Consensus optimization
======================
Suppose we have a convex optimization problem with :math:`N` terms in
the objective
.. math::
\begin{array}{ll} \mbox{minimize} & \sum_{i=1}^N f_i(x)\\
\end{array}
For example, we might be fitting a model to data and :math:`f_i` is the
loss function for the :math:`i`\ th block of training data.
We can convert this problem into consensus form
.. math::
\begin{array}{ll} \mbox{minimize} & \sum_{i=1}^N f_i(x_i)\\
\mbox{subject to} & x_i = z
\end{array}
We interpret the :math:`x_i` as local variables, since they are
particular to a given :math:`f_i`. The variable :math:`z`, by contrast,
is global. The constraints :math:`x_i = z` enforce consistency, or
consensus.
We can solve a problem in consensus form using the Alternating Direction
Method of Multipliers (ADMM). Each iteration of ADMM reduces to the
following updates:
.. math::
\begin{array}{lll}
% xbar, u parameters in prox.
% called proximal operator.
x^{k+1}_i & := & \mathop{\rm argmin}_{x_i}\left(f_i(x_i) + (\rho/2)\left\|x_i - \overline{x}^k + u^k_i \right\|^2_2 \right) \\
% u running sum of errors.
u^{k+1}_i & := & u^{k}_i + x^{k+1}_i - \overline{x}^{k+1}
\end{array}
where :math:`\overline{x}^k = (1/N)\sum_{i=1}^N x^k_i`.
The following code carries out consensus ADMM, using CVXPY to solve the
local subproblems.
We split the :math:`x_i` variables across :math:`N` different worker
processes. The workers update the :math:`x_i` in parallel. A master
process then gathers and averages the :math:`x_i` and broadcasts
:math:`\overline x` back to the workers. The workers update :math:`u_i`
locally.
.. code::
from cvxpy import *
import numpy as np
from multiprocessing import Process, Pipe
# Number of terms f_i.
N = ...
# A list of all the f_i.
f_list = ...
def run_worker(f, pipe):
xbar = Parameter(n, value=np.zeros(n))
u = Parameter(n, value=np.zeros(n))
f += (rho/2)*sum_squares(x - xbar + u)
prox = Problem(Minimize(f))
# ADMM loop.
while True:
prox.solve()
pipe.send(x.value)
xbar.value = pipe.recv()
u.value += x.value - xbar.value
# Setup the workers.
pipes = []
procs = []
for i in range(N):
local, remote = Pipe()
pipes += [local]
procs += [Process(target=run_process, args=(f_list[i], remote))]
procs[-1].start()
# ADMM loop.
for i in range(MAX_ITER):
# Gather and average xi
xbar = sum(pipe.recv() for pipe in pipes)/N
# Scatter xbar
for pipe in pipes:
pipe.send(xbar)
[p.terminate() for p in procs]