Extension of the PAC framework to finite and countable Markov chains
Gamarnik, D.
IBM T. J. Watson Res. Center, Yorktown Heights, NY, USA;
This paper appears in: Information Theory, IEEE Transactions on
Publication Date: Jan 2003
Volume: 49,
Issue: 1
On page(s): 338- 345
ISSN: 0018-9448
INSPEC Accession Number: 7525480
Digital Object Identifier: 10.1109/TIT.2002.806131
Current Version Published: 2003-01-14
Abstract
We consider a model of learning in which the successive observations follow a certain Markov chain. The observations are labeled according to a membership to some unknown target set. For a Markov chain with finitely many states we show that, if the target set belongs to a family of sets with a finite Vapnik-Chervonenkis (1995) dimension, then probably approximately correct (PAC) learning of this set is possible with polynomially large samples. Specifically for observations following a random walk with a state space 𝒳 and uniform stationary distribution, the sample size required is no more than Ω(t0/1-λ2log(t0|χ|1/δ)), where δ is the confidence level, λ2 is the second largest eigenvalue of the transition matrix, and t0 is the sample size sufficient for learning from independent and identically distributed (i.i.d.) observations. We then obtain similar results for Markov chains with countably many states using Lyapunov function technique and results on mixing properties of infinite state Markov chains.
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