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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 <inline-formula> <tex-math notation="LaTeX">$\epsilon &gt;0$ </tex-math></inline-formula> that controls the magnitude of transaction costs. When <inline-formula> <tex-math notation="LaTeX">$\epsilon $ </tex-math></inline-formula> is small, we can implement an NN algorithm based on the expansion of the solution in powers of <inline-formula> <tex-math notation="LaTeX">$\epsilon $ </tex-math></inline-formula>. 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.