In this topic, we’ll take a deep dive into both the potential and the challenges of neural networks. We start with the Universal Approximation Theorem (UAT), a powerful result that shows mathematically how a neural network, with enough neurons and a non-linear activation function, can approximate any continuous function. It’s a fundamental idea that underscores the vast capability of neural networks to model complex relationships.

However, as we’ll see, theory and practice are often different. The topic then explores how, in real-world scenarios, depth (the number of layers) often matters more than width (the number of neurons in each layer). Understanding this distinction is crucial for building networks that are both efficient and effective.

The second half of the topic introduces the Bias-Variance Tradeoff, a critical concept that helps us balance the desire to learn as much as possible from data while avoiding overfitting—learning too much and capturing noise instead of true patterns. Together, these ideas remind us that the real power of neural networks doesn’t just come from their mathematical capabilities, but from how they’re trained, constrained, and guided to generalize effectively to new, unseen data.

• What does the Universal Approximation Theorem truly guarantee—and what does it leave unanswered?

• Why are deep networks often more practical than single, wide-layer networks?

• How can we recognize and manage the tension between bias and variance during model training?