5.b) Explain the challenges that occur in neural network optimization in detail.
Answer:
Challenges in Neural Network Optimization
- Optimization in general is an extremely difficult task.
- Traditionally, machine learning has avoided the difficulty of general optimization by carefully designing the objective function and constraints to ensure that the optimization problem is convex.
- Whentraining neural networks, we must confront the general non-convex case.
- Evenconvexoptimization is not without its complications.
- In this section, we summarize several of the most prominent challenges involved in optimization for training deep models.
Convex and non-convex functions are important concepts in machine learning, particularly in optimization problems. Convex functions have a unique global minimum, making optimization easier and more reliable. Non-convex functions, on the other hand, can have multiple local minima, making optimization more challenging.
1. Ill-Conditioning
2. Local Minima
3. Plateaus, Saddle Points and Other Flat Regions
4. Cliffs and Exploding Gradients
5. Long-Term Dependencies
6. Inexact Gradients
7. Poor Correspondence between Local and Global Structure
Neural network training often struggles due to poor alignment between local improvements and the global objective. Even when local challenges like saddle points, cliffs, or poor conditioning are overcome, the optimization process can remain inefficient if local steps fail to guide toward regions of significantly lower cost.
The training trajectory may follow unnecessarily long paths around complex loss landscapes, such as mountain-like structures, increasing training time. Furthermore, neural networks often do not converge to critical points, as their loss functions may asymptotically approach low values without reaching minima. This behavior can result in high computational costs and suboptimal solutions.
Local optimization methods like gradient descent are limited by their reliance on small, local moves. These moves may fail in flat regions, at discontinuities, or if initialization places the model in an unfavorable area of the loss surface. Research is increasingly focused on finding effective initializations and strategies to address these global challenges, emphasizing better characterization of learning trajectories and the development of optimization methods that extend beyond local adjustments.
8. Theoretical Limits of Optimization
Theoretical results indicate fundamental limits on the performance of any optimization algorithm for neural networks. These results often highlight the intractability of solving certain problem classes or finding exact solutions for networks of fixed size. However, these theoretical constraints usually have limited practical impact.
In practice, optimization is feasible because neural network units typically output smooth values that enable local search methods, and larger networks allow many acceptable parameter configurations. Moreover, exact minimization is rarely the goal in neural network training; instead, the objective is to sufficiently reduce the loss to achieve good generalization. Developing realistic performance bounds for optimization algorithms remains an ongoing challenge in machine learning research.