We generalize stochastic subgradient descent methods to situations in which we do not receive independent samples from the distribution over which we optimize, instead receiving samples 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, decision problems with dependent data, and stochastic optimization problems over combinatorial spaces.