For classification tasks, the performance of a deep neural network is determined by the structure of its decision boundary, whose geometry directly affects essential properties of the model, including accuracy and robustness. Motivated by a classical tube formula due to Weyl, we introduce a method to measure the decision boundary of a neural network through local surface volumes, providing a theoretically justifiable and efficient measure enabling a geometric interpretation of the effectiveness of the model applicable to the high dimensional feature spaces considered in deep learning. A smaller surface volume is expected to correspond to lower model complexity and better generalisation. We verify, on a number of image processing tasks with convolutional architectures that decision boundary volume is inversely proportional to classification accuracy. Meanwhile, the relationship between local surface volume and generalisation for fully connected architecture is observed to be less stable between tasks. Therefore, for network architectures suited to a particular data structure, we demonstrate that smoother decision boundaries lead to better performance, as our intuition would suggest.

Remark Since we apply the row-wise softmax in Eq. (7), P jpij = 1 i and pij 0 (i,j) is alwaysfulfilled.If C(P)=0,allbutoneentryinacolumn pi, are0andtheotherentryis1. Hence,P ipij = 1 j isfulfilled. Synthetic random graph generation To generate train and test graph datasets we utilized the pythonpackage NetworkX[1]. Ego graphs extracted from Binominal graphs (p (0.2,0.6))selecting all neighbours of onerandomnode. Training Details We did not perform an extensive hyperparameter evaluation for the different experiments and mostly followed [2]for hyperparameter selection. We set the graph embedding dimension to 64.

Our development of ToolQA involved a scalable, automated process for dataset curation, along with 13 specialized tools designed for interaction with external knowledge in order to answer questions.

Furthermore, we prove a corresponding non-asymptotic result.

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