Sparsity is a highly desired feature in deep neural networks (DNNs) since it ensures numerical efficiency, improves the interpretability of models (due to the smaller number of relevant features), and robustness. In machine learning approaches based on linear models, it is well known that there exists a connecting path between the sparsest solution in terms of the $\ell^1$ norm (i.e., zero weights) and the non-regularized solution, which is called the regularization path. Very recently, there was a first attempt to extend the concept of regularization paths to DNNs by means of treating the empirical loss and sparsity ($\ell^1$ norm) as two conflicting criteria and solving the resulting multiobjective optimization problem. However, due to the non-smoothness of the $\ell^1$ norm and the high number of parameters, this approach is not very efficient from a computational perspective. To overcome this limitation, we present an algorithm that allows for the approximation of the entire Pareto front for the above-mentioned objectives in a very efficient manner. We present numerical examples using both deterministic and stochastic gradients. We furthermore demonstrate that knowledge of the regularization path allows for a well-generalizing network parametrization.

翻译：稀疏性是深度神经网络（DNNs）中一个非常理想的特性，因为它能确保数值效率、提升模型的可解释性（由于相关特征数量更少）以及鲁棒性。在基于线性模型的机器学习方法中，众所周知，存在一条连接$\ell^1$范数意义下的最稀疏解（即零权重）与非正则化解的路径，这被称为正则化路径。最近，通过将经验损失和稀疏性（$\ell^1$范数）视为两个相互冲突的目标，并求解由此产生的多目标优化问题，首次尝试将正则化路径的概念扩展到DNNs。然而，由于$\ell^1$范数的非光滑性以及参数数量庞大，该方法的计算效率并不理想。为克服这一局限，我们提出了一种算法，能够以极高效率逼近上述目标的整个帕累托前沿。我们利用确定性梯度和随机梯度给出了数值示例，并进一步证明，对正则化路径的掌握能够实现具有良好泛化能力的网络参数化。