P2:

Understanding Global Minimum in Error Functions

• The goal in optimization is to settle the error function at a global minimum.
• The values of weights (denoted as w) at the global minimum are used to approximate functions.
• The main aim is to minimize the error function through proper weight values.

Types of Neural Networks and Learning Approaches

• Different learning paradigms:
• Supervised Learning: Learn from labeled data.
• Unsupervised Learning: No labeled data available.
• Self-Supervised Learning: Uses the intrinsic structure in the data for supervision.
• Reinforcement Learning: More focused on optimization rather than approximations.
• Most neural network approaches utilize function approximations rather than ensuring convergence to a global minimum.

Challenge of Finding Optimal Minimums

• High-dimensional error surfaces can contain multiple local and flat minimums, making it difficult to identify the global minimum.
• Many algorithms only reach local minimums, which could be flat or sharp.
• Initial weight settings significantly influence reachable minimums during optimization.

Local and Flat Minimums

• Local Minimum: A point where the error function is lower than its immediate neighbors, but not necessarily the lowest overall (global minimum).
• Flat Minimum vs. Sharp Minimum:
• Flat Minimum: Wide basin that is less sensitive to changes, yielding better generalization with lower training/testing error deviations.
• Sharp Minimum: Narrow basin which might lead to larger discrepancies between training and testing errors.

Weight Initialization

• The point where weights are initialized before training has a profound impact on optimization outcomes.
• Poor initialization can lead to suboptimal minima compared to more strategic initializations.
• The Beta algorithm and noise addition can help to explore different paths in the error surface to potentially escape local minima.

Training and Testing Distributions

• It's crucial for the training and test data to share the same distribution (IID) to minimize performance discrepancies.
• The error surfaces for training and testing will rarely be identical, though attempting to minimize the deviation is essential.

Error Function Dynamics

• The model's training error and testing error need to be analyzed for possible overfitting or underfitting.
• A smaller deviation between training and testing errors (like 5%) can be indicative of being in a flat minimum, while larger deviations hint at sharp minima.

Learning Algorithm Flexibility

• The choice of learning algorithm, step size adjustments, and noise influences the path taken through the error landscape.
• Various approaches, including mini-batch learning, can be employed to enhance exploration of the error surface.

Activation Functions and Optimization

• Activation functions themselves can influence optimization.
• Selecting appropriate activation functions helps improve approximations of functions in neural networks.
• Strategies exist to make optimization easier, e.g., keeping surfaces relatively symmetric for easier updates.

Conclusion

• Overall, understanding the error function landscape, weight initializations, and optimizing learning strategies is crucial for neural network training.
• Aim for methods that guide the optimization process towards flat minima for robust and reliable models without excessive deviations between training and testing errors.