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
Trust Region Methods
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- Numerical Optimization, Jorge Nocedal, Stephen J. Wright, 2006 (Springer) DOI: 10.1007/978-0-387-40065-5 - A standard graduate-level textbook offering a comprehensive introduction to optimization algorithms, including a detailed treatment of trust region methods, their theoretical foundations, and practical algorithms.
- Trust-Region Methods, Andrew R. Conn, Nicholas I. M. Gould, Philippe L. Toint, 2000 Vol. 1 (MPS-SIAM Series on Optimization) (Society for Industrial and Applied Mathematics) DOI: 10.1137/1.9780898719857 - This authoritative book focuses solely on trust region methods, providing an in-depth analysis of their theory, algorithms, and applications.
- The conjugate gradient method and trust regions, Trond Steihaug, 1983 SIAM Journal on Numerical Analysis, Vol. 20 (Society for Industrial and Applied Mathematics Publications) DOI: 10.1137/0720042 - This seminal paper introduced the truncated conjugate gradient method for solving the trust region subproblem, widely known as the Steihaug-Toint method.
- A new algorithm for unconstrained optimization, Michael J. D. Powell, 1970 Nonlinear Programming (Academic Press) DOI: 10.1016/B978-0-12-597050-1.50006-3 - This foundational paper, a chapter within 'Nonlinear Programming', introduces the Dogleg method, a technique for approximately solving the trust region subproblem.