Finite-Time Error Bounds For Linear Stochastic Approximation and TD Learning

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