Machine Learning Beyond Statistical Consistency for Hybrid AI

We propose a new definition of machine learning, according to which a machine learns a function when it computes the function successfully, that is, when the value it returns for each input has a proper derivation in a formal system. We extend Gödel's Representability Theorem, which equates computation with derivation, to empirical targets in two ways: the theorem is extended to the computable reals, and the data are admitted into the system as empirical definitions that record what has been observed. We prove that a machine learns a true probability if and only if two conditions are met. The Truth Condition is that its estimate converges to the true probability, which is statistical consistency. The Assurance Condition is that the machine properly derives its own convergence inside the system. Learning thus lies beyond statistical consistency, for it requires a derivation of the convergence from the record. A global assumption about the data, such as the i.i.d. assumption, does not supply such a derivation, because it leads only to a circular one, as we show for PAC learning. On the other hand, the two conditions are met for the probability of a sentence over a finite ideal corpus. A real machine can approximate a true probability only if an ideal machine can learn it precisely, that is, only in a domain that meets the two conditions. Modern AI works in such a domain, and we offer this as a clue to its success. Modern AI is hybrid as well, in two senses: First, it performs two kinds of tasks, namely probabilistic ones such as predicting the next word and deductive ones such as proving and coding. Second, it combines two parts of different kinds, a statistical proposer and a deductive verifier, for proving and coding. Our framework explains both senses of hybridity in one formal system.