The theory of full conformal prediction uses deterministic non-conformity measure, but modern usage of full conformal prediction often relies on machine learning training, making stochasticity inevitable. A simple sufficient condition of almost sure permutation invariance of the non-conformity measure can be too restrictive, so many have suggested the relaxation to permutation in distribution as a condition for full conformal prediction validity. We, however, show that this commonly known condition is actually insufficient. We then provide a correct sufficient condition: Conditional Independence & Permutation Invariance in Distribution, which encompasses several stochastic settings that may be used in machine learning.

Since the attention mechanism is rarely used within the GP literature, we provide a brief review of the topic in this section. Below we follow the description of attention as given by Vaswani et al. [8], including extensions to self-attention and multi-head self-attention. The dot-product attention mechanism [8] takes as input a set of queries, keys and values. The queries and keys have dimension Dz and the values have dimension Dv which may differ from Dz. The operation of dot-product attention then generates weights from the queries and keys which are used to produce a linear mapping of the input values.

In many applications, we desire neural networks to exhibit invariance or equivari-ance to certain groups due to symmetries inherent in the data. Recently, frame-averaging methods emerged to be a unified framework for attaining symmetries efficiently by averaging over input-dependent subsets of the group, i.e., frames. What we currently lack is a principled understanding of the design of frames.

Since the attention mechanism is rarely used within the GP literature, we provide a brief review of thetopicinthissection.