# Exact Solutions of a Deep Linear Network

@article{Ziyin2022ExactSO, title={Exact Solutions of a Deep Linear Network}, author={Liu Ziyin and Botao Li and Xiangmin Meng}, journal={ArXiv}, year={2022}, volume={abs/2202.04777} }

This work finds the analytical expression of the global minima of a deep linear network with weight decay and stochastic neurons, a fundamental model for understanding the landscape of neural networks. Our result implies that zero is a special point in deep neural network architecture. We show that weight decay strongly interacts with the model architecture and can create bad minima at zero in a network with more than $1$ hidden layer, qualitatively different from a network with only $1$ hidden…

## 7 Citations

### The Probabilistic Stability of Stochastic Gradient Descent

- Computer Science
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Only under the lens of probabilistic stability does SGD exhibit rich and practically relevant phases of learning, such as the phases of the complete loss of stability, incorrect learning, convergence to low-rank saddles, and correct learning.

### SGD WITH A C ONSTANT L ARGE L EARNING R ATE C AN C ONVERGE TO L OCAL M AXIMA

- Computer Science
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This work constructs worst-case optimization problems illustrating that, when not in the regimes that the previous works often assume, SGD can exhibit many strange and potentially undesirable behaviors.

### SGD with a Constant Large Learning Rate Can Converge to Local Maxima

- Computer Science
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This work constructs worst-case optimization problems illustrating that, when not in the regimes that the previous works often assume, SGD can exhibit many strange and potentially undesirable behaviors.

### Sparsity by Redundancy: Solving L1 with a Simple Reparametrization

- Computer ScienceArXiv
- 2022

The results lead to a simple algorithm, spred, that seamlessly integrates L 1 regularization into any modern deep learning framework, and bridges the gap in understanding the inductive bias of the redundant parametrization common in deep learning and conventional statistical learning.

### What shapes the loss landscape of self-supervised learning?

- Computer ScienceArXiv
- 2022

In this theory, the causes of the dimensional collapse are identified, the effect of normalization and bias is studied, and the interpretability afforded by the analytical theory is leveraged to understand how dimensional collapse can be beneficial and what affects the robustness of SSL against data imbalance.

### Exact Phase Transitions in Deep Learning

- Computer ScienceArXiv
- 2022

It is proved that the competition between prediction error and model complexity in the training loss leads to the second-order phase transition for nets with one hidden layer and the first-orderphase transition fornets with more than onehidden layer.

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- Computer ScienceNeurIPS
- 2022

The existence and cause of a type of posterior collapse that frequently occurs in the Bayesian deep learning practice are identified and the result suggests that posterior collapse may be related to neural collapse and dimensional collapse and could be a subclass of a general problem of learning for deeper architectures.

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The results lead to a simple algorithm, spred, that seamlessly integrates L 1 regularization into any modern deep learning framework, and bridges the gap in understanding the inductive bias of the redundant parametrization common in deep learning and conventional statistical learning.