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Adapting Newton's Method to Neural Networks through a Summary of Higher-Order Derivatives

When training large models, such as neural networks, the full derivatives of order 2 and beyond are usually inaccessible, due to their computational cost. This is why, among the second-order optimization methods, it is very common to bypass the computation of the Hessian by using first-order information, such as the gradient of the parameters (e.g., quasi-Newton methods) or the activations (e.g., K-FAC). In this paper, we focus on the exact and explicit computation of projections of the Hessian and higher-order derivatives on well-chosen subspaces, which are relevant for optimization. Namely, for a given partition of the set of parameters, it is possible to compute tensors which can be seen as "higher-order derivatives according to the partition", at a reasonable cost as long as the number of subsets of the partition remains small. Then, we propose an optimization method exploiting these tensors at order 2 and 3 with several interesting properties, including: it outputs a learning rate per subset of parameters, which can be used for hyperparameter tuning; it takes into account long-range interactions between the layers of the trained neural network, which is usually not the case in similar methods (e.g., K-FAC); the trajectory of the optimization is invariant under affine layer-wise reparameterization. Code available at https://github.com/p-wol/GroupedNewton/ .

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