Nonconvex Nonsmooth Multicomposite Optimization and Its Applications to Recurrent Neural Networks

We consider a class of nonconvex nonsmooth multicomposite optimization problems where the objective function consists of a Tikhonov regularizer and a composition of multiple nonconvex nonsmooth component functions. Such optimization problems arise from tangible applications in machine learning and beyond. To define and compute its first-order and second-order d(irectional)-stationary points effectively, we first derive the closed-form expression of the tangent cone for the feasible region of its constrained reformulation. Building on this, we establish its equivalence with the corresponding constrained and ${\mathrm{\ell }}_1$-penalty reformulations in terms of global optimality and d-stationarity. The equivalence offers indirect methods to attain the first-order and second-order d-stationary points of the original problem in certain cases. We apply our results to the training process of recurrent neural networks (RNNs).