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
Challenges in Non-Convex Optimization
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- Deep Learning, Ian Goodfellow, Yoshua Bengio, and Aaron Courville, 2016 (MIT Press) - Classic textbook with a dedicated chapter explaining optimization challenges in deep learning models.
- Identifying and Attacking the Saddle Point Problem in High-Dimensional Non-Convex Optimization, Yann N. Dauphin, Razvan Pascanu, Caglar Gulcehre, Kyunghyun Cho, Surya Ganguli, and Yoshua Bengio, 2014 Advances in Neural Information Processing Systems, Vol. 27 (Advances in Neural Information Processing Systems) - Foundational paper discussing the prevalence of saddle points over local minima in high-dimensional non-convex optimization.
- How to Escape Saddle Points Efficiently, Chi Jin, Rong Ge, Praneeth Netrapalli, Sham M. Kakade, and Michael I. Jordan, 2017 International Conference on Machine Learning (ICML), Vol. 70 (Proceedings of Machine Learning Research (PMLR)) DOI: 10.5555/3305890.3306000 - This work presents algorithms and theoretical guarantees for efficiently escaping saddle points in non-convex optimization.
- Optimization Algorithms for Deep Learning, Xiangxiang Zhang, Anna Choromanska, and Yann LeCun, 2019 arXiv preprint arXiv:1904.12260 - Comprehensive review of optimization techniques and associated challenges in deep learning.