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.

翻译：许多机器学习算法依赖于不确定性表示的迭代更新，涵盖变分推断、期望最大化、强化学习、持续学习以及多智能体学习等领域。在存在不精确性和模糊性的情况下，信度集——即概率分布的闭凸集——已成为表示不精确概率信念的常用框架。在此类不精确性条件下，不精确概率机器学习中的许多学习问题可视为对信度集连续应用更新规则的过程。这自然引出一个问题：该迭代过程是否收敛于稳定不动点？更一般地说，更新机制在何种条件下能保证此类不动点的存在性及其可达性？我们首次对该问题进行了系统性分析，并以信度贝叶斯深度学习为具体案例阐释了研究发现。我们的工作表明，将不精确性纳入学习过程不仅丰富了不确定性的表征方式，同时揭示了稳定性涌现的结构性条件，从而为不精确条件下的迭代学习动力学提供了新的理论洞见。