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]