  This paper is concerned with the problem of policy evaluation with linear
function approximation in discounted infinite horizon Markov decision
processes. We investigate the sample complexities required to guarantee a
predefined estimation error of the best linear coefficients for two widely-used
policy evaluation algorithms: the temporal difference (TD) learning algorithm
and the two-timescale linear TD with gradient correction (TDC) algorithm. In
both the on-policy setting, where observations are generated from the target
policy, and the off-policy setting, where samples are drawn from a behavior
policy potentially different from the target policy, we establish the first
sample complexity bound with high-probability convergence guarantee that
attains the optimal dependence on the tolerance level. We also exhihit an
explicit dependence on problem-related quantities, and show in the on-policy
setting that our upper bound matches the minimax lower bound on crucial problem
parameters, including the choice of the feature maps and the problem dimension.
