Finite-Time Analysis of Stochastic Gradient Descent under Markov Randomness
This work addresses a fundamental issue in reinforcement learning and machine learning by providing convergence guarantees for SGD under Markov randomness without restrictive assumptions, which is incremental but important for practical applications.
The paper tackles the problem of analyzing stochastic gradient descent (SGD) when gradients are sampled from Markov processes, which introduces bias and non-independence, and shows that SGD converges nearly at the same rate as with independent samples, differing only by a logarithmic factor related to the mixing time.
Motivated by broad applications in reinforcement learning and machine learning, this paper considers the popular stochastic gradient descent (SGD) when the gradients of the underlying objective function are sampled from Markov processes. This Markov sampling leads to the gradient samples being biased and not independent. The existing results for the convergence of SGD under Markov randomness are often established under the assumptions on the boundedness of either the iterates or the gradient samples. Our main focus is to study the finite-time convergence of SGD for different types of objective functions, without requiring these assumptions. We show that SGD converges nearly at the same rate with Markovian gradient samples as with independent gradient samples. The only difference is a logarithmic factor that accounts for the mixing time of the Markov chain.