
On the Global Convergence of Training Deep Linear ResNets
We study the convergence of gradient descent (GD) and stochastic gradien...
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Effects of Depth, Width, and Initialization: A Convergence Analysis of Layerwise Training for Deep Linear Neural Networks
Deep neural networks have been used in various machine learning applicat...
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On the Explicit Role of Initialization on the Convergence and Implicit Bias of Overparametrized Linear Networks
Neural networks trained via gradient descent with random initialization ...
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Directional Convergence Analysis under Spherically Symmetric Distribution
We consider the fundamental problem of learning linear predictors (i.e.,...
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Revisiting "Qualitatively Characterizing Neural Network Optimization Problems"
We revisit and extend the experiments of Goodfellow et al. (2014), who s...
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Gradient descent with identity initialization efficiently learns positive definite linear transformations by deep residual networks
We analyze algorithms for approximating a function f(x) = Φ x mapping ^d...
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LowRank plus Sparse Decomposition of Covariance Matrices using Neural Network Parametrization
This paper revisits the problem of decomposing a positive semidefinite m...
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A Convergence Analysis of Gradient Descent for Deep Linear Neural Networks
We analyze speed of convergence to global optimum for gradient descent training a deep linear neural network (parameterized as x W_N ... W_1x) by minimizing the ℓ_2 loss over whitened data. Convergence at a linear rate is guaranteed when the following hold: (i) dimensions of hidden layers are at least the minimum of the input and output dimensions; (ii) weight matrices at initialization are approximately balanced; and (iii) the initial loss is smaller than the loss of any rankdeficient solution. The assumptions on initialization (conditions (ii) and (iii)) are necessary, in the sense that violating any one of them may lead to convergence failure. Moreover, in the important case of output dimension 1, i.e. scalar regression, they are met, and thus convergence to global optimum holds, with constant probability under a random initialization scheme. Our results significantly extend previous analyses, e.g., of deep linear residual networks (Bartlett et al., 2018).
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