Proximal gradient method

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Proximal gradient methods are a generalized form of projection used to solve non-differentiable convex optimization problems. Many interesting problems can be formulated as convex optimization problems of form

where are convex functions defined from where some of the functions are non-differentiable, this rules out our conventional smooth optimization techniques like Steepest descent method, conjugate gradient method etc. There is a specific class of algorithms which can solve the above optimization problem. These methods proceed by splitting, in that the functions are used individually so as to yield an easily implementable algorithm. They are called proximal because each non smooth function among is involved via its proximity operator. Iterative Shrinkage thresholding algorithm, projected Landweber, projected gradient, alternating projections, alternating-direction method of multipliers, alternating split Bregman are special instances of proximal algorithms. Details of proximal methods are discussed in Combettes and Pesquet.[1] For the theory of proximal gradient methods from the perspective of and with applications to statistical learning theory, see proximal gradient methods for learning.

Notations and terminology[edit]

Let , the -dimensional euclidean space, be the domain of the function . Suppose is a non-empty convex subset of . Then, the indicator function of is defined as

-norm is defined as ( )

The distance from to is defined as

If is closed and convex, the projection of onto is the unique point such that .

The subdifferential of is given by

Projection onto convex sets (POCS)[edit]

One of the widely used convex optimization algorithms is POCS (Projection Onto Convex Sets). This algorithm is employed to recover/synthesize a signal satisfying simultaneously several convex constraints. Let be the indicator function of non-empty closed convex set modeling a constraint. This reduces to convex feasibility problem, which require us to find a solution such that it lies in the intersection of all convex sets . In POCS method each set is incorporated by its projection operator . So in each iteration is updated as

However beyond such problems projection operators are not appropriate and more general operators are required to tackle them. Among the various generalizations of the notion of a convex projection operator that exist, proximity operators are best suited for other purposes.

Definition[edit]

The proximity operator of a convex function at is defined as the unique solution to

and is denoted .

Note that in the specific case where is the indicator function of some convex set

showing that the proximity operator is indeed a generalisation of the projection operator.

The proximity operator of is characterized by inclusion

If is differentiable then above equation reduces to

Examples[edit]

Special instances of Proximal Gradient Methods are

See also[edit]

Notes[edit]

  1. ^ Combettes, Patrick L.; Pesquet, Jean-Christophe (2009). "Proximal Splitting Methods in Signal Processing". arXiv:0912.3522Freely accessible. 
  2. ^ "Beck, A; Teboulle, M (2009). "A fast iterative shrinkage-thresholding algorithm for linear inverse problems". SIAM J. Imaging Science. 2. pp. 183–202. 

References[edit]

  • Rockafellar, R. T. (1970). Convex analysis. Princeton: Princeton University Press. 
  • Combettes, Patrick L.; Pesquet, Jean-Christophe (2011). Springer's Fixed-Point Algorithms for Inverse Problems in Science and Engineering. 49. pp. 185–212. 

External links[edit]