Behavior of Analogical Inference w.r.t. Boolean Functions

It has been observed that a particular form of analogical inference, based on analogical proportions, yields competitive results in classification tasks. Using the algebraic normal form of Boolean functions, it has been shown that analogical prediction is always exact iff the labeling function is affine. We point out that affine functions are also meaningful when using another view of analogy. We address the accuracy of analogical inference for arbitrary Boolean functions and show that if a function is ε-close to an affine function, then the probability of making a wrong prediction is upper bounded by 4ε. This result is confirmed by an empirical study showing that the upper bound is tight. It highlights the specificity of analogical inference, also characterized in terms of the Hamming distance.