Adapting to Mixing Time in Stochastic Optimization with Markovian Data
This addresses a key bottleneck for practitioners in machine learning and reinforcement learning who face unknown mixing times in real-world applications, offering a novel solution with broad applicability.
The paper tackles the problem of stochastic optimization with Markovian data where mixing time is unknown, proposing the first method that achieves optimal asymptotic convergence rates for convex problems without requiring this knowledge. It also extends to non-convex optimization and temporal difference learning, maintaining obliviousness to mixing time.
We consider stochastic optimization problems where data is drawn from a Markov chain. Existing methods for this setting crucially rely on knowing the mixing time of the chain, which in real-world applications is usually unknown. We propose the first optimization method that does not require the knowledge of the mixing time, yet obtains the optimal asymptotic convergence rate when applied to convex problems. We further show that our approach can be extended to: (i) finding stationary points in non-convex optimization with Markovian data, and (ii) obtaining better dependence on the mixing time in temporal difference (TD) learning; in both cases, our method is completely oblivious to the mixing time. Our method relies on a novel combination of multi-level Monte Carlo (MLMC) gradient estimation together with an adaptive learning method.