Higher-Order Newton Methods with Polynomial Work per Iteration · HackerPapers discussions

Higher-Order Newton Methods with Polynomial Work per Iteration — mentioned by magicalhippo

Tue, 25 Mar 2025 09:59:59 GMT

The paper has some examples, and a very clear one for the univariate case[1]. Yes, weird of Quanta to not include such an example. [1]: https://arxiv.org/html/2311.06374v2#S5.F1

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Higher-Order Newton Methods with Polynomial Work per Iteration — mentioned by magicalhippo

Tue, 25 Mar 2025 08:45:23 GMT

In the paper[1], they note that their method seems much more well-behaved for multi-variate problems. In the case they numerically examined, the regular Newton's method had a fractal pattern of which starting points converged, while their method had a nice contiguous area with fairly sharp boundary. That is, if using Newton's method one could start close to the solution but still not converge, while their method seems much more well-behaved. I guess that might be a more attractive property than "just" being faster. However not an expert so not sure if other, cheaper alternatives also has this…

Paper and discussion history. A mention is not an endorsement.

Higher-Order Newton Methods with Polynomial Work per Iteration — mentioned by bbor

Tue, 25 Mar 2025 02:53:48 GMT

Very, very cool -- thanks for posting! Quanta is yet again proving itself to be the very best casual science publication in the English-speaking world. This is a short one, but as always, clean, original graphics + big pictures of the participants is a winning combo IMO. Other iterative methods, like gradient descent, converge toward the true minimum at a linear rate. Newton’s method converges toward it... at an [exponential] rate... The rate of convergence would scale with the number of derivatives used: Just as using two derivatives allowed Newton to approach the true minimum at a quadratic…

Paper and discussion history. A mention is not an endorsement.