Fixed-Point Automatic Differentiation of Forward--Backward Splitting
Algorithms for Partly Smooth Functions
- URL: http://arxiv.org/abs/2208.03107v1
- Date: Fri, 5 Aug 2022 11:27:55 GMT
- Title: Fixed-Point Automatic Differentiation of Forward--Backward Splitting
Algorithms for Partly Smooth Functions
- Authors: Sheheryar Mehmood and Peter Ochs
- Abstract summary: We show that under partial smoothness, Automatic Differentiation converges to the derivative of the solution mapping.
We numerically illustrate the convergence and convergence rates of AD and FPAD on Lasso and Group Lasso problems.
- Score: 8.680676599607123
- License: http://creativecommons.org/licenses/by/4.0/
- Abstract: A large class of non-smooth practical optimization problems can be written as
minimization of a sum of smooth and partly smooth functions. We consider such
structured problems which also depend on a parameter vector and study the
problem of differentiating its solution mapping with respect to the parameter
which has far reaching applications in sensitivity analysis and parameter
learning optmization problems. We show that under partial smoothness and other
mild assumptions, Automatic Differentiation (AD) of the sequence generated by
proximal splitting algorithms converges to the derivative of the solution
mapping. For a variant of automatic differentiation, which we call Fixed-Point
Automatic Differentiation (FPAD), we remedy the memory overhead problem of the
Reverse Mode AD and moreover provide faster convergence theoretically. We
numerically illustrate the convergence and convergence rates of AD and FPAD on
Lasso and Group Lasso problems and demonstrate the working of FPAD on
prototypical practical image denoising problem by learning the regularization
term.
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