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In the context of deep learning models, attention has recently been paid tostudying the surface of the loss function in order to better understandtraining with methods based on gradient descent. This search for an appropriatedescription, both analytical and topological, has led to numerous efforts toidentify spurious minima and characterize gradient dynamics. Our work aims tocontribute to this field by providing a topological measure to evaluate losscomplexity in the case of multilayer neural networks. We compare deep andshallow architectures with common sigmoidal activation functions by derivingupper and lower bounds on the complexity of their loss function and revealinghow that complexity is influenced by the number of hidden units, trainingmodels, and the activation function used. Additionally, we found that certainvariations in the loss function or model architecture, such as adding an$\ell_2$ regularization term or implementing skip connections in a feedforwardnetwork, do not affect loss topology in specific cases.

A topological description of loss surfaces based on Betti Numbers