  Linear fixed point equations in Hilbert spaces arise in a variety of
settings, including reinforcement learning, and computational methods for
solving differential and integral equations. We study methods that use a
collection of random observations to compute approximate solutions by searching
over a known low-dimensional subspace of the Hilbert space. First, we prove an
instance-dependent upper bound on the mean-squared error for a linear
stochastic approximation scheme that exploits Polyak--Ruppert averaging. This
bound consists of two terms: an approximation error term with an
instance-dependent approximation factor, and a statistical error term that
captures the instance-specific complexity of the noise when projected onto the
low-dimensional subspace. Using information theoretic methods, we also
establish lower bounds showing that both of these terms cannot be improved,
again in an instance-dependent sense. A concrete consequence of our
characterization is that the optimal approximation factor in this problem can
be much larger than a universal constant. We show how our results precisely
characterize the error of a class of temporal difference learning methods for
the policy evaluation problem with linear function approximation, establishing
their optimality.
