### Abstract

We provide a bound on the first moment of the error in the Q-function estimate resulting from fixed step-size algorithms applied to finite state-space, discounted reward Markov decision problems. Motivated by Tsitsiklis' proof for the decreasing step-size case, we decompose the Q-learning update equations into a dynamical system driven by a noise sequence and another dynamical system whose state variable is the Q-learning error, i.e., the difference between the true Q-function and the estimate. A natural persistence of excitation condition allows us to sample the system periodically and derive a simple scalar difference equation from which the convergence properties and bounds on the error of the Q-learning algorithm can be derived.

Original language | English (US) |
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Pages (from-to) | 1203-1208 |

Number of pages | 6 |

Journal | Systems and Control Letters |

Volume | 61 |

Issue number | 12 |

DOIs | |

State | Published - Nov 13 2012 |

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### Keywords

- Markov decision processes
- Q-learning
- Stochastic approximation

### ASJC Scopus subject areas

- Control and Systems Engineering
- Computer Science(all)
- Mechanical Engineering
- Electrical and Electronic Engineering