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
minimize x ∈ R N f 1 ( x ) + f 2 ( x ) + ⋯ + f n − 1 ( x ) + f n ( x ) where f 1 , f 2 , . . . , f n are convex functions defined from f : R N → R 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 f 1 , . . . , f n are used individually so as to yield an easily implementable algorithm. They are called proximal because each non smooth function among f 1 , . . . , f n 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. 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
Let R N , the N -dimensional euclidean space, be the domain of the function f : R N → ( − ∞ , + ∞ ] . Suppose C is a non-empty convex subset of R N . Then, the indicator function of C is defined as
i C : x ↦ { 0 if x ∈ C + ∞ if x ∉ C p -norm is defined as (
∥ ⋅ ∥ p )
∥ x ∥ p = ( | x 1 | p + | x 2 | p + ⋯ + | x N | p ) 1 / p The distance from x ∈ R N to C is defined as
D C ( x ) = min y ∈ C ∥ x − y ∥ If C is closed and convex, the projection of x ∈ R N onto C is the unique point P C x ∈ C such that D C ( x ) = ∥ x − P C x ∥ 2 .
The subdifferential of f is given by
∂ f = { u ∈ R N ∣ ∀ y ∈ R N , ( y − x ) T u + f ( x ) ≤ f ( y ) . } 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 f i be the indicator function of non-empty closed convex set C i 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 C i . In POCS method each set C i is incorporated by its projection operator P C i . So in each iteration x is updated as
x k + 1 = P C 1 P C 2 ⋯ P C n x k 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.
The proximity operator of a convex function f at x is defined as the unique solution to
minimize y ∈ C f ( y ) + 1 2 ∥ x − y ∥ 2 2 and is denoted prox f ( x ) .
prox f ( x ) : R N → R N The proximity operator of f is characterized by inclusion
p = prox f ( x ) ⇔ x − p ∈ ∂ f ( p ) ( ∀ ( x , p ) ∈ R N × R N ) If f is differentiable then above equation reduces to
p = prox f ( x ) ⇔ x − p = ∇ f ( p ) ( ∀ ( x , p ) ∈ R N × R N ) Special instances of Proximal Gradient Methods are
Projected LandweberAlternating projectionAlternating-direction method of multipliersFast Iterative Shrinkage Thresholding Algorithm (FISTA)