  The paper focuses on mean-field type multi-agent control problems where the
dynamics and cost structures are symmetric and homogeneous, and are affected by
the distribution of the agents. A standard solution method for these problems
is to consider the infinite population limit as an approximation and use
symmetric solutions of the limit problem to achieve near optimality. The
control policies, and in particular the dynamics, depend on the population
distribution in the finite population setting, or the marginal distribution of
the state variable of a representative agent for the infinite population
setting. Hence, learning and planning for these control problems generally
require estimating the reaction of the system to all possible state
distributions of the agents. To overcome this issue, we consider linear
function approximation for the control problem and provide several coordinated
and independent learning methods. We rigorously establish error upper bounds
for the performance of learned solutions. The performance gap stems from (i)
the mismatch due to estimating the true model with a linear one, and (ii) using
the infinite population solution in the finite population problem as an
approximate control. The provided upper bounds quantify the impact of these
error sources on the overall performance.
