Choosing one regularization weight per feature is a bilevel program with a nonsmooth lower level and thousands of upper-level variables. The standard hypergradient — implicit differentiation on the active support — assumes strict complementarity, and returns exactly zero at biactive coordinates, so the outer loop can never move them. Using an exact forward–backward reformulation and the closed-form coderivative of soft-thresholding, we build a hyper-subgradient whose only freedom is a sign per biactive coordinate, and a sign-consistent oracle to choose it. Its selection test is precisely a descent test, and it reduces exactly to the classical hypergradient under strict complementarity.