When Do Credal Sets Stabilize? Fixed-Point Theorems for Credal Set Updates
This work addresses a foundational issue in iterative learning under imprecision, relevant for algorithms in variational inference, reinforcement learning, and multi-agent systems, but it is incremental as it builds on existing credal set frameworks.
The paper tackles the problem of whether iterative updates of credal sets in imprecise probabilistic machine learning converge to stable fixed points, and provides the first analysis of conditions for existence and attainment of such fixed points, illustrated with Credal Bayesian Deep Learning.
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.