  We present the Stochastic alternate Linearization Method (StochaLM), a
token-based method for distributed optimization. This algorithm finds the
solution of a consensus optimization problem by solving a sequence of
subproblems where some components of the cost are linearized around specific
anchor points. StochaLM can be interpreted as a dual block coordinate ascent
method whose block components are selected using the state of an ergodic Markov
chain. This sampling process is neither essentially cyclic nor independent over
time, preventing us from using proofs of convergence of dual block coordinate
ascent methods done in previous works. The proof of convergence of our method
is, therefore, also novel. We show that, if the cost is strongly convex and the
network is fully connected, then, with probability one, the primal sequence
generated by StochaLM converges to the solution. Our method is
application-friendly, as it has no global hyperparameters to tune; any
hyperparameters can be tuned locally by the agents, using information regarding
their private cost and neighbourhood. Our method is, therefore, decentralized
in the truest sense. Numerical experiments evidence that our method converges
to the solution faster than other token-based methods, even when these methods'
hyperparameters are tuned for optimal performance.
