  We analyze a fixed-point algorithm for reinforcement learning (RL) of optimal
portfolio mean-variance preferences in the setting of multivariate generalized
autoregressive conditional-heteroskedasticity (MGARCH) with a small penalty on
trading. A numerical solution is obtained using a neural network (NN)
architecture within a recursive RL loop. A fixed-point theorem proves that NN
approximation error has a big-oh bound that we can reduce by increasing the
number of NN parameters. The functional form of the trading penalty has a
parameter $\epsilon>0$ that controls the magnitude of transaction costs. When
$\epsilon$ is small, we can implement an NN algorithm based on the expansion of
the solution in powers of $\epsilon$. This expansion has a base term equal to a
myopic solution with an explicit form, and a first-order correction term that
we compute in the RL loop. Our expansion-based algorithm is stable, allows for
fast computation, and outputs a solution that shows positive testing
performance.
