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
Limitations of Fixed Learning Rates
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- Deep Learning, Ian Goodfellow, Yoshua Bengio, and Aaron Courville, 2016 (MIT Press) - Chapters 4 and 8 provide a comprehensive explanation of gradient descent optimization, including the role of the learning rate and its practical challenges in deep learning.
- Adam: A Method for Stochastic Optimization, Diederik P. Kingma, Jimmy Ba, 2015 International Conference for Learning Representations DOI: 10.48550/arXiv.1412.6980 - This paper introduces Adam, a widely adopted adaptive learning rate algorithm that addresses many limitations of fixed learning rates by dynamically adjusting step sizes for individual parameters.
- Adaptive Subgradient Methods for Online Learning and Stochastic Optimization, John Duchi, Elad Hazan, and Yoram Singer, 2011 Journal of Machine Learning Research (JMLR), Vol. 12 - Presents AdaGrad, an early and influential adaptive learning rate algorithm that scales learning rates based on the historical sum of squared gradients for each parameter.
- Lecture 6.5 - RMSprop: Divide the gradient by a running average of its recent magnitude, Geoffrey Hinton, Nitish Srivastava, and Kevin Swersky, 2012 Coursera Lecture Notes (University of Toronto) (University of Toronto) - The original lecture slides introducing RMSprop, an adaptive learning rate method developed to mitigate AdaGrad's aggressively diminishing learning rates.