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Register a new userPure and Spurious Critical Points: a Geometric Study of Linear Networks
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Added by
Matthew Trager
Publication date
2019
fields
Informatics Engineering
and research's language is
English
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The critical locus of the loss function of a neural network is determined by the geometry of the functional space and by the parameterization of this space by the networks weights. We introduce a natural distinction between pure critical points, which only depend on the functional space, and spurious critical points, which arise from the parameterization. We apply this perspective to revisit and extend the literature on the loss function of linear neural networks. For this type of network, the functional space is either the set of all linear maps from input to output space, or a determinantal variety, i.e., a set of linear maps with bounded rank. We use geometric properties of determinantal varieties to derive new results on the landscape of linear networks with different loss functions and different parameterizations. Our analysis clearly illustrates that the absence of bad local minima in the loss landscape of linear networks is due to two distinct phenomena that apply in different settings: it is true for arbitrary smooth convex losses in the case of architectures that can express all linear maps (filling architectures) but it holds only for the quadratic loss when the functional space is a determinantal variety (non-filling architectures). Without any assumption on the architecture, smooth convex losses may lead to landscapes with many bad minima.
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We study the family of functions that are represented by a linear convolutional neural network (LCN). These functions form a semi-algebraic subset of the set of linear maps from input space to output space. In contrast, the families of functions represented by fully-connected linear networks form algebraic sets. We observe that the functions represented by LCNs can be identified with polynomials that admit certain factorizations, and we use this perspective to describe the impact of the networks architecture on the geometry of the resulting function space. We further study the optimization of an objective function over an LCN, analyzing critical points in function space and in parameter space, and describing dynamical invariants for gradient descent. Overall, our theory predicts that the optimized parameters of an LCN will often correspond to repeated filters across layers, or filters that can be decomposed as repeated filters. We also conduct numerical and symbolic experiments that illustrate our results and present an in-depth analysis of the landscape for small architectures.
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