• Batch normalization
  • Makes the objective function and gradients smoother (more Lipschitz) (Santurkar 2018 - How Does Batch Normalization Help Optimization?. Another perspective: see section 3 of De 2020.)
• Weight normalization
  • Improves the conditioning of the optimization problem (Salimans & Kingma 2016)
• Lipschitz Constant
  • Qi et al 2023 - Understanding Optimization of Deep Learning via Jacobian Matrix and Lipschitz Constant Talks about the effect of the Lipschitz constant on optimizing deep neural networks.

• Choromanska et al 2014 - The Loss Surfaces of Multilayer Networks
• Du et al 2018 - Gradient Descent Provably Optimizes Over-parameterized Neural Networks “We show that as long as m is large enough and no two inputs are parallel, randomly initialized gradient descent converges to a globally optimal solution at a linear convergence rate for the quadratic loss function, for an m hidden node shallow neural network with ReLU activation and n training data.”
• Goldblum et al 2020 - Truth or Backpropaganda? An Empirical Investigation of Deep Learning Theory Examines optimization issues in deep learning and their relation to ML theory

• Sussman 1991 - Uniqueness of the weights for minimal feedforward nets with a given input-output map Also points out some discrete symmetries in neural networks: swapping any two neurons in the same layer, negating the input weights and output weights for any neuron (if using tahn activation), etc.

• Yao et al 2019 - PYHESSIAN: Neural Networks Through the Lens of the Hessian

If the weights are unbounded, the Lipschitz constant will be unbouned (true even for logistic regression).

• 2019 - Efficient and Accurate Estimation of Lipschitz Constants for Deep Neural Networks
• 2018 - Lipschitz regularity of deep neural networks: analysis and efficient estimation

• Bengio et al 2013 - Estimating or Propagating Gradients Through Stochastic Neurons for Conditional Computation Introduces straight-through estimator
• REINFORCE
• Peng et al 2018 - Backpropagating through Structured Argmax using a SPIGOT

• Mosbach 2020 - ON THE STABILITY OF FINE-TUNING BERT: MISCONCEPTIONS, EXPLANATIONS, AND STRONG BASELINES catastrophic forgetting and small size of the fine-tuning datasets fail to explain the fine-tuning instability, which is caused by optimization difficulties and differences in generalization.

See also Soheil Feizi's lecture The implicit bias of gradient descent.

• Soudry et al 2018 - The Implicit Bias of Gradient Descent on Separable Data
• Nacson et al 2018 - Stochastic Gradient Descent on Separable Data: Exact Convergence with a Fixed Learning Rate

• Balduzzi et al 2017 - The Shattered Gradients Problem: If resnets are the answer, then what is the question?
• Orhan & Pitkow 2017 - Skip Connections Eliminate Singularities

• Daniel Soudry

• Optimization