Proximal gradient method: Difference between revisions

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{{Short description|Form of projection}}

{{more footnotes|date=November 2013}}

'''Proximal gradient methods''' are a generalized form of projection used to solve non-differentiable [[convex optimization]] problems.

[[File:Frank_Wolfe_vs_Projected_Gradient.webm|thumb|A comparison between the iterates of the projected gradient method (in red) and the [[Frank–Wolfe algorithm|Frank-Wolfe method]] (in green).]]

Many interesting problems can be formulated as convex optimization problems of the form:

<math>

\operatornamemin_{min}\limits_mathbf{x} \in \mathbb{R}^Nd} \sum_{i=1}^n f_i(\mathbf{x})

</math>

where <math>f_i: \mathbb{R}^d \rightarrow \mathbb{R},\ i = 1, \dots, n</math> are possibly non-differentiable [[convex functions]]. The lack of differentiability rules out conventional smooth optimization techniques like the [[Gradient descent|steepest descent method]] and the [[conjugate gradient method]], but proximal gradient methods can be used instead.

where <math>f_i,\ i = 1, \dots, n</math> are [[convex functions]] defined from <math>f: \mathbb{R}^N \rightarrow \mathbb{R} </math>

where some of the functions are non-differentiable, this rules out our conventional smooth optimization techniques like

[[Gradient descent|Steepest descent method]], [[conjugate gradient method]] etc. Proximal gradient methods can be used instead. These methods proceed by splitting, in that the functions <math>f_1, . . . , f_n</math> are used individually so as to yield an easily [[wikt:implementable|implementable]] algorithm.

They are called [[proximal]] because each non [[smooth function]] among <math>f_1, . . . , f_n</math> is involved via its proximity

operator. Iterative Shrinkage thresholding algorithm,<ref>

{{cite news | last1="Daubechies | first1=I | last2=Defrise | first2 = M | last3 = De Mol| first3 = C| title=An iterative thresholding algorithm for linear inverse problems with a sparsity constraint

|journal=A Journal Issued by the Courant Institute of Mathematical Sciences|volume=57 |year=2004|pages=1413–1457}}</ref> [[Landweber iteration|projected Landweber]], projected

gradient, [[alternating projection]]s, [[Alternating direction method of multipliers#Alternating direction method of multipliers|alternating-direction method of multipliers]], alternating

split [[Bregman method|Bregman]] are special instances of proximal algorithms. Details of proximal methods are discussed in Combettes and Pesquet.<ref>

{{cite arXiv |last1=Combettes |first1=Patrick L. |last2= Pesquet |first2=Jean-Christophe |title=Proximal Splitting Methods in Signal Processing|page=|year=2009 |arxiv=0912.3522}}</ref> For the theory of proximal gradient methods from the perspective of and with applications to [[statistical learning theory]], see [[proximal gradient methods for learning]].

Proximal gradient methods starts by a splitting step, in which the functions <math>f_1, . . . , f_n</math> are used individually so as to yield an easily [[wikt:implementable|implementable]] algorithm. They are called [[proximal]] because each non-differentiable function among <math>f_1, . . . , f_n</math> is involved via its [[Proximal operator|proximity operator]]. Iterative shrinkage thresholding algorithm,<ref>

== Notations and terminology ==

{{cite journal | last1=Daubechies | first1=I | last2=Defrise | first2 = M | last3 = De Mol| first3 = C|author3-link= Christine De Mol | title=An iterative thresholding algorithm for linear inverse problems with a sparsity constraint |journal= Communications on Pure and Applied Mathematics|volume=57 | issue=11 |year=2004|pages=1413–1457| bibcode=2003math......7152D | arxiv=math/0307152 |doi=10.1002/cpa.20042}}</ref> [[Landweber iteration|projected Landweber]], projected gradient, [[alternating projection]]s, [[Alternating direction method of multipliers#Alternating direction method of multipliers|alternating-direction method of multipliers]], alternating

Let <math>\mathbb{R}^N</math>, the <math>N</math>-dimensional [[euclidean space]], be the ___domain of the function

split [[Bregman method|Bregman]] are special instances of proximal algorithms.<ref>Details of proximal methods are discussed in {{cite arXiv |last1=Combettes |first1=Patrick L. |last2= Pesquet |first2=Jean-Christophe |title=Proximal Splitting Methods in Signal Processing|page=|year=2009 |eprint=0912.3522|class=math.OC }}</ref>

<math> f: \mathbb{R}^N \rightarrow (-\infty,+\infty]</math>. Suppose <math>C</math> is a non-empty

convex subset of <math>\mathbb{R}^N</math>. Then, the indicator function of <math>C</math> is defined as

For the theory of proximal gradient methods from the perspective of and with applications to [[statistical learning theory]], see [[proximal gradient methods for learning]].

: <math> \iota_C : x \mapsto

\begin{cases}

0 & \text{if } x \in C \\

+ \infty & \text{if } x \notin C

\end{cases}

</math>

: <math>p</math>-norm is defined as ( <math>\| \cdot \|_p</math> )

: <math>

\|x\|_p = ( |x_1|^p + |x_2|^p + \cdots + |x_N|^p )^{1/p}

</math>

The distance from <math>x \in \mathbb{R}^N</math> to <math>C</math> is defined as

: <math>

D_C(x) = \min_{y \in C} \|x - y\|_2

</math>

If <math>C</math> is closed and convex, the projection of <math>x \in \mathbb{R}^N</math> onto <math>C</math> is the unique point

<math>P_Cx \in C</math> such that <math>D_C(x) = \| x - P_Cx \|_2 </math>.

The [[subdifferential]] of <math>f</math> at <math>x</math> is given by

: <math>

\partial f = \{ u \in \mathbb{R}^N \mid \forall y \in \mathbb{R}^N, (y-x)^\mathrm{T}u+f(x) \leq f(y).\}

</math>

== Projection onto convex sets (POCS) ==

One of the widely used convex optimization algorithms is [[projections onto convex sets]] (POCS). This algorithm is employed to recover/synthesize a signal satisfying simultaneously several convex constraints. Let <math>f_i</math> be the indicator function of non-empty closed convex set <math>C_i</math> 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 <math>C_i</math>. In POCS method each set <math>C_i</math> is incorporated by its [[projection operator]] <math>P_{C_i}</math>. So in each [[iteration]] <math>x</math> is updated as

One of the widely used convex optimization algorithms is POCS ([[Projections onto convex sets|Projection Onto Convex Sets]]).

This algorithm is employed to recover/synthesize a signal satisfying simultaneously several convex constraints.

Let <math>f_i</math> be the indicator function of non-empty closed convex set <math>C_i</math> 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 <math>C_i</math>. In POCS method each set <math>C_i</math> is incorporated by its [[projection operator]]

<math>P_{C_i}</math>. So in each [[iteration]] <math>x</math> is updated as

: <math>

x_{k+1} = P_{C_1} P_{C_2} \cdots P_{C_n} x_k

</math>

However beyond such problems [[projection operator]]s 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, proximal operators are best suited for other purposes.

However beyond such problems [[projection operator]]s 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 ==

The [[proximal operator|proximity operator]] of a convex function <math>f</math> at <math>x</math> is defined as the unique solution to

:<math>

\operatorname{argmin}\limits_y \bigg( f(y) + \frac{1}{2} \left\| x-y \right\|_2^2 \bigg)

</math>

and is denoted <math>\operatorname{prox}_f(x)</math>.

: <math>

\operatorname{prox}_f(x) :\mathbb{R}^N \rightarrow \mathbb{R}^N

</math>

Note that in the specific case where <math>f</math> is the indicator function <math>\iota_C</math> of some convex set <math>C</math>

: <math>

\begin{align}

\operatorname{prox}_{\iota_C}(x)

&= \operatorname{argmin}\limits_y

\begin{cases}

\frac{1}{2} \left\| x-y \right\|_2^2 & \text{if } y \in C \\

+ \infty & \text{if } y \notin C

\end{cases} \\

&=\operatorname{argmin}\limits_{y \in C} \frac{1}{2} \left\| x-y \right\|_2^2 \\

&= P_C(x)

\end{align}

</math>

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

The proximity operator of <math>f</math> is characterized by inclusion

: <math>

p=\operatorname{prox}_f(x) \Leftrightarrow x-p \in \partial f(p) \qquad (\forall(x,p) \in \mathbb{R}^N \times \mathbb{R}^N)

</math>

If <math>f</math> is differentiable then above equation reduces to

: <math>

p=\operatorname{prox}_f(x) \Leftrightarrow x-p = \nabla f(p) \quad (\forall(x,p) \in \mathbb{R}^N \times \mathbb{R}^N)

</math>

== Examples ==

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*[[Alternating projection]]

*[[Alternating direction method of multipliers#Alternating direction method of multipliers|Alternating-direction method of multipliers]]

*Fast Iterative Shrinkage Thresholding Algorithm (FISTA)<ref>

{{cite news | last1="Beck | first1=A | last2=Teboulle | first2 = M | title=A fast iterative shrinkage-thresholding algorithm for linear inverse problems

|journal=SIAM Journal on Imaging Sciences|volume=2 |year=2009|pages=183–202}}</ref>

== See also ==

* [[Alternating projection]]

* [[Convex optimization]]

* [[Frank–Wolfe algorithm]]

* [[Proximal operator]]

* [[Proximal gradient methods for learning]]

* [[Frank–Wolfe algorithm]]

== Notes ==

{{reflist}}

<references/>

== References ==

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| last2 = Pesquet

| first2 = Jean-Christophe

| title = Springer's Fixed-Point Algorithms for Inverse Problems in Science and Engineering

| volume = 49

| year = 2011

Line 151 ⟶ 62:

==External links==

* Stephen Boyd and Lieven Vandenberghe Book, [httphttps://wwwweb.stanford.edu/~boyd/cvxbook/ ''Convex optimization'']

* [httphttps://wwwweb.stanford.edu/class/ee364a/ EE364a: Convex Optimization I] and [httphttps://wwwweb.stanford.edu/class/ee364b/ EE364b: Convex Optimization II], Stanford course homepages

* [httphttps://wwwpeople.eecs.berkeley.edu/~elghaoui/Teaching/EE227A/lecture18.pdf EE227A: Lieven Vandenberghe Notes] Lecture 18

* [https://github.com/kul-forbes/ProximalOperators.jl ProximalOperators.jl]: a [[Julia (programming language)|Julia]] package implementing proximal operators.

* [https://github.com/kul-forbes/ProximalAlgorithms.jl ProximalAlgorithms.jl]: a [[Julia (programming language)|Julia]] package implementing algorithms based on the proximal operator, including the proximal gradient method.