Peter Hinz

Peter Hinz

For fixed training data and network parameters in the other layers the L1 loss of a ReLU neural network as a function of the first layer's parameters is a piece-wise affine function. We use the Deep ReLU Simplex algorithm to iteratively minimize the loss monotonically on adjacent vertices and analyze the trajectory of these vertex positions. We empirically observe that in a neighbourhood around a local minimum, the iterations behave differently such that conclusions on loss level and proximity of the local minimum can be made before it has been found: Firstly the loss seems to decay exponentially slow at iterated adjacent vertices such that the loss level at the local minimum can be estimated from the loss levels of subsequently iterated vertices, and secondly we observe a strong increase of the vertex density around local minima. This could have far-reaching consequences for the design of new gradient-descent algorithms that might improve convergence rate by exploiting these facts.

Peter Hinz

Several current bounds on the maximal number of affine regions of a ReLU feed-forward neural network are special cases of the framework [1] which relies on layer-wise activation histogram bounds. We analyze and partially solve a problem in algebraic topology the solution of which would fully exploit this framework. Our partial solution already induces slightly tighter bounds and suggests insight in how parameter initialization methods can affect the number of regions. Furthermore, we extend the framework to allow the composition of subnetwork instead of layer-wise activation histogram bounds to reduce the number of required compositions which negatively affect the tightness of the resulting bound.

Dominik Alfke, Weston Baines, Jan Blechschmidt et al.

We present a novel technique based on deep learning and set theory which yields exceptional classification and prediction results. Having access to a sufficiently large amount of labelled training data, our methodology is capable of predicting the labels of the test data almost always even if the training data is entirely unrelated to the test data. In other words, we prove in a specific setting that as long as one has access to enough data points, the quality of the data is irrelevant.

Peter Hinz, Sara van de Geer

We present a framework to derive upper bounds on the number of regions that feed-forward neural networks with ReLU activation functions are affine linear on. It is based on an inductive analysis that keeps track of the number of such regions per dimensionality of their images within the layers. More precisely, the information about the number regions per dimensionality is pushed through the layers starting with one region of the input dimension of the neural network and using a recursion based on an analysis of how many regions per output dimensionality a subsequent layer with a certain width can induce on an input region with a given dimensionality. The final bound on the number of regions depends on the number and widths of the layers of the neural network and on some additional parameters that were used for the recursion. It is stated in terms of the $L1$-norm of the last column of a product of matrices and provides a unifying treatment of several previously known bounds: Depending on the choice of the recursion parameters that determine these matrices, it is possible to obtain the bounds from Montúfar (2014), (2017) and Serra et. al. (2017) as special cases. For the latter, which is the strongest of these bounds, the formulation in terms of matrices provides new insight. In particular, by using explicit formulas for a Jordan-like decomposition of the involved matrices, we achieve new tighter results for the asymptotic setting, where the number of layers of the same fixed width tends to infinity.