Gradient Descent Algorithm in Hilbert Spaces under Stationary Markov Chains with $φ$- and $β$-Mixing
This work addresses theoretical convergence guarantees for optimization algorithms in stochastic settings, which is incremental as it extends existing analyses to more general mixing conditions.
The paper tackles the convergence analysis of gradient descent in Hilbert spaces under stationary Markov chains with φ- and β-mixing, deriving probabilistic upper bounds based on exponential and polynomial decay of mixing coefficients.
In this paper, we study a strictly stationary Markov chain gradient descent algorithm operating in general Hilbert spaces. Our analysis focuses on the mixing coefficients of the underlying process, specifically the $φ$- and $β$-mixing coefficients. Under these assumptions, we derive probabilistic upper bounds on the convergence behavior of the algorithm based on the exponential as well as the polynomial decay of the mixing coefficients.