A Gradient Method for Multilevel Optimization
- URL: http://arxiv.org/abs/2105.13954v1
- Date: Fri, 28 May 2021 16:22:10 GMT
- Title: A Gradient Method for Multilevel Optimization
- Authors: Ryo Sato, Mirai Tanaka, Akiko Takeda
- Abstract summary: We have developed a gradient-based algorithm for multilevel $n$ levels based on Franceschi et al.'s idea.
As far as we know, this is one of the first algorithms with some theoretical guarantee for multilevel optimization.
- Score: 8.80899367147235
- License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
- Abstract: Although application examples of multilevel optimization have already been
discussed since the '90s, the development of solution methods was almost
limited to bilevel cases due to the difficulty of the problem. In recent years,
in machine learning, Franceschi et al. have proposed a method for solving
bilevel optimization problems by replacing their lower-level problems with the
$T$ steepest descent update equations with some prechosen iteration number $T$.
In this paper, we have developed a gradient-based algorithm for multilevel
optimization with $n$ levels based on their idea and proved that our
reformulation with $n T$ variables asymptotically converges to the original
multilevel problem. As far as we know, this is one of the first algorithms with
some theoretical guarantee for multilevel optimization. Numerical experiments
show that a trilevel hyperparameter learning model considering data poisoning
produces more stable prediction results than an existing bilevel hyperparameter
learning model in noisy data settings.
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