Optimal Sets and Solution Paths of ReLU Networks
This provides foundational insights into the optimization landscape of ReLU networks, benefiting researchers in deep learning theory and practitioners seeking efficient network compression.
The paper tackles the non-convex training problem of ReLU neural networks by reformulating it as a convex program, showing that global optima form a polyhedral set and characterizing all critical points, leading to an optimal pruning algorithm for minimal networks and conditions for continuous regularization paths.
We develop an analytical framework to characterize the set of optimal ReLU neural networks by reformulating the non-convex training problem as a convex program. We show that the global optima of the convex parameterization are given by a polyhedral set and then extend this characterization to the optimal set of the non-convex training objective. Since all stationary points of the ReLU training problem can be represented as optima of sub-sampled convex programs, our work provides a general expression for all critical points of the non-convex objective. We then leverage our results to provide an optimal pruning algorithm for computing minimal networks, establish conditions for the regularization path of ReLU networks to be continuous, and develop sensitivity results for minimal ReLU networks.