Truncated Newton method

In this article, we will delve into the world of Truncated Newton method, exploring its many facets and manifestations. From its origins to its relevance today, Truncated Newton method has captured the attention of people of all ages and backgrounds. Through this comprehensive analysis, we will seek to better understand what makes Truncated Newton method so fascinating and what impact it has on our society. Regardless of whether you are an expert on the subject or simply curious to learn more, this article will provide you with the information necessary to fully understand the phenomenon of Truncated Newton method. Get ready to immerse yourself in a journey of discovery and knowledge!

The truncated Newton method, originated in a paper by Ron Dembo and Trond Steihaug,[1] also known as Hessian-free optimization,[2] are a family of optimization algorithms designed for optimizing non-linear functions with large numbers of independent variables. A truncated Newton method consists of repeated application of an iterative optimization algorithm to approximately solve Newton's equations, to determine an update to the function's parameters. The inner solver is truncated, i.e., run for only a limited number of iterations. It follows that, for truncated Newton methods to work, the inner solver needs to produce a good approximation in a finite number of iterations;[3] conjugate gradient has been suggested and evaluated as a candidate inner loop.[2] Another prerequisite is good preconditioning for the inner algorithm.[4]

References

  1. ^ Dembo, Ron S.; Steihaug, Trond (1983). "Truncated-Newton algorithms for large-scale unconstrained optimization". Mathematical Programming. 26 (2). Springer: 190–212. doi:10.1007/BF02592055. S2CID 40537623.. Convergence results for this algorithm can be found in Dembo, Ron S.; Eisenstat, Stanley C.; Steihaug, Trond (1982). "Inexact newton methods". SIAM Journal on Numerical Analysis. 19 (2): 400–408. Bibcode:1982SJNA...19..400D. doi:10.1137/0719025. JSTOR 2156954..
  2. ^ a b Martens, James (2010). Deep learning via Hessian-free optimization (PDF). Proc. International Conference on Machine Learning.
  3. ^ Nash, Stephen G. (2000). "A survey of truncated-Newton methods". Journal of Computational and Applied Mathematics. 124 (1–2): 45–59. Bibcode:2000JCoAM.124...45N. doi:10.1016/S0377-0427(00)00426-X.
  4. ^ Nash, Stephen G. (1985). "Preconditioning of truncated-Newton methods" (PDF). SIAM J. Sci. Stat. Comput. 6 (3): 599–616. doi:10.1137/0906042.

Further reading

  • Grippo, L.; Lampariello, F.; Lucidi, S. (1989). "A Truncated Newton Method with Nonmonotone Line Search for Unconstrained Optimization". J. Optimization Theory and Applications. 60 (3): 401–419. CiteSeerX 10.1.1.455.7495. doi:10.1007/BF00940345. S2CID 18990650.
  • Nash, Stephen G.; Nocedal, Jorge (1991). "A numerical study of the limited memory BFGS method and the truncated-Newton method for large scale optimization". SIAM J. Optim. 1 (3): 358–372. CiteSeerX 10.1.1.474.3400. doi:10.1137/0801023.


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