  Mean-variance optimization (MVO) is known to be sensitive to estimation error
in its inputs. Norm penalization of MVO programs is a regularization technique
that can mitigate the adverse effects of estimation error. We augment the
standard MVO program with a convex combination of parameterized $L_1$ and
$L_2$-norm penalty functions. The resulting program is a parameterized
quadratic program (QP) whose dual is a box-constrained QP. We make use of
recent advances in neural network architecture for differentiable QPs and
present a data-driven framework for optimizing parameterized norm-penalties to
minimize the downstream MVO objective. We present a novel technique for
computing the derivative of the optimal primal solution with respect to the
parameterized $L_1$-norm penalty by implicit differentiation of the dual
program. The primal solution is then recovered from the optimal dual variables.
Historical simulations using US stocks and global futures data demonstrate the
benefit of the data-driven optimization approach.
