Learning PDFA with Asynchronous Transitions
Lecture Notes in Computer Science
In this paper we extend the PAC learning algorithm due to Clark and Thollard for learning distributions generated by PDFA to automata whose transitions may take varying time lengths, governed by exponential distributions. Motivation The problem of learning (distributions generated by) probabilistic automata and related models has been intensely studied by the grammatical inference community; see [4, 12, 13] and references therein. The problem has also been studied in variants of the PAC model.
... n particular, it has been observed that polynomial-time learnability of PDFA is feasible if one allows polynomiality not only in the number states but also in other measures of the target automaton complexity. Specifically, Ron et al.  showed that acyclic PDFA can be learned w.r.t. the Kullback-Leibler (KL) divergence in time polynomial in alphabet size, 1/ , 1/δ, number of target states, and 1/µ, where µ denotes the distinguishability of the target automaton. Clark and Thollard extended the result to general PDFA by considering also as a parameter the expected length of the strings L generated by the automaton  . Their algorithm, a state merge-split method, was in turn extended or refined in subsequent work [6, 7, 5, 2] . Here we consider what we call asynchronous PDFA (AsPDFA), in which each transition has an associated exponential distribution. We think of this distribution as indicating the 'time' or duration of the transition. Note that there are several models of timed automata in the literature with other meanings, for example automata with timing constraints on the transitions. Our model is rather the finite-state and deterministic restriction of so-called semi-Markov processes; a widely-studied particular case of the latter are continuous-time Markov chains, in which times between transitions are exponentially distributed. We show a general expression for the KL divergence between two given AsPDFA similar to that in  for PDFA. Based on this expression and a variant of the Clark-Thollard algorithm from , we show that AsPDFA are learnable w.r.t. the KL divergence. Technically, the algorithm requires bounds on the largest and smallest possible values of the parameters of the exponential distributions, which can be thought as defining the 'time-scale' of the target AsPDFA. Full proofs are omitted in this version and will appear elsewhere. The result above is motivated by the importance of modeling temporal components in many scenarios where probabilistic automata or HMM's are used as modeling tools. We in particular were brought to this problem by the work of one of the authors and other collaborators on modeling users' access patterns to websites    . Models similar to (visibleor hidden-state) Markov Models have been used for this purpose in marketing circles and are called Customer Behavior Model Graphs. After the work in [8-10], we noted that the time among successive web clicks, the user think time, was extremely informative to discriminate among different user types and predict their future behavior, and this information is not captured by standard PFA.