### Abstract

We develop deterministic necessary and sufficient conditions on individual noise sequences of a stochastic approximation algorithm for the error of the iterates to converge at a given rate. Specifically, suppose {ρ_{n}} is a given positive sequence converging monotonically to 0. Consider a stochastic approximation algorithm x_{n+1} = x_{n}-a_{n}(A_{n}x_{n}-b_{n})+a_{n}e_{n}, where {x_{n}} is the iterate sequence, {a_{n}} is the step size sequence, {e_{n}} is the noise sequence, and x* is the desired zero of the function f(x) = Ax-b. We show that x_{n}-z* = o(ρ_{n}) if and only if the sequence {e_{n}} satisfies one of five equivalent conditions. These conditions are based on well known for formulas for noise sequences found in the literature.

Original language | English (US) |
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Pages (from-to) | 2279-2280 |

Number of pages | 2 |

Journal | Proceedings of the IEEE Conference on Decision and Control |

Volume | 3 |

State | Published - Dec 1 1997 |

Externally published | Yes |

Event | Proceedings of the 1997 36th IEEE Conference on Decision and Control. Part 1 (of 5) - San Diego, CA, USA Duration: Dec 10 1997 → Dec 12 1997 |

### All Science Journal Classification (ASJC) codes

- Control and Systems Engineering
- Modeling and Simulation
- Control and Optimization

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## Cite this

*Proceedings of the IEEE Conference on Decision and Control*,

*3*, 2279-2280.