Consensus optimization

Suppose we have a convex optimization problem with \(N\) terms in the objective

\[\begin{split}\begin{array}{ll} \mbox{minimize} & \sum_{i=1}^N f_i(x)\\ \end{array}\end{split}\]

For example, we might be fitting a model to data and \(f_i\) is the loss function for the \(i\)th block of training data.

We can convert this problem into consensus form

\[\begin{split}\begin{array}{ll} \mbox{minimize} & \sum_{i=1}^N f_i(x_i)\\ \mbox{subject to} & x_i = z \end{array}\end{split}\]

We interpret the \(x_i\) as local variables, since they are particular to a given \(f_i\). The variable \(z\), by contrast, is global. The constraints \(x_i = z\) 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:

\[\begin{split}\begin{array}{lll} % xbar, u parameters in prox. % called proximal operator. x^{k+1}_i & := & \mathop{\rm argmin}_{x_i}\left(f_i(x_i) + (\rho/2)\left\|x_i - \overline{x}^k + u^k_i \right\|^2_2 \right) \\ % u running sum of errors. u^{k+1}_i & := & u^{k}_i + x^{k+1}_i - \overline{x}^{k+1} \end{array}\end{split}\]

where \(\overline{x}^k = (1/N)\sum_{i=1}^N x^k_i\).

The following code carries out consensus ADMM, using CVXPY to solve the local subproblems.

We split the \(x_i\) variables across \(N\) different worker processes. The workers update the \(x_i\) in parallel. A master process then gathers and averages the \(x_i\) and broadcasts \(\overline x\) back to the workers. The workers update \(u_i\) locally.

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]