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 can be interpreted either in terms of Stein's
method to obtain bounds on steady-state performance or in terms of Lyapunov
stability theory for linear ODEs. 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 open 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.

考虑由 Markovian 噪声驱动的线性随机逼近算法的动态特性，通过考虑适当选择的 Lyapunov 函数的漂移，获得常数步长算法的有限时间误差的二次矩的有限时间界限。我们还对逼近误差 2 范数的平方的矩进行了全面的处理。