Newton's Method Examples

Optimization · Axiom Academy

EXAMPLE Newton's Method for Optimization Master quadratic convergence through three complete worked examples Excellent work! You've mastered Newton's method for optimization. Here's what we learned: Newton's Method uses second-order information: By incorporating the Hessian (second derivatives), Newton's method achieves quadratic convergence near the optimum, making it much faster than gradient descent. Perfect for quadratic functions: For quadratic objectives, Newton's method converges in exactly one step because the quadratic approximation is exact. Requires careful initialization: Newton's method can fail or converge to saddle points when started far from the optimum or when the Hessian is indefinite. Line search or trust regions help stabilize it. Computational trade-offs: While Newton converges in fewer iterations, each iteration requires computing and inverting the Hessian, which costs O(n³) for n-dimensional problems. Gradient descent is more robust: When Newton struggles with poor initialization, gradient descent with appropriate step sizes provides safer, more reliable convergence. Newton's method is the foundation for many modern optimization algorithms. Understanding when it excels and when it struggles helps you choose the right optimizer for your machine learning models!

This is the written version of the interactive lesson above. See the full Optimization course.