Advanced Optimization Techniques for Machine Learning
Chapter 1: Foundations of Optimization in Machine Learning
Revisiting Gradient Descent Variants
The Role of Convexity
Understanding Loss Surfaces
Convergence Analysis Fundamentals
Challenges in Non-Convex Optimization
Numerical Stability Considerations
Practice: Analyzing Convergence Behavior
Chapter 2: Second-Order Optimization Methods
Newton's Method: Theory and Derivation
The Hessian Matrix: Computation and Properties
Challenges with Newton's Method
Quasi-Newton Methods: Approximating the Hessian
BFGS Algorithm Explained
Limited-memory BFGS (L-BFGS)
Trust Region Methods
Hands-on Practical: Implementing L-BFGS
Chapter 3: Adaptive Learning Rate Algorithms
Limitations of Fixed Learning Rates
AdaGrad: Adapting Rates Based on Past Gradients
RMSprop: Addressing AdaGrad's Diminishing Rates
Adam: Combining Momentum and RMSprop
Adamax and Nadam Variants
AMSGrad: Improving Adam's Convergence
Understanding Learning Rate Schedules
Hands-on Practical: Comparing Adaptive Optimizers
Chapter 4: Optimization for Large-Scale Datasets
Stochastic Gradient Descent Revisited: Variance Reduction
Stochastic Average Gradient (SAG)
Stochastic Variance Reduced Gradient (SVRG)
Mini-batch Gradient Descent Trade-offs
Asynchronous Stochastic Gradient Descent
Data Parallelism Strategies
Hands-on Practical: Implementing SVRG
Chapter 5: Distributed Optimization Strategies
Motivation for Distributed Training
Parameter Server Architectures
Synchronous vs. Asynchronous Updates
Communication Bottlenecks and Strategies
All-Reduce Algorithms
Federated Learning Optimization Principles
Hands-on Practical: Simulating Distributed SGD
Chapter 6: Optimization Challenges in Deep Learning
Characteristics of Deep Learning Loss Landscapes
Impact of Network Architecture on Optimization
Normalization Techniques and Optimization
Gradient Clipping and Explosion/Vanishing Gradients
Initialization Strategies and Their Effect
Regularization Methods as Implicit Optimization
Practice: Tuning Optimizers for Deep Networks
Chapter 7: Advanced and Specialized Optimization Topics
Constrained Optimization Fundamentals
Lagrangian Duality and KKT Conditions
Projected Gradient Methods
Derivative-Free Optimization Overview
Bayesian Optimization for Hyperparameter Tuning
Optimization for Reinforcement Learning Policies
Practice: Implementing Projected Gradient Descent
Gradient Clipping and Explosion/Vanishing Gradients
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- Learning Long-Term Dependencies with Recurrent Neural Networks, Yoshua Bengio, Patrice Simard, Paolo Frasconi, 1994 Advances in Neural Information Processing Systems 6 (NIPS 1993) - Essential paper identifying the vanishing gradient problem in recurrent neural networks.
- On the difficulty of training Recurrent Neural Networks, Razvan Pascanu, Tomas Mikolov, Yoshua Bengio, 2013 Proceedings of the 30th International Conference on Machine Learning (ICML), Vol. 28 - Introduces gradient clipping as a technique to stabilize training of recurrent neural networks by addressing exploding gradients.
- Deep Learning, Ian Goodfellow, Yoshua Bengio, Aaron Courville, 2016 (MIT Press) - Comprehensive textbook offering foundational understanding of deep learning, including detailed explanations of gradient issues and their solutions.
- CS224n: Natural Language Processing with Deep Learning Lecture Notes, Diyi Yang, Tatsunori Hashimoto, 2024 (Stanford University) - University course materials offering practical and theoretical insights into training deep learning models, especially recurrent neural networks, covering gradient stability.