representations, ranging from variational inference and expectation-maximization, to reinforcement learning, continual learning, and multi-agent learning. In the presence of imprecision and ambiguity, credal sets -- closed, convex sets of probability distributions -- have emerged as a popular framework for representing imprecise probabilistic beliefs. Under such imprecision, many learning problems in imprecise probabilistic machine learning (IPML) may be viewed as processes involving successive applications of update rules on credal sets. This naturally raises the question of whether this iterative process converges to stable fixed points -- or, more generally, under what conditions on the updating mechanism such fixed points exist, and whether they can be attained. We provide the first analysis of this problem and illustrate our findings using Credal Bayesian Deep Learning as a concrete example. Our work demonstrates that incorporating imprecision into the learning process not only enriches the representation of uncertainty, but also reveals structural conditions under which stability emerges, thereby offering new insights into the dynamics of iterative learning under imprecision.

Figures (2)

• Figure 1: An illustration of Theorem \ref{['fixed-point1']} with sequences of credal sets defined in a 2d probability simplex. As we apply our Hausdorff continuous update rule $f$ to a credal set repeatedly, it eventually arrives at a fixed "point".
• Figure 2: Illustration of finitely generated credal sets reaching their fixed points. Starting with a credal set generated by finitely many extreme distributions, we repeatedly apply the updating map $f$. With each iteration, the distance between the parameters of the distributions located at the extreme corners of the set decreases. This contraction effect continues until the successive distances vanish, at which point the credal set collapses to its fixed point.

Theorems & Definitions (24)

• Remark 1: On Finite Additivity
• Theorem 1: Fixed Point Theorem for Credal Updating
• Example 1: On the Importance of a Continuous Updating Rule
• Example 2: Credal Bayesian Deep Learning, CBDL
• Proposition 2
• Corollary 2.1: Unique Fixed Point Theorem for Credal Updating
• proof
• Corollary 2.2: Convergence of $(\mathcal{P}_n)$
• proof
• Remark 2: A Weaker Requirement on $\mathcal{X}$