Computational hardness of agnostic ERM for GL(d,2)

Determine whether agnostic empirical risk minimization for the class of invertible linear maps GL(d,2) acting on F_2^d is computationally hard.

Background

For S_n acting on [n], the paper shows that agnostic ERM reduces to maximum-weight bipartite matching and is polynomial-time solvable. For GL(d,2), agnostic ERM asks for an invertible matrix satisfying as many noisy equations Ax_i=y_i as possible, suggesting a connection with decoding noisy linear equations over F_2.

The computational complexity of this optimization problem is left unresolved.