The interactive film is in the book's Companion

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The Newton Step: Jumping to the Taylor Bowl's Bottom

Chapter 11 · Mathematics for AI and Machine Learning · Advanced · 1:26

An interactive live film: gradient descent zig-zags in an ill-conditioned valley, while the Newton step jumps directly to the bottom of the local Taylor bowl in a single leap.

Narration

Here is an ill-conditioned valley, steeper along x one than along x zero, with a starting point x on the side.

A gradient descent step, the red arrow, follows steepest descent: it points almost straight across the valley, bouncing off the steep walls.

The rest of the narration is in the book's Companion.

More from Chapter 11

Interactive1:54Backpropagation, One VJP per LayerChapter 11Preview1:27Partial Derivatives Are Two SlicesChapter 11Preview1:34The Gradient Is the Steepest DirectionChapter 11Preview1:42Jacobian Matrix: Linearizing a MapChapter 11Preview1:43The Derivative of the Matrix InverseChapter 11Preview1:54Forward or Reverse: Filling the JacobianChapter 11Preview1:32Taylor Expansion, One Term at a TimeChapter 11Preview1:30Second-Order Taylor: Where the Quadratic Model Can Be TrustedChapter 11Preview1:38A Hessian-Vector Product from Two GradientsChapter 11Preview1:34The Gauss–Newton MatrixChapter 11Preview1:44Hessian Matrix: Bowls, Saddles and Curvature DirectionsChapter 11Preview1:37Why the Hessian Is SymmetricChapter 11Preview2:36Corners, Maxima and Steps in the Chain RuleChapter 11Preview1:59GPTQ: Rounding with the HessianChapter 11Preview2:12AWQ: Activation-Aware Weight QuantizationChapter 11Preview1:41The Implicit Function TheoremChapter 11