Analyzing Inexact Hypergradients for Bilevel Learning
- URL: http://arxiv.org/abs/2301.04764v2
- Date: Wed, 15 Nov 2023 01:07:38 GMT
- Title: Analyzing Inexact Hypergradients for Bilevel Learning
- Authors: Matthias J. Ehrhardt and Lindon Roberts
- Abstract summary: We introduce a unified framework for computing hypergradients that generalizes existing methods based on the implicit function theorem and automatic differentiation/backpropagation.
Our numerical results show that, surprisingly, for efficient bilevel optimization, the choice of hypergradient algorithm is at least as important as the choice of lower-level solver.
- Score: 0.09669369645900441
- License: http://creativecommons.org/licenses/by/4.0/
- Abstract: Estimating hyperparameters has been a long-standing problem in machine
learning. We consider the case where the task at hand is modeled as the
solution to an optimization problem. Here the exact gradient with respect to
the hyperparameters cannot be feasibly computed and approximate strategies are
required. We introduce a unified framework for computing hypergradients that
generalizes existing methods based on the implicit function theorem and
automatic differentiation/backpropagation, showing that these two seemingly
disparate approaches are actually tightly connected. Our framework is extremely
flexible, allowing its subproblems to be solved with any suitable method, to
any degree of accuracy. We derive a priori and computable a posteriori error
bounds for all our methods, and numerically show that our a posteriori bounds
are usually more accurate. Our numerical results also show that, surprisingly,
for efficient bilevel optimization, the choice of hypergradient algorithm is at
least as important as the choice of lower-level solver.
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