  Sorting is one of the most fundamental algorithms in computer science.
Recently, Learned Sorts, which use machine learning to improve sorting speed,
have attracted attention. While existing studies show that Learned Sort is
experimentally faster than classical sorting algorithms, they do not provide
theoretical guarantees about its computational complexity. We propose PCF
Learned Sort, a theoretically guaranteed Learned Sort algorithm. We prove that
the expected complexity of PCF Learned Sort is $O(n \log \log n)$ under mild
assumptions on the data distribution. We also confirm experimentally that PCF
Learned Sort has a computational complexity of $O(n \log \log n)$ on both
synthetic and real datasets. This is the first study to theoretically support
the experimental success of Learned Sort, and provides evidence for why Learned
Sort is fast.
