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
Characteristics of Deep Learning Loss Landscapes
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- On Large-Batch Training for Deep Learning: Generalization Gap and Sharp Minima, Nitish Shirish Keskar, Dheevatsa Mudigere, Jorge Nocedal, Mikhail Smelyanskiy, and Ping Tak Peter Tang, 2016 ICLR 2017 DOI: 10.48550/arXiv.1609.04836 - This paper examines the relationship between batch size, the sharpness of minima found by optimizers, and the generalization performance of deep learning models.
- Deep Learning, Ian Goodfellow, Yoshua Bengio, and Aaron Courville, 2016 (MIT Press) - Chapter 8 of this authoritative textbook provides a comprehensive overview of optimization challenges, including the characteristics of loss landscapes in deep learning.
- Visualizing the Loss Landscape of Neural Networks, Hao Li, Zheng Xu, Gavin Taylor, Christoph Studer, Tom Goldstein, 2018 Advances in Neural Information Processing Systems, Vol. 31 (Neural Information Processing Systems Foundation) DOI: 10.48550/arXiv.1712.09913 - This paper introduces methods for effectively visualizing the high-dimensional loss landscapes of neural networks, helping to illustrate concepts like sharp and flat minima.
- Optimization Methods for Large-Scale Machine Learning, Léon Bottou, Frank E. Curtis, and Jorge Nocedal, 2018 SIAM Review, Vol. 60 (Society for Industrial and Applied Mathematics) DOI: 10.1137/16M1080173 - This review article surveys a wide range of optimization methods, discussing their applicability and challenges in large-scale machine learning, with specific attention to deep learning's non-convexity and high-dimensionality.