Syllabus Map

• Study map: Syllabus Study Map

Core Idea

• Taylor expansion approximates functions locally using derivatives.

Key Formula

First-order approximation near $x_0$:

Practical Notes

Optimization steps are locally valid, not globally.

• A good local approximation can still fail far from the expansion point.

Second-order terms capture curvature.

• Useful for understanding Newton-style methods.

Why This Matters for ML

• Local linear/quadratic approximations explain why gradient-based steps work.
• Curvature-aware reasoning guides step-size choice and second-order method intuition.
• Loss-landscape analysis around minima uses Taylor expansions conceptually.
• Approximation error intuition helps interpret optimizer behavior away from optimum.