## Abstract

We consider the dynamics of a linear stochastic approximation algorithm driven by Markovian noise, and derive finite-time bounds on the moments of the error, i.e., deviation of the output of the algorithm from the equilibrium point of an associated ordinary differential equation (ODE). We obtain finite-time bounds on the mean-square error in the case of constant step-size algorithms by considering the drift of an appropriately chosen Lyapunov function. The Lyapunov function is a standard Lyapunov function used to study the stability of linear ODEs, but can also be interpreted in terms of Stein’s method, which is used to obtain bounds on steady-state performance bounds. We also provide a comprehensive treatment of the moments of the square of the 2-norm of the approximation error. Our analysis yields the following results: (i) for a given step-size, we show that the lower-order moments can be made small as a function of the step-size and can be upper-bounded by the moments of a Gaussian random variable; (ii) we show that the higher-order moments beyond a threshold may be infinite in steady-state; and (iii) we characterize the number of samples needed for the finite-time bounds to be of the same order as the steady-state bounds. As a by-product of our analysis, we also solve the problem of obtaining finite-time bounds for the performance of temporal difference learning algorithms with linear function approximation and a constant step-size, without requiring a projection step or an i.i.d. noise assumption.

Original language | English (US) |
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Pages (from-to) | 2803-2830 |

Number of pages | 28 |

Journal | Proceedings of Machine Learning Research |

Volume | 99 |

State | Published - 2019 |

Event | 32nd Conference on Learning Theory, COLT 2019 - Phoenix, United States Duration: Jun 25 2019 → Jun 28 2019 |

## ASJC Scopus subject areas

- Artificial Intelligence
- Software
- Control and Systems Engineering
- Statistics and Probability