SIAM Journal on Optimization (SIOPT), Volume 22, No. 4, 2012, pages 1549-1578. Posted on arXiv, May 2011.
SIOPT Final copy. In SIAM Journal on Optimization.
Extended abstract in 2011 Allerton Conference on Communication, Control, and Computing. Slides for talk (pdf).
We generalize stochastic subgradient descent methods to situations in which we do not receive independent samples from the distribution over which we optimize, but instead receive samples that are coupled over time. We show that as long as the source of randomness is suitably ergodic--it converges quickly enough to a stationary distribution--the method enjoys strong convergence guarantees, both in expectation and with high probability. This result has implications for stochastic optimization in high-dimensional spaces, peer-to-peer distributed optimization schemes, and stochastic optimization problems over combinatorial spaces.