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
Limited-memory BFGS (L-BFGS)
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- Updating Quasi-Newton Matrices with Limited Storage, Jorge Nocedal, 1980 Mathematics of Computation, Vol. 35 (American Mathematical Society) DOI: 10.1090/S0025-5718-1980-0572855-7 - The original academic paper introducing the L-BFGS algorithm.
- Numerical Optimization, Jorge Nocedal and Stephen J. Wright, 2006 (Springer) DOI: 10.1007/978-0-387-40065-5 - A widely recognized textbook providing a rigorous and thorough treatment of L-BFGS and other optimization methods.
- On the limited memory BFGS method for large scale optimization, Dong C. Liu and Jorge Nocedal, 1989 Mathematical Programming, Vol. 45 (Springer) DOI: 10.1007/BF01589116 - This paper elaborates on the L-BFGS-B algorithm, an extension that handles bound constraints, and provides further algorithmic analysis.
- Optimization for Machine Learning (Lecture Notes), Jonathan Richard Shewchuk, 2005 (University of California, Berkeley) - Educational notes offering a clear derivation and explanation of the L-BFGS algorithm, including its two-loop recursion.