The duality structure gradient descent algorithm: analysis and applications to neural networks
This work addresses the optimization bottleneck in deep learning by providing a theoretically grounded method for neural network training, though it appears incremental as it builds on layer-wise coordinate descent.
The authors tackled the problem of training deep neural networks with gradient descent under weak assumptions by proposing the duality structure gradient descent (DSGD) algorithm, which achieves non-asymptotic convergence to approximate stationary points in both deterministic and stochastic settings.
The training of machine learning models is typically carried out using some form of gradient descent, often with great success. However, non-asymptotic analyses of first-order optimization algorithms typically employ a gradient smoothness assumption (formally, Lipschitz continuity of the gradient) that is too strong to be applicable in the case of deep neural networks. To address this, we propose an algorithm named duality structure gradient descent (DSGD) that is amenable to non-asymptotic performance analysis, under mild assumptions on the training set and network architecture. The algorithm can be viewed as a form of layer-wise coordinate descent, where at each iteration the algorithm chooses one layer of the network to update. The decision of what layer to update is done in a greedy fashion, based on a rigorous lower bound on the improvement of the objective function for each choice of layer. In the analysis, we bound the time required to reach approximate stationary points, in both the deterministic and stochastic settings. The convergence is measured in terms of a parameter-dependent family of norms that is derived from the network architecture and designed to confirm a smoothness-like property on the gradient of the training loss function. We empirically demonstrate the behavior of DSGD in several neural network training scenarios.