# High-dimensional dynamics of generalization error in neural networks

@article{Advani2020HighdimensionalDO, title={High-dimensional dynamics of generalization error in neural networks}, author={Madhu S. Advani and Andrew M. Saxe}, journal={Neural Networks}, year={2020}, volume={132}, pages={428 - 446} }

## 300 Citations

An analytic theory of shallow networks dynamics for hinge loss classification

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This paper study in detail the training dynamics of a simple type of neural network: a single hidden layer trained to perform a classification task, and shows that in a suitable mean-field limit this case maps to a single-node learning problem with a time-dependent dataset determined self-consistently from the average nodes population.

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This work relies on the so-called Neural Tangent Kernel, which connects large neural nets to kernel methods, to show that the initialization causes finite-size random fluctuations of the neural net output function f N around its expectation, which affects the generalization error for classification.

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The exact population risk of the unregularized least squares regression problem with two-layer neural networks when either the first or the second layer is trained using a gradient flow under different initialization setups is derived.

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This work investigates the origins of the less studied epoch-wise double descent in which the test error undergoes two non-monotonous transitions, or descents as the training time increases, and derives closed-form analytical expressions for the evolution of generalization error over training.

A Dynamical View on Optimization Algorithms of Overparameterized Neural Networks

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It is shown from a dynamical system perspective that the Heavy Ball method can converge to global minimum on mean squared error (MSE) at a linear rate (similar to GD); however, the Nesterov accelerated gradient descent (NAG) only converges toglobal minimum sublinearly.

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This work analyzes the performance of random feature regression with features F = f ( WX + B ) for a random weight matrix W and bias vector B, obtaining exact formulae for the asymptotic training and test errors for data generated by a linear teacher model.

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This work provides a general framework to characterize the asymptotic generalization error for single-layer neural networks (i.e., generalized linear models) with arbitrary non-linearities, making it applicable to regression as well as classification problems.

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Intriguingly, it is found that a mixture of nonlinearities can outperform the best single nonlinearity on the noisy autoecndoing task, suggesting that mixtures of non linearities might be useful for approximate kernel methods or neural network architecture design.

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