Revealing the Structure of Deep Neural Networks via Convex Duality
We study regularized deep neural networks (DNNs) and introduce a convex analytic framework to characterize the structure of the hidden layers. We show that a set of optimal hidden layer weights for a norm regularized DNN training problem can be explicitly found as the extreme points of a convex set. For the special case of deep linear networks, we prove that each optimal weight matrix aligns with the previous layers via duality. More importantly, we apply the same characterization to deep ReLU networks with whitened data and prove the same weight alignment holds. As a corollary, we also prove that norm regularized deep ReLU networks yield spline interpolation for one-dimensional datasets which was previously known only for two-layer networks. Furthermore, we provide closed-form solutions for the optimal layer weights when data is rank-one or whitened. The same analysis also applies to architectures with batch normalization even for arbitrary data. Therefore, we obtain a complete explanation for a recent empirical observation termed Neural Collapse where class means collapse to the vertices of a simplex equiangular tight frame.
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NSF-PAR ID:
10310559
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International Conference on Machine Learning (ICML) 2021
2. We develop a convex analytic approach to analyze finite width two-layer ReLU networks. We first prove that an optimal solution to the regularized training problem can be characterized as extreme points of a convex set, where simple solutions are encouraged via its convex geometrical properties. We then leverage this characterization to show that an optimal set of parameters yield linear spline interpolation for regression problems involving one dimensional or rank-one data. We also characterize the classification decision regions in terms of a kernel matrix and minimum 1-norm solutions. This is in contrast to Neural Tangent Kernel which is unable to explain predictions of finite width networks. Our convex geometric characterization also provides intuitive explanations of hidden neurons as auto-encoders. In higher dimensions, we show that the training problem can be cast as a finite dimensional convex problem with infinitely many constraints. Then, we apply certain convex relaxations and introduce a cutting-plane algorithm to globally optimize the network. We further analyze the exactness of the relaxations to provide conditions for the convergence to a global optimum. Our analysis also shows that optimal network parameters can be also characterized as interpretable closed-form formulas in some practically relevant special cases.
3. We study training of Convolutional Neural Networks (CNNs) with ReLU activations and introduce exact convex optimization formulations with a polynomial complexity with respect to the number of data samples, the number of neurons, and data dimension. More specifically, we develop a convex analytic framework utilizing semi-infinite duality to obtain equivalent convex optimization problems for several two- and three-layer CNN architectures. We first prove that two-layer CNNs can be globally optimized via an 2 norm regularized convex program. We then show that multi-layer circular CNN training problems with a single ReLU layer are equivalent to an 1 regularized convex program that encourages sparsity in the spectral domain. We also extend these results to three-layer CNNs with two ReLU layers. Furthermore, we present extensions of our approach to different pooling methods, which elucidates the implicit architectural bias as convex regularizers.