Random coordinate descent explained

Randomized (Block) Coordinate Descent Method is an optimization algorithm popularized by Nesterov (2010) and Richtárik and Takáč (2011). The first analysis of this method, when applied to the problem of minimizing a smooth convex function, was performed by Nesterov (2010). In Nesterov's analysis the method needs to be applied to a quadratic perturbation of the original function with an unknown scaling factor. Richtárik and Takáč (2011) give iteration complexity bounds which do not require this, i.e., the method is applied to the objective function directly. Furthermore, they generalize the setting to the problem of minimizing a composite function, i.e., sum of a smooth convex and a (possibly nonsmooth) convex block-separable function:

F(x) = f(x) + \Psi(x),

where

\Psi(x) = \sum ^{n} _{i =1} \Psi ^{(i)} _{i(x} ), x\in R ^{N} is decomposed into n blocks of variables/coordinates: x = (x ^{(1)} , ... ,x ^{(n)} ) and \Psi _{1,...,} \Psi _{n} are (simple) convex functions.

Example (block decomposition): If

x = (x _{1,x} _{2,...,x} _{5)} \in R ^{5} and n = 3 , one may choose x ^{(1)} = (x _{1,x} _{3),} x ^{(2)} = (x _{2,x} _{5)} and x ^{(3)} = x _{4} .

Example (block-separable regularizers):

n = N; \Psi(x) = \|x\| _{1} = \sum ^{n} _{i =1} |x _{i|} N = N _{1} + N _{2} + ... + N _{n;} \Psi(x) = \sum ^{n} _{i =1} \|x ^{(i)} \| _{2} , where x ^{(i)} \in R ^{N _{i}} and \| ⋅ \| _{2} is the standard Euclidean norm.

Algorithm

Consider the optimization problem

min _{x \in R ^{n}} f(x),

where

f is a convex and smooth function.

Smoothness: By smoothness we mean the following: we assumethe gradient of

f is coordinate-wise Lipschitz continuous with constants L _{1,} L _{2,} ... , L _{n} . That is, we assume that|\nabla _{i} f(x + h e _{i)} - \nabla _{i} f(x)| \leq L _{i} |h|,

for all

x \in R ^{n} and h \in R , where \nabla _{i} denotes the partial derivative with respect to variable x ^{(i)} .

Nesterov, and Richtarik and Takac showed that the following algorithm converges to the optimal point: Input:

x _{0} \in R ^{n} //starting point Output: x set x := x_0 for k := 1, ... do choose coordinate i\in \{ 1 , 2 , ... ,n\} , uniformly at random update x ^{(i)} = x ^{(i)} - (1 {L)/(_{i}}) \nabla _{i} f(x) end for

Convergence rate

Since the iterates of this algorithm are random vectors, a complexity result would give a bound on the number of iterations needed for the method to output an approximate solution with high probability. It was shown in that if

k\geq (2 n R _{L(x} _{0)})/(\epsilon) log \left( (f(x ^{*} _{0 ) - f})/(\epsilon \rho) \right) , where R _{L(x)=max} _{y} max _{x ^{*} \in X ^{*}} \{ \|y - x ^{*\|} _{L} : f(y)\leq f(x) \} , f ^{*} is an optimal solution (f ^{*} = min _{x\in R ^{n}} \{f(x)\} ),\rho\in( 0 , 1 ) is a confidence level and \epsilon> 0 is target accuracy,then Prob(f(x ^{*>} _{k) - f} \epsilon) \leq \rho .

Example on particular function

The following Figure showshow

x _{k} develops during iterations, in principle.The problem isf(x) = \tfrac{ 1 }{ 2 } x ^{T} \left(\begin{array}{cc} 1 & 0 . 5 \ 0 . 5 & 1 \end{array} \right) x - \left(\begin{array}{cc} 1 . 5 & 1 . 5 \end{array} \right)x, x _{0=\left(\begin{array}{} cc} 0 & 0 \end{array} \right) ^{T}

Extension to block coordinate setting

One can naturally extend this algorithm not only just to coordinates, but to blocks of coordinates. Assume that we have space

R ^{5} . This space has 5 coordinate directions, concretelye _{1} = ( 1 , 0 , 0 , 0 , 0 ) ^{T, e} _{2} = ( 0 , 1 , 0 , 0 , 0 ) ^{T, e} _{3} = ( 0 , 0 , 1 , 0 , 0 ) ^{T, e} _{4} = ( 0 , 0 , 0 , 1 , 0 ) ^{T, e} _{5} = ( 0 , 0 , 0 , 0 , 1 ) ^{T} in which Random Coordinate Descent Method can move. However, one can group some coordinate directions into blocks and we can have instead of those 5 coordinate directions 3 block coordinate directions (see image).

See also

• Coordinate descent
• Gradient descent
• Mathematical optimization