A Lower Bound for Learning Distributions Generated by Probabilistic Automata ## AbstractKnown 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.
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