A Lower Bound for Learning Distributions Generated by Probabilistic Automata
Borja Balle, Jorge Castro and Ricard Gavaldà
Lecture Notes in Computer Science
Known algorithms for learning PDFA can only be shown to
run in time polynomial in the so-called distinguishability μ of the target
machine, besides the number of states and the usual accuracy and con-
fidence parameters. We show that the dependence on μ is necessary for
every algorithm whose structure resembles existing ones. As a technical
tool, a new variant of Statistical Queries termed L∞ -queries is defined.
We show how these queries can be simulated from samples and observe
that known PAC algorithms for learning PDFA can be rewritten to access
its target using L∞ -queries and standard Statistical Queries. Finally, we
show a lower bound: every algorithm to learn PDFA using queries with
a resonable tolerance needs a number of queries larger than (1/μ)c for
every c < 1.