On the Conditioning Consistency Gap in Conditional Neural Processes
Provides theoretical understanding of why CNPs work despite violating consistency conditions, benefiting researchers in meta-learning and probabilistic modeling.
The paper defines the conditioning consistency gap for conditional neural processes (CNPs) and proves it scales as O(1/n^2) with context size n, showing CNPs approximate valid stochastic processes well for moderate contexts but poorly in few-shot settings.
Neural processes are meta-learning models that map context sets to predictive distributions. While inspired by stochastic processes, NPs do not generally satisfy the Kolmogorov consistency conditions required to define a valid stochastic process. This inconsistency is widely acknowledged but poorly understood. Practitioners note that NPs work well despite the violation, without quantifying what this means. We address this gap by defining the conditioning consistency gap, a KL divergence measuring how much a conditional neural process's (CNP) predictions change when a point is added to the context versus conditioned upon. Our main results show that for CNPs with bounded encoders and Lipschitz decoders, the consistency gap is $O(1/n^2)$ in context size $n$, and that this rate is tight. These bounds establish the precise sense in which CNPs approximate valid stochastic processes. The inconsistency is negligible for moderate context sizes but can be significant in the few-shot regime.