Embeddings as Epistemic States: Limitations on the Use of Pooling Operators for Accumulating Knowledge
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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 then study the implications of these constraints, starting from the idea that we should be able to verify whether an arbitrary propositional formula is satisfied in the epistemic state encoded by a given vector. 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. Finally, we also study an extension of the epistemic pooling principle to weighted epistemic states, where max-pooling emerges as the most suitable operator.
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