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State-Space Models (SSMs) have recently been shown to achieve strong empirical performance on a variety of long-range sequence modeling task… (see more)s while remaining efficient and highly-parallelizable. However, the theoretical understanding of their expressive power remains limited. In this work, we study the expressivity of input-Dependent Complex-valued Diagonal (DCD) State-Space Models (SSMs) on sequential state-tracking tasks for abstract groups. It is easy to show that a single DCD SSM layer with a universal decoder can track any Abelian group at finite precision by decomposing it into a product of cyclic groups. We show that this is tight by proving that such a model cannot track any non-Abelian group at finite precision. We further establish the expressivity of multi-layer DCD SSMs. We show that a
2025-12-31
International Conference on Learning Representations (Accept (Poster))
Recent work has formalized the reward hypothesis through the lens of expected utility theory, by interpreting reward as utility. Hausner's f… (see more)oundational work showed that dropping the continuity axiom leads to a generalization of expected utility theory where utilities are lexicographically ordered vectors of arbitrary dimension. In this paper, we extend this result by identifying a simple and practical condition under which preferences cannot be represented by scalar rewards, necessitating a 2-dimensional reward function. We provide a full characterization of such reward functions, as well as the general d-dimensional case, in Markov Decision Processes (MDPs) under a memorylessness assumption on preferences. Furthermore, we show that optimal policies in this setting retain many desirable properties of their scalar-reward counterparts, while in the Constrained MDP (CMDP) setting -- another common multiobjective setting -- they do not.
The von Neumann-Morgenstern (VNM) utility theorem shows that under certain axioms of rationality, decision-making is reduced to maximizing t… (see more)he expectation of some utility function. We extend these axioms to increasingly structured sequential decision making settings and identify the structure of the corresponding utility functions. In particular, we show that memoryless preferences lead to a utility in the form of a per transition reward and multiplicative factor on the future return. This result motivates a generalization of Markov Decision Processes (MDPs) with this structure on the agent's returns, which we call Affine-Reward MDPs. A stronger constraint on preferences is needed to recover the commonly used cumulative sum of scalar rewards in MDPs. A yet stronger constraint simplifies the utility function for goal-seeking agents in the form of a difference in some function of states that we call potential functions. Our necessary and sufficient conditions demystify the reward hypothesis that underlies the design of rational agents in reinforcement learning by adding an axiom to the VNM rationality axioms and motivates new directions for AI research involving sequential decision making.
2022-06-27
Proceedings of the 39th International Conference on Machine Learning (published)
Pixelizations of Platonic solids such as the cube and icosahedron have been widely used to represent spherical data, from climate records to… (see more) Cosmic Microwave Background maps. Platonic solids have well-known global symmetries. Once we pixelize each face of the solid, each face also possesses its own local symmetries in the form of Euclidean isometries. One way to combine these symmetries is through a hierarchy. However, this approach does not adequately model the interplay between the two levels of symmetry transformations. We show how to model this interplay using ideas from group theory, identify the equivariant linear maps, and introduce equivariant padding that respects these symmetries. Deep networks that use these maps as their building blocks generalize gauge equivariant CNNs on pixelized spheres. These deep networks achieve state-of-the-art results on semantic segmentation for climate data and omnidirectional image processing. Code is available at https://git.io/JGiZA.
2021-06-30
Proceedings of the 38th International Conference on Machine Learning (published)
