Ergodic Mirror Descent

TL;DR

This paper extends stochastic subgradient descent to dependent data scenarios with ergodic sampling, providing strong convergence guarantees applicable to high-dimensional, distributed, and combinatorial stochastic optimization problems.

Contribution

It introduces a generalized ergodic mirror descent method that converges under ergodic sampling, broadening the applicability of stochastic optimization techniques.

Findings

Convergence guarantees hold under ergodic sampling conditions.

Method applicable to high-dimensional and distributed optimization.

Effective with dependent data and combinatorial spaces.

Abstract

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, decision problems with dependent data, and stochastic optimization problems over combinatorial spaces.