Non-attracting Regions of Local Minima in Deep and Wide Neural Networks

Understanding the loss surface of neural networks is essential for the design of models with predictable performance and their success in applications. Experimental results suggest that sufficiently deep and wide neural networks are not negatively impacted by suboptimal local minima. Despite recent progress, the reason for this outcome is not fully understood. Could deep networks have very few, if at all, suboptimal local optima? or could all of them be equally good? We provide a construction to show that suboptimal local minima (i.e., non-global ones), even though degenerate, exist for fully connected neural networks with sigmoid activation functions. The local minima obtained by our construction belong to a connected set of local solutions that can be escaped from via a non-increasing path on the loss curve. For extremely wide neural networks of decreasing width after the wide layer, we prove that every suboptimal local minimum belongs to such a connected set. This provides a partial explanation for the successful application of deep neural networks. In addition, we also characterize under what conditions the same construction leads to saddle points instead of local minima for deep neural networks

Corrigendum to “Regularity for Stably Projectionless, Simple C ∗ -Algebras”

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On the regularization of Wasserstein GANs

Since their invention, generative adversarial networks (GANs) have become a popular approach for learning to model a distribution of real (unlabeled) data. Convergence problems during training are overcome by Wasserstein GANs which minimize the distance between the model and the empirical distribution in terms of a different metric, but thereby introduce a Lipschitz constraint into the optimization problem. A simple way to enforce the Lipschitz constraint on the class of functions, which can be modeled by the neural network, is weight clipping. It was proposed that training can be improved by instead augmenting the loss by a regularization term that penalizes the deviation of the gradient of the critic (as a function of the network's input) from one. We present theoretical arguments why using a weaker regularization term enforcing the Lipschitz constraint is preferable. These arguments are supported by experimental results on toy data sets.Comment: Published as a conference paper at ICLR 2018. * Henning Petzka and Asja Fischer contributed equally to this work (11 pages +13 pages appendix

Comparison properties of the Cuntz semigroup and applications to C*-algebras

We study comparison properties in the category Cu aiming to lift results to the C*-algebraic setting. We introduce a new comparison property and relate it to both the CFP and -comparison. We show differences of all properties by providing examples, which suggest that the corona factorization for C*-algebras might allow for both finite and infinite projections. In addition, we show that R{\o}rdam's simple, nuclear C*-algebra with a finite and an infinite projection does not have the CFP

Corrigendum to “Regularity for stably projectionless, simple C⁎-algebras” [J. Funct. Anal. 263 (2012) 1382–1407]

Research partially supported by EPSRC (grant no. EP/N002377/1), NSERC (PDF, held by AT), and by the DFG (SFB 878).Peer reviewedPostprin