Find a witness or shatter: the landscape of computable PAC learning
This work provides a complete landscape for CPAC learnability, addressing foundational problems in computational learning theory for researchers in that field.
The paper tackles open questions in computable PAC (CPAC) learning by proving that every improperly CPAC learnable class is contained in a properly learnable one with polynomial sample complexity, constructing a decidable class that is properly learnable but with uncomputably fast growing sample complexity, and showing a decidable class of finite Littlestone dimension that is not improperly learnable.
This paper contributes to the study of CPAC learnability -- a computable version of PAC learning -- by solving three open questions from recent papers. Firstly, we prove that every improperly CPAC learnable class is contained in a class which is properly CPAC learnable with polynomial sample complexity. This confirms a conjecture by Agarwal et al (COLT 2021). Secondly, we show that there exists a decidable class of hypothesis which is properly CPAC learnable, but only with uncomputably fast growing sample complexity. This solves a question from Sterkenburg (COLT 2022). Finally, we construct a decidable class of finite Littlestone dimension which is not improperly CPAC learnable, strengthening a recent result of Sterkenburg (2022) and answering a question posed by Hasrati and Ben-David (ALT 2023). Together with previous work, our results provide a complete landscape for the learnability problem in the CPAC setting.