Embeddings as Epistemic States: Limitations on the Use of Pooling Operators for Accumulating Knowledge
This work addresses limitations in using pooling operators for knowledge accumulation in neural networks, particularly for reasoning tasks, and is incremental as it builds on existing theory to provide specific constraints and exceptions.
The paper investigates the conditions under which standard pooling operators in neural networks satisfy the epistemic pooling principle, which assumes embeddings encode evidence, and finds that this only holds under specific constraints like high dimensionality or non-negativity. It shows that when satisfied, most pooling operators prevent verification of propositional formulas with linear scoring functions, except for max-pooling with upper-bounded embeddings and Hadamard pooling with non-negative embeddings, clarifying performance issues in tasks like Graph Neural Networks.
Various neural network architectures rely on pooling operators to aggregate information coming from different sources. It is often implicitly assumed in such contexts that vectors encode epistemic states, i.e. that vectors capture the evidence that has been obtained about some properties of interest, and that pooling these vectors yields a vector that combines this evidence. We study, for a number of standard pooling operators, under what conditions they are compatible with this idea, which we call the epistemic pooling principle. While we find that all the considered pooling operators can satisfy the epistemic pooling principle, this only holds when embeddings are sufficiently high-dimensional and, for most pooling operators, when the embeddings satisfy particular constraints (e.g. having non-negative coordinates). We furthermore show that these constraints have important implications on how the embeddings can be used in practice. In particular, we find that when the epistemic pooling principle is satisfied, in most cases it is impossible to verify the satisfaction of propositional formulas using linear scoring functions, with two exceptions: (i) max-pooling with embeddings that are upper-bounded and (ii) Hadamard pooling with non-negative embeddings. This finding helps to clarify, among others, why Graph Neural Networks sometimes under-perform in reasoning tasks. Finally, we also study an extension of the epistemic pooling principle to weighted epistemic states, which are important in the context of non-monotonic reasoning, where max-pooling emerges as the most suitable operator.