Fast Convex Optimization for Two-Layer ReLU Networks: Equivalent Model Classes and Cone Decompositions
This work addresses the challenge of efficient and robust training for neural networks, offering a convex alternative to non-convex optimization, which is incremental as it builds on existing convex reformulation ideas.
The paper tackled the problem of training two-layer ReLU networks by developing fast convex optimization algorithms, showing that these methods are faster than standard non-convex training heuristics like SGD and outperform commercial solvers, with experimental validation on MNIST and CIFAR-10 datasets.
We develop fast algorithms and robust software for convex optimization of two-layer neural networks with ReLU activation functions. Our work leverages a convex reformulation of the standard weight-decay penalized training problem as a set of group-$\ell_1$-regularized data-local models, where locality is enforced by polyhedral cone constraints. In the special case of zero-regularization, we show that this problem is exactly equivalent to unconstrained optimization of a convex "gated ReLU" network with non-singular gates. For problems with non-zero regularization, we show that convex gated ReLU models obtain data-dependent approximation bounds for the ReLU training problem. To optimize the convex reformulations, we develop an accelerated proximal gradient method and a practical augmented Lagrangian solver. We show that these approaches are faster than standard training heuristics for the non-convex problem, such as SGD, and outperform commercial interior-point solvers. Experimentally, we verify our theoretical results, explore the group-$\ell_1$ regularization path, and scale convex optimization for neural networks to image classification on MNIST and CIFAR-10.