Binno: A 1st-order method for Bi-level Nonconvex Nonsmooth Optimization for Matrix Factorizations

Nonconvex and nonsmooth bi-level optimization poses critical theoretical challenges, while arising in several applications. In this work, we develop a method for nonconvex, nonsmooth bi-level optimization and introduce Binno, a first-order method that builds on proximal-gradient updates within the the proximal alternate minimization framework with descent conditions from variational analysis. Binno couples the two levels via a descent-driven averaging mechanism, extending single-level proximal schemes to the nonconvex nonsmooth bi-level setting. We show thatBinno induces a descent property for a suitable surrogate of the bi-level objective. Each iteration performs blockwise proximal-gradient updates for the upper and lower problems separately, then forms a calibrated, block-diagonal convex combination of the two iterates. A linesearch selects combination weights enforcing simultaneous descent of both objectives. We give conditions ensuring that both these weights and the descent directions induced by the associated proximal-gradient maps exist. We apply Binno to sparse low-rank factorization, where the upper level uses elementwise l1 penalties and the lower level uses nuclear norms, coupled via a Frobenius data term. We test Binno on synthetic matrices and a real traffic-video dataset, achieving lower reconstruction error and higher peak signal-to-noise ratio than standard methods. We also validate it on a regularized market-clearing problem, where it selects policy-preferred equilibria, and compare it with bi-level baseline, showing consistent improvements